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Augmentations and link group representations

We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different types. We also study properties of the augmentation representation.


Introduction
A link is a disjoint union of simple closed curves. A link invariant is an algebraic construction associated to links which is well-defined within each isotopy class. Research on link invariants serves the goal of not only distinguishing links, but also understanding fundamental properties of links and related subjects.
Link invariants appear in various guises. Considering the complement space of a link and taking its fundamental group, we get a group as a link invariant which is known as the link group. In a more complicated form, another algebraic construction we study in this paper is the framed cord algebra, which is a non-commutative algebra generated by paths beginning and ending on the framing longitudes of the link.
They are powerful invariants. In the case of knots, both invariants in their further enhanced forms become complete invariants [W, ENS], meaning that distinct isotopy classes result in non-isomorphic invariants. However, robust invariants can be impractical to distinguish knots or links, because sometimes it is difficult to show that two groups or two non-commutative algebras are not isomorphic.
A possible trade-off is to study more computable invariants, such as representations of the group or the algebra. An augmentation is a rank one representation of the framed cord algebra. We want to understand augmentations in terms of representations of the link group, since a group appears simpler than a non-commutative algebra. We approach this goal by constructing the augmentation representation.
Theorem 1.1. Let K be an oriented link with its Seifert framing. Let Cord(K) be the framed cord algebra and π K be the link group.
(Theorem-Definition 2.16) Let : Cord(K) → k be an augmentation of the framed cord algebra. By writing K as a braid closure, we construct a representation of the link group ρ : π K → GL(V ).
(Theorem 2.18) Up to isomorphism, (ρ , V ) is well-defined for the augmentation . In particular, it does not depend on the choice of the braid in the construction.
The slogan of the construction is "action by interpolation". Placing the link as the closure of a braid, we can select a set of standard cords and arrange their augmented values into a square matrix. The column vectors of the matrix span a vector space underlining the augmentation representation. A based loop acts on a matrix entry by interpolating the standard cord with the based loop, see In this example, we start on the left with a based loop (black) in the fundamental group and a standard cord (red). After interpolation we get another framed cord on the right. The action of the based loop takes the augmented value of the standard cord to the augmented value of the interpolated cord.
In the next part of the paper, we look into properties of the augmentation representation.
Proposition 1.2. Let K be an oriented link with its Seifert framing, and : Cord(K) → k an augmentation of the framed cord algebra. The augmentation representation (ρ , V ) satisfy the following properties.
Suppose K ⊂ K is a sublink. If (γ) = 0 for either (1) every framed cord γ starting from K , or (2) every framed cord γ ending on K , then ρ(m ) = id V for any meridian m of K .
Suppose K = K 1 K 2 is the union of two sublinks. If (γ) = 0 for all mixed cords between K 1 and K 2 , then (ρ , V ) is a direct sum of two representation (ρ 1 , V 1 ) and (ρ 2 , V 2 ), where each (ρ i , V i ) is a representation of π K i . These properties can be reinterpreted in terms of microlocal sheaf theory. In microlocal theory, one considers the "micro-support" in the cotangent bundle, generalizing the usual notion of the support of a sheaf on the base manifold. The augmentation representation is defined over the link group, which is equivalent to a locally constant sheaf on the link complement. Taking the underived push-forward, we obtain a sheaf F microsupported within the conormal bundle of the link.
The three properties of (ρ , V ) can be rephrased in terms of F. The first property, (ρ , V ) being microlocally simple, is equivalent to say F is a simple sheaf along its micro-support (in the convention of [KS]), or microlocal rank 1 along its micro-support (in the convention of [STZ]). The vanishing property states a sufficient condition when F is microsupported outside the conormal of the sublink K . The separability states a sufficient condition when F splits into two subsheaves, each microsupported along a sublink.
The results of augmentation representations outreach to several directions. First off, the construction explicitly connects rank 1 representations of the framed cord algebra to higher rank representations of the link group. The framed cord algebra can be thought of as being generated by a subcategory of the fundamental groupoid of the link complement. One may guess from functoriality that representations should be pulled back in some sense. Our construction is a concrete way to realizes this idea.
A character variety is a moduli space of representations of a finitely generated group. The character variety of the link group is the space of flat bundles on the link complement, which plays an important role in knot theory and three manifolds. The SL(2)-character variety has been intensively studied. We mention [CS, KM] and the A-polynomial discussed in the next paragraph. The SL(n)-character variety in higher ranks becomes increasingly complicated [AH, GTZ, GW, HMP, MP]. Our results show that the "full augmentation variety", which is the moduli space of augmentations, cuts off a closed subvariety in the link group character variety. It suggests a direction of future research to understand the subvariety characterized by microlocally simple representations.
Introduced by Copper et al. [CCGLS], the A-polynomial of a knot defines a complex plane curves (which we call the "A-variety" for now) which parametrizes the SL(2)-character variety projected to a torus determined by the peripheral subgroup. It can be used to detect knots [DG, BZ, NiZh]. The A-polynomial is closely related to the augmentation polynomial, whose vanishing locus is the moduli space of augmentations projected to the same torus, or the "augmentation variety". When K is a knot, both polynomials have two variables, denoted as A K (λ, µ) and Aug K (λ, µ). Ng proved that A K (λ, µ) divides Aug K (λ, µ 2 ) [Ng3]. It is conjectured in [AENV] that the augmentation polynomial as a generalization of the A-polynomial produces a new notion of mirror symmetry. To see the relation between two polynomials using our result, augmentation representations are microlocally simple, and a generic rank 2 microlocally simple representation is in one-to-one correspondence with SL(2)representations. When K is a link, we no long have polynomials since the ideals of vanishing function are not principal, but a similar result holds for the same reason -the A-variety is a closed subvariety of the augmentation variety.
Finally the augmentation representation builds up to the correspondence between augmentations and sheaves for links. The correspondence was proven for Legendrian links [NRSSZ] and connected Legendrain surfaces defined from knot conormals [Ga2] or cubic graphs (by Sackel in appendix of [CM]). For a brief motivation, augmentations of a framed cord algebra correspond to augmentations of the conormal tori dga [Ng3], whose geometric counterparts are Lagrangian filling in the sense of SFT [El, EGH]. Fillings are sheaves through microlocalization [Na, NaZa], and further determine link group representations via the Radon transform [Ga1]. For knots, the correspondence in both ways are constructed, see [Ga2]. A technical reason why the construction is more complicated in the case of links than knots is that the framed cord algebra of a knot is generated by elements in the knot group, but the similar statement does not hold for links. In this paper, we focus on the geometric origin of the theory and construct the augmentation representation. As explain earlier in the slogan and Figure 1.1, the action of the link group underlines a geometric meaning, namely based loops acts by interpolating the framed cords. This construction makes a direct connection from augmentations of a framed cord algebra to simple sheaves, without the lengthy detour in the motivation story. This paper is organized in a simple fashion. We construct the augmentation representation is Section 2 and study its properties in Section 4.

Augmentation representation
The goal of this section is to construct the augmentation representation. It is a link group representation associated to an augmentation. We will first introduce some preliminary concepts, including the framed cord algebra and the augmentation associated to the framed cord algebra, then we construct the augmentation representation.
We fix some conventions. Let k be a commutative field. It is the ground field where we will define augmentations and group representations. Throughout the paper, we set X = R 3 or S 3 . Let K be an oriented r-component link in X. We label the components as Suppose p : [0, 1] → X is a path or a loop, we write p −1 for the reversed path: We denote by p 1 · p 2 the concatenation of two composable paths. Namely if p 1 (1) = p 2 (0), we define We work with paths up to homotopy, hence the concatenation induces an associative product.
2.1. Braids. To construct an augmentation representation, we need to represent the link as the closure of a braid. A braid can be expressed as an element in Artin's braid group of n strands: Geometrically, a braid is a collection of strands in a solid cylinder, with endpoints fixed on boundary disks. Braids can be realized via the mapping class group of a configuration space. Let D be an oriented disk with sufficiently large radius, equipped with two-tuple coordinates (−, −). Let y 1 = (1, 0), · · · , y n = (n, 0) be n marked points, and D • ⊂ D be the n-punctured disk with marked points removed. The braid group Br n is isomorphic to the mapping class group MCG(D • ) := π 0 (Diff + (D • )), where Diff + (D • ) is the topological group of orientation preserving diffeomorphisms of D • . For each [h] ∈ MCG(D • ), we can extend h to a homeomorphismh : Example 2.1. In Figure 2.1, we plot a 3-strand braid whose braid word is The closure of the braid is a two-component link, known as the Whitehead link. The configuration disk D with n marked points will be extensively used in the upcoming constructions. In practice, one can plot the figure in a more symmetric fashion. We keep the coordinates for easier description.
Given a braid B ∈ D × [0, 1], one can close the solid torus by identifying D × {0} and D × {1}. Since the marked points are fixed on the boundaries, the braid can be closed to a one dimensional compact submanifold B in the solid torus. If we take a sectional disk, in particular which we glued, we obtain the configuration disk where the marked points equal to D ∩ B . If the solid torus is further embedded in X = R 3 or S 3 as the tubular neighborhood of a unknot, we obtain a link K = B ⊂ X. A braid carries a natural orientation (induced from the orientation of [0, 1]), and hence the braid closure is an orientated link. Alexander's theorem asserts that every oriented link can be expressed (not in a unique way) as a braid closure. See [Al] for the original construction, or [Ya, Vo] for the improved Yamada-Vogel algorithm. Suppose an r-component link K = K 1 · · · K r is the closure of an n-strand braid B. The strands in B can be indexed by the first coordinate in the transverse disk D. We define the component function to be a map such that the strand t of the braid belongs to the component K {t} .
The strand index admits a natural linear ordering by the x-coordinate of the marked points in the configuration disk. A partition of a linearly ordered set is ordered if for any two parts P 1 , P 2 of the partition, one has either a < b for all a ∈ P 1 , b ∈ P 2 , or a > b for all a ∈ P 1 , b ∈ P 2 . Lemma 2.2. If K is the closure of an n-strand braid, then it can be represented by an nstrand braid which admits an ordered partition where strands lie in the same part if and only if they belong to the same component of K.
The conjugated braid σ −1 i Bσ i is also an n-strand braid, with i-th and (i + 1)-th strands switched. We can conjugate the braid finitely many times and obtain the desired braid.
Remark 2.3. Suppose an m-component link K = K 1 · · · K m is the closure of an n strand braid B. By Lemma 2.2, we can assume there exist integers 0 = n 0 < n 1 < · · · < n m−1 < n m = n, such that the closure of the strands {n i−1 + 1, n i−1 + 2, · · · , n i } is the component K i of K. In other words, we can assume the component function is non-decreasing.
2.2. Link group. Let X = R 3 or S 3 . Let K ⊂ X be an oriented r-component link, K = K 1 · · · K r . The link group π K is the fundamental group of the link complement, i.e. π K = π 1 (X \ K). The link group has the following properties: • It is finitely generated by meridians, and finitely presented.
• There are r conjugacy classes, labelled by components of the link. These properties follow easily from the Wirtinger presentation of the link group, (for example, see [Ro]). We recall the construction here. Thinking of an oriented link by it two-dimensional diagram with under-crossings, in a generic position, the diagram has finite arcs and finite under-crossings. We take the base point of the fundamental group far away from the plane. Each arc corresponding to a loop in the fundamental group, which travels from the base point to the plane, wraps around the arc and then travels back. After fixing a convention, the orientation of the loop is determined by the orientation of the link. Note this loop is a meridian, which is by definition the boundary of a disk intersecting the link transversely at a point. Finally, each under-crossing imposes a conjugation relation among the meridian generators, giving the Wirtinger presentation.
When the link K is the the closure of an n-strand braid, the link group π K can be necessarily generated by n meridians, (though n may not be the minimum number of meridian generators). To see this, we scan the braid diagram from left to right. The braid is given by a word of half twists. At the beginning, each of the n strands determines a meridian, which we denote m t for 1 ≤ t ≤ n. Each half twist introduces an under-crossing, and therefore a new meridian in the Wirtinger presentation. The new meridian can be expressed as a word of previous meridians. Iterating the procedure, we see that the set of meridians {m t } 1≤t≤n generates the whole link group. In the rest of the paper, we will call the generating set of meridians of an n-strand braid closure.
Recall the configuration disk D with n marked points. If the base point of the link group is (0, −δ) ∈ D ⊂ X, where δ is a small positive real number, then the meridian generators {m t } 1≤t≤n can be plotted on the configuration disk. Namely, m t is the loop in D wrapping around the marked point y t = (t, 0).
To be compatible with the convention in [CELN], we fix the orientation of the knot in the configuration disk pointing inward to the paper, and the meridian generators wrap clockwisely around marked points. See  There are five strands, giving five meridian generators in the Wirtinger presentation of the link group. Each crossing in the diagram gives a conjugation relation. Therefore the link group is isomorphic to π K = m 1 , m 2 , m 3 , m 4 , m 5 / ∼, modulo relations We have seen that the Whitehead link can be expressed as the closure of a 3-strand braid, as in Example 2.1. Therefore the link group can be generated by three meridians instead of five. As for the Wirtinger presentation we just computed, we can write m 3 , m 5 in terms of m 1 , m 2 , m 4 , reducing the number of meridian generators to three.
2.3. Framed cord algebra. The cord algebra first appeared in [Ng1,Ng2]. The framed version was introduced in [Ng3], which models the degree zero knot contact homology [EENS].
Suppose K ⊂ X is an oriented knot, and n(K) is a small tubular neighborhood of K. A framing of K is a push off the knot to the boundary of the tubular neighborhood. In other words, a curve ⊂ ∂(n(K)) whose homology class in H 1 (n(K)) agrees with [K] ∈ H 1 (n(K)). A Seifert surface is an oriented surface S with ∂S = K. The Seifert framing is = S ∩ n(K), which is up to homotopy independent from the choice of the Seifert surface. The Seifert framing has zero linking number with K.
A framing of a link K = K 1 · · · K r is a choice of a framing i for each component K i . We decorate each i with a marked point * i ∈ i . We write * := { * 1 , · · · , * r }, and L = 1 · · · r . We write (K, L) for a framed link, namely an oriented link with a choice of a framing.
Upcoming, we define the framed cord algebra of a framed link, which is a mild generalization of the definition for knots as in [CELN,Definition 2.5].
Definition 2.5. Suppose (K, L) ⊂ X is a framed link.
A framed cord of K is a continuous map c : [0, 1] → X \ K such that c(0), c(1) ∈ L \ * . Two framed cords are homotopic if they are homotopic through framed cords. We write [c] the homotopy class of cord c.
We now construct a noncommutative unital ring A as follows: as a ring, A is freely generated by homotopy classes of framed cords and extra generators λ ±1 Thus A is generated as a Z-module by noncommutative words in homotopy classes of cords and powers of λ i and µ j . The powers of the λ i and µ i commute with each other, but do not with any cords.
The framed cord algebra is the quotient ring Cord(K, L) = A/I, where I is the two-sided ideal of A generated by the following relations: Remark 2.6. The definition of a framed cord algebra, at first glance, depends on the choices of decorations * i on framings. Different choices of decorations give isomorphic algebras. Consider the following argument. Since the generators are framed cords up to cord homotopy, we can choose a base point on each framing and homotope the endpoints of each cord to the base point. For example, we can choose base points to be a positive push off of marked points * i along each component of the framing. For two different sets of marked points, one set of marked points can be homotoped to the other set along the framing. Therefore we can use the same homotopy to move base points along the framing, underlining an isomorphism of the framed cord algebra.
Remark 2.7. For different choices of framings L 1 , L 2 , the resulting framed cord algebras are isomorphic as Z-algebras. We can construct by keeping µ i and framed cords, and sending λ i to λ i · µ . Here lk i (K, L) is the linking number between K i and its framing.
If L is the Seifert framing, we simply write the framed cord algebra as Cord(K).
Remark 2.8. If we take the quotient of Cord(K, L) where λ i , µ i commute with everything, the the result Cord r ]-algebras. Remark 2.9. There are various versions of the (framed) cord algebra in literature C K [Ng3], Cord c (K) [CELN], and P K [Ng4]. Each cord algebra is a quotient ring of the free noncommutative algebra generated by some version of cords over the same coefficient ring In each version, the free algebra quotients out four relations: normalizations, meridian relations, longitude relations, and skein relations. We shall compare their definitions with a focus on skein relations. ( (3) [Ng4] P K is defined for a knot K, namely r = 1. A cord [γ] ∈ P K is represented by a based loop γ ∈ π K . In particular, we forget about the group structure in π K and consider it as a set, then take the set elements as free generators. Let m be the meridian in the peripheral subgroup of π K . The skein relations are given by We remind the reader again that the P K version of the framed cord algebra is only defined for knots, but not links. If L is the Seifert framing, then (1) = (2). If in addition that K is a single component knot, From C K to Cord c (K), one can push the end points of a cord off the link to the framing. There are different choices of the push off which leads to ambiguity. One can fix this issue by multiply copies of µ i depending on the linking number. From Cord c (K) to P K , one can choose a base point on each framing, then move the points of each framed cord to the base point along the oriented of the framing.
Cords in a framed cord algebra can be classified into pure and mixed cords depending on the points where the cord starts and ends.
Definition 2.10. Let K be a link.
Suppose K ⊂ K is a sublink. A cord is a pure cord of K if it starts and ends on the framing of K .
Suppose K 1 , K 2 ⊂ K are two disjoint sublinks. A cord is a mixed cord between K 1 and K 2 if either (1) it starts on the framing of K 1 and ends on the framing of K 2 , or (2) it starts on the framing of K 2 and ends on the framing of K 1 .
If we simply say a pure cord or a mixed cord without decorations, it means the underlying sublink is a knot.
Recall the configuration disk D with n-marked points * . It can be regarded as a locally closed submanifold in X with * = D ∩ K. We can perturb the framing such that The orientation of the link K induces an orientation of the framing L. After a suitable perturbation, the framing {i} can be viewed as a (framed) cord, with the choice of end points . We call it a framing cord. Recall that we set the based point of π K at x 0 = (0, −δ). Define the capping path p i to be the linear map from x 0 to x i , for i = 1, · · · , n.
If we move the based point of π K to one of x i , a meridian generator m t can also be reviewed as a framed cord, which we term as a meridian cord. A meridian based at different longitudes are different framed cords in Cord(K). Hence it is necessary to remember the base point. Let m (i) t be the meridian cord based at x i wrapping around y t . Base point change subjects to the following relation: i . Definition 2.11. Define the standard cord to be γ ij := p −1 i · p j , for i, j ∈ {1, · · · , n}. Standard cords satisfy the following relations: In terms of standard cords, conjugations of meridian cords satisfy The relations of framed cords on the configuration disk can be written in the following way. Let c ij denote a framed cord from y 1 y 2 y 3 y 4 y 5 The longitude relation is not really visible in the configuration disk, but we write it down anyway. Note a longitude is a word of meridian generators in the configuration disk. For most of calculations over the disk, it suffices to use the other three relations.
Lemma 2.12. Standard cords generate the framed cord algebra.
Proof. By generation, we mean that every element in Cord(K) can be written as a sum of words consisting of standard cords and λ ±1 i , µ ±1 i . We start by proving two claims. (1) Every framed cord is cord homotopic to one from There exists x i such that {i} = s. Take a path p ⊂ s connecting x i and x, without passing through point * s . The concatenation [p · c] is isomorphic to c in the framed cord algebra, and the staring point of p · c is x i . Similarly one can similarly homotope the end point to some x j .
(2) Every framed cord from x i to x j is cord homotopic to the concatenation of a loop γ (i) ∈ π 1 (X \ K, x i ) and the standard cord γ ij . Suppose , proving the statement. Now we prove the lemma. By statements (1) and (2), it sufficed to prove the assertion for a framed cord in the form of it is generated by meridians. We prove by induction on the word length of γ (i) . The initial step is trivial, since [c] = [γ ij ] is already a standard cord.
Suppose the induction hypothesis that [γ (i) · γ ij ] is generated by standard cords holds for any loop γ (i) ∈ π K with word length less or equal to s ∈ N, we need to show that the induction hypothesis also holds for word length s + 1. It is equivalent to prove that [(m can also be generated by standard cords. There is The second equality uses the skein relation, and the other equalities are cord identities. Similarly we have The last equation uses the meridian relation. We complete the induction, as well as the proof of the lemma.
Remark 2.13. It follows the lemma that meridian cords are generated by standard cords.
t ] is a meridian cord wrapping around strand t, and based at x b . That is to say, the underlining loop is Here {t} is the component function defined in (2.1). It is evident from the equations that a meridian cord depends on the base point as well as the homotopy class of the underlining loop.

2.4.
Augmentations. An augmentation of the framed cord algebra Cord(K) of an oriented link K is a unit preserving algebra morphism where k is any commutative field.
Recall we have a variant framed cord algebra i commutes with cords. We can similarly define its augmentation as a unit preserving algebra morphism Since k is commutative, the image of any non-commutative generators in either Cord(K) or Cord c (K) becomes commutative as elements in k. Therefore, the set of augmentations for Cord(K) and Cord c (K) are canonically isomorphic.
Remark 2.14. In Remark 2.7 we explained that different framings give isomorphic Z-algebras but not Z[λ ±1 1 , µ ±1 1 , · · · , λ ±1 r , µ ±1 r ]-algebras. Hence there is a bijection between augmentations in one framing and augmentations in the other, but if we restricts to augmentations sending λ i , µ i to particular values in k * , then there is no bijection.
Given a framed cord [c], its augmented value ([c]) will be abbreviated as (c). This abbreviation should cause no confusions. For example, an augmentation applied to the skein relation within a configuration disk (2.2) can be expressed as Suppose K is the closure of an n-strand braid and : Cord(K) → k is an augmentation. We can organize augmented values of standard cords into an n × n square matrix R, by setting R ij = (γ ij ). Namely, Remark 2.15. Since Lemma 2.12 asserts that standard cords generate Cord(K), it is natural to ask whether the matrix R in turn determines the augmentation . The answer is sometimes. For example when K is a knot as in [Ga2,Theorem 4.16], in the first two cases one can reconstruct the augmentation from the associated matrix R, while in the third case one needs to specify in addition the data of (λ) to recover from R.
2.5. Augmentation representation. We defined a matrix R out of an augmentation : We adopt the following convention to represent a column vector of size n. For any path Suppose K is an oriented link equipped with its Seifert framing. Let : Cord(K) → k be an augmentation of its framed cord algebra. The following map defines a link group representation ρ : In this to be verified representation, it may be difficult to compute the action for a general γ, but the action of meridians on standard cords takes a much simpler form. Check out the following lemma.
We recall some notations and identities. For 1 ≤ t, i, j ≤ n, p i is a capping path from x 0 to x i , γ ij = p −1 i · p j is a standard cord from x i to x j , m t is a meridian based at x 0 , and m (t) t = p −1 t · m t · p t is a meridian cord based at x t . To verify the first identity, by definition and interpolating t · γ tj ). By the skein relation, there is Combining these equations, we conclude that In a similar fashion, we can derive for the second identity in the assertion: From the meridian relation, we know that ((m Now we are ready to prove that the construction is indeed a representation. Proof of Theorem-Definition 2.16. We write (ρ, V ) instead of (ρ , V ) in this proof.
(1) To see that ρ(γ) is a closed linear map, it suffices to prove for meridian generators. For γ = m ±1 t and any vector R j , there is ρ(m t )R j ∈ V . By Lemma 2.17, we have The identity is straightforward to check. Let e be the identity loop based at x 0 , then ρ(e)R j = R j for all j by definition.
(3) To verify the composition in the group action, one needs ρ(γ 1 )ρ(γ 2 ) = ρ(γ 1 · γ 2 ). Since each loop in π K can be expressed as a word of meridian generators, we can prove by induction on the word length of γ 2 .
For the initial step, we take γ 1 ∈ π K , and γ 2 = m ±1 t . It suffices to prove that for any R j , . The first equation is by definition (2.3) and the second equation is by skein relations. Comparing the results, we see that ρ(γ 1 )ρ(m t )R j = ρ(γ 1 · m t )R j for any j = 1, · · · n.
The proof for the case γ 2 = m −1 t is similar, except the skein relation is applied in the other way. In brief, we get Proceeding to the induction step, we assume that ρ(γ 1 )ρ(γ 2 ) = ρ(γ 1 · γ 2 ) for any γ 1 ∈ π K , and any γ 2 ∈ π K that can be written as a word of meridian generators with length less or equal to s. Assume γ 2 = γ 2 m ±1 t has word length less or equal to s + 1, then . Each of the three equalities follows from the induction hypothesis. Therefore we complete the induction and show that ρ(γ 1 )ρ(γ 2 ) = ρ(γ 1 · γ 2 ).
(4) We prove that ρ(γ) is a well-defined linear map, namely if I ⊂ {1, · · · , n} is a subset and i∈I a i R i = 0 for some constants a i , then i∈I a i ρ(γ)R i = 0 as well. If γ = m t , by Lemma 2.17, there is The first summand is zero by hypothesis. In the second summand, i∈I a i (γ ti ) equals to the t-th row of i∈I a i R i = 0, which is also zero. The argument for γ = m −1 t is similar. Continuing by induction on word length of γ in terms of meridian generators, we prove the well-definedness.
Since a framed link isotopy can be extended to an ambient isotopy and every framed cord finds its counterpart by the isotopy, it is clear that Cord(K) is a framed link invariant. An augmentation : Cord(K) → k, algebraically defined as a algebra morphism, is also well-defined for isotopy classes of framed links.
However, to construct an associated augmentation representation (ρ , V ), one needs to choose a braid, which is additional data. It is not immediately clear about the dependent of this choice from the definition. In the following theorem, we address this question.
Theorem 2.18. Suppose K is an oriented link equipped with its Seifert framing. Let : Cord(K) → k be an augmentation of its framed cord algebra. Up to isomorphism, the augmentation representation (ρ , V ) is well-defined for the augmentation . In particular, it does not depend on the choice of the braid in the construction.
Proof. Suppose B is a braid whose closure is K. Let (ρ , V ) be the augmentation representation of constructed for B as in Theorem-Definition 2.16. SupposeB is another braid whose closure is also K, and let (ρ ,Ṽ ) be the representation constructed with respect toB. We shall prove that (ρ , V ) and (ρ ,Ṽ ) are isomorphic as π K -representations.
In the rest of the proof, we drop the subscript and write (ρ, V ), (ρ,Ṽ ) for (ρ , V ), (ρ ,Ṽ ) To prove the isomorphism, we will construct a linear map T : V →Ṽ which intertwines with link group actions. Namely, for any γ ∈ π K , there should be a commutative diagram, Since the link group is generated by a set of meridian generators, it suffices to check the commutativity for γ being meridian generators.
By Markov's theorem, two braids have isotopic closures if and only if they are related by a sequence of equivalence relations: (1) they are equivalent braids, (2) they are conjugate braids, (3) one braid is a positive/negative stabilization of the other braid. It is clear that the construction of the augmentation representation is well-defined within an equivalence class of braids. We will verify the well-definedness in other cases.

Conjugation.
Suppose B is an n-strand braid. Let σ s , 1 ≤ s ≤ n − 1 be the positive half twists in the braid group Br n . LetB = σ s Bσ −1 s be a conjugation. We write γ ij for the standard cords in the configuration disk D that is used to define (ρ, V ), andγ ij (abbreviated from a more rigorous notationγ˜ij) for the standard cords in the configuration diskD that is used to define (ρ,Ṽ ). These standard cords are related as s · γ s,s+1 . We related the two sets of meridian generators {m t } 1≤t≤n , {m t } 1≤t≤n in a similar way: Again,m t is an abbreviation ofmt.
We define matrices R,R by R ij = (γ ij ),R ij = (γ ij ). Following the construction of the augmentation representation, we define We first verify that V andṼ have the same rank, which is not obvious from the definition. We will argue thatR equals to R after a sequence of row operations and column operations, namely there exist n × n matrices M L and M R such thatR = M L RM R . The isomorphism follows from the fact that these operations do not change the rank of the space spanned by column vectors. We claim where M L and M R are 2 × 2 matrices We verify that M L RM R =R at each entry. Observe that M L , M R only affect rows and columns indexed by s, s + 1. If neither the row index nor column index is in {s, s + 1}, it is clear that If only one of the row or column index is in {s, s + 1}, then Finally we verify for the 2 × 2 sub-matricies whose indices are contained in [s, s + 1] 2 . We write A [s,s+1] 2 for a 2 × 2 submatrix of A consisting entries with both row and column indices in {s, s + 1}. Let (γ s+1,s ) = a, (γ s,s+1 ) = b.
where the last equality is because of (γ s,s+1 ) = (γ s+1,s · (m (s) We have verified that V andṼ are isomorphic as vector spaces. We define a linear map T : V →Ṽ . We define T over the spanning vectors: The inverse map T −1 is thusly determined: In terms of the spanning vectors, T and T −1 can be expressed as n × n matrices: Recall that V andṼ are related by row and column operations. Since row operations do not change the linear dependence relations among the column vectors and the matrix M R for column operations is consistent with T −1 , we conclude that the map T descents to a linear transform from V toṼ .
Finally we show thatρ(γ) • T = T • ρ(γ) for γ = m t , t = 1, · · · , n. We introduce some notations to prepare the calculation. Following the calculations in Lemma 2.17, ρ(m t ) and ρ(m t ) can be written as n×n matrices respecting formal linear combinations of their spanning vectors: where E ij is the matrix having 1 at the (i, j)-th entry and 0 elsewhere. We use Kroneker delta δ i j to denote a number which equals to 1 when i = j and 0 otherwise. We do not use Einstein's convention for contractions. All summations will be indicated explicitly.
Introducing new matrices N t := I n − M t andÑ t := I n −M t , then (2.9) We are ready to proceed into details. Recall our goal is to verify We remind the reader that on the left hand side, it isρ(m t ) instead ofρ(m t ). We will consider the following three cases based on the value of t: (A) t s, s + 1, (B) t = s and (C) t = s + 1.
• Case (C). If t = s + 1, then m s+1 =m −1 s+1 ·m s ·m s+1 . This time equation (2.10) is more complicated, becomingM −1 s+1M sMs+1 T = T M s+1 , which further simplifies to (2.11)M −1 s+1Ñ sMs+1 T = T N s+1 . We first compute the right hand side of (2.11). At the (i, j)-entry, there is Observe that row s + 1 of T is mostly zero except i = s, s + 1. Therefore the right hand side of (2.11) reads (2.12) Next we compute the left hand side of (2.11). We make a claim that the the (i, j) entry admits the following expansion Proof of claim (2.13): Using (2.9), we find that Continuing with the term (γ se )(M s+1 ) ef in the summand, we have Combining these terms and we obtain (2.13).
We have verified equation (2.13). Observing the result is the product of two term, one depends only on i and the other depends only on j, we calculate them separately.
Consider the term with i, (M −1 s+1 ) is . Given the expression ofM t , it is straightforward to compute its inversẽ We set t = s + 1 and get s+1 ). Consider the term involving j. We first present the answer, which is Multiplying it with the term involving i, we immediately see that the result matches (2.12), proving (2.11).
We verify (2.14) in the following cases.
-If j s, s + 1, then T f j = δ f j , and Here the second equality follows from (2.6), and the last equality follows from the skein relations.
In the past two pages, we have verified Case (C) of (2.10). Together with the other two cases, we have proven (2.10), which says that the vector space isomorphism T respects group actions. To summarize, the augmentation representations of a braid before and after a conjugation are isomorphic as representations.
Positive/Negative stabilization. A stabilization of a braid closure is given by adding one strand and perform a positive/negative a half twist with the outmost strand. The braid closure of a stabilized braids is equivalence to the closure of the original braid by a Reidemeister I move. Therefore stabilizations do not change the isotopy class of a link.
We fix conventions. Let ι : Br n → Br n+1 be the natural inclusion by adding a trivial strand labelled n + 1. Suppose B is an n-strand braid, its positive stabilization is σ n ι(B) and its its negative stabilization is σ Let B be an n-strand braid whose closure is K, letB = σ n ι(B) be its positive stabilization. We adopt similar notations as before. Coefficients involving µ, are related bỹ µ {n} =μ {n+1} = µ {n} , µ {t} = µ {t} for t n, n + 1. (2.15) Let γ ij , 1 ≤ i, j ≤ n andγ ij , 1 ≤ i, j ≤ n + 1 be standard cords for B andB. Different from the case of conjugations, this time B andB no longer have the same number of strands. As a consequence, the indices for the two sets of framed cords are different. For i, j n, n + 1, there areγ n ,γ n+1,n+1 = γ nn . The meridian generators are related by: We define matrices R,R by R ij = (γ ij ),R ij = (γ ij ). Following the construction of the augmentation representation, we define Different from the previous case of conjugations, this time the two matrices have different sizes. Out first goal is to show that V andṼ have the same rank. Recall the identities Now it is clear that V andṼ have the same rank. Define a linear transform T : V →Ṽ over spanning vectors: Because of the linear relation µ −1 {n}R n +R n+1 = 0 inṼ , the linear map T is a surjection, and further an isomorphism of vector spaces. In terms of spanning vectors, T can be written as a (n + 1) × n matrix: Finally we prove that T is an isomorphism of representations, namely for 1 ≤ t ≤ n, If t n, then m t =m t , γ tj =γ tj . It is equivalent to check . By a straightforward calculation, both hand sides equal to a (n + 1) × n matrix where the (t, j) entry equals to (γ tj ) and zero otherwise.
If t = n, then we take m t =m n+1 . It remains to check (I n+1 −M n+1 ) • T = T • (I n − M n ). Both hand sides equal to a (n + 1) × n matrix where the (n + 1, j) entry equals to (γ nj ) and zero otherwise.
We have proven that the group actions of the meridian generators are compatible on both V andṼ , therefore so is the entire link group π K . We conclude that the construction of the augmentation representations before and after a positive stabilization are isomorphic.
The proof for a negative stabilization is similar, which we do not repeat here. For reference, we record some identities and the matrixR after a negative stabilization. For 1 ≤ i, j ≤ n−1, the framed cords are identified by the following relations the meridian generators are identified in the same way as in (2.17), and the matrixR is So far we have proven that the augmentation representation constructed from a braid is invariant under either a positive or a negative stabilization.

Summary.
In previous arguments, we discussed in different cases that the augmentation representation ρ : π K → GL(V ) constructed from : Cord(K) → k, where the link K is represented by a braid closure, is invariant under a conjugation or a positive/negative stabilization of the braid. Together with the relatively obvious case of braid equivalences, we can apply Markov's theorem and prove that the augmentation representation (ρ , V ) is well-defined up to isomorphism, namely it is independent from the choice of the braid representative of the link.
Remark 2.19. In this remark, we compare the construction in this paper and that in [Ga2].
In the case when K is a knot, we can either write it as a braid closure and apply Theorem-Definition 2.16 to construct the augmentation representation for : Cord(K) → k, or we can choose a set of meridian generators and apply [Ga2,Propsition 4.11] to construct a representation for : P K → k.
Both constructions require additional data, and in both cases, the representation does not depend on additional data. The arguments for independence are different. In this paper, we use Marko's theorem to prove that Markov moves give isomorphic representations. In [Ga2], we use the correspondence between augmentations and sheaves (which is the main theorem of loc.cit.) to argue that the construction does not depend on the choice of the generating set of meridians. Now we explain that the current paper manifests the underlining geometry of the construction in [Ga2]. It is not an immediate identification of the two constructions using the Z-algebra isomorphism Cord c (K) P K in Remark 2.9. Thanks to the fact that in neither constructions the representation depends on the additional data, we can make a preferred choice for the position of the knot as well as the set of meridian generators.
Suppose knot K is the braid closure of an n-strand braid B and let {m t } 1≤t≤n be the set of meridian generators in the configuration disk associated to B. We choose x 1 in the configuration disk to be the base point of the knot group π K , then our preferred set of meridian generators is {m (1) t } 1≤t≤n . Since any two meridians are conjugate to each other in knot group, there exists g t ∈ π K for each t, such that m 1 · g t . Both constructions starts with a square matrix R. Given our set up, both R matrices are of size n. For Cord(K), the R matrix is given by augmented values of standard cords, For P K , the R matrix is given by (R P K ) ij = P K (g i · g −1 j ). We present a entry-wise identification of the two R matrices. The orientation of knot K induces an orientation of its Seifert framing . Recall that * ∈ is the marked point used in the longitude relation in the cord algebra. For each t ∈ {1, · · · , n}, there is a unique path d t1 from x t to x 1 that is contained in L \ { * } with compatible orientation. There are 1 · g t , we can solve for (2.18) Note the g t here is really a choice we made to fit into the equation m 1 · g t . This choice is not unique in general. For example, the longitude commutes with the meridian in the peripheral subgroup. The concatenation of a choice of g t with the longitude gives another choice. Fix the chosen g t in (2.18), we have It is similar to check that the group actions also match.
As a conclusion, the construction in this paper explains the underlining geometry of the correspondence between augmentations and sheaves for knots.

A microlocal digression
Given an augmentation of the framed cord algebra of a link, we defined the associated augmentation representation in the previous section. It is a representation of the fundamental group of the link complement, which is equivalent to a locally system on the link complement. We will study properties of the augmentation representation in the next section. It is easier to understand these properties from the perspective of microlocal sheaf theory. Indeed, a local system can be viewed as a locally constant sheaf, and we can proceed to study it using microlocal methods. In this section, we give a quick introduction to microlocal sheaf theory.
A key concept is the micro-support of a sheaf. We introduce the definition in a general setting. Let k be a commutative field. Let Y be a smooth manifold. Let M od(Y ) be the abelian category of sheaves of k-modules on Y , and Sh(Y ) be the bounded dg derived category. For any object F ∈ Sh(Y ), its microsupport SS(F) ⊂ T * Y is a closed conic subset, point-wisely defined as follows (also see [KS,Definition 5.1.1]).
Definition 3.1. Let F ∈ Sh(Y ) and let p = (y 0 , ξ 0 ) ∈ T * Y . We say that p SS(F) if there exists an open neighborhood U of p such that for any y ∈ Y and any real C 1 function φ on Y satisfying dφ(y 0 ) ∈ U and φ(y 0 ) = 0, we have Kashiwara-Schapira defined the microlocal hom bifunctor [KS,Definition 4 giving a quantitative description of the micro-support. Suppose F, G ∈ Sh(Y ), then Let T ∞ Y be the unit cosphere bundle of Y . It admits a natural a contact induced form from the canonical form of the cotangent bundle. Let Λ ⊂ T ∞ Y be a (not necessarily connected) smooth Legendrian submanifold. Define a dg subcategory Following a result of Guillermou-Kashiwara-Schapira [GKS], Sh Λ (Y ) is a Legendrian isotopy invariant. The abelian category M od(Y ) can be viewed as a dg subcategory of Sh(Y ) consisting of objects concentrated in homological degree zero. Then M od Λ (Y ) := Sh Λ (Y )∩M od(Y ) is also a Legendrian isotopy invariant of Λ.
A sheaf F ∈ Sh Λ (Y ) is simple along Λ if one of the following equivalent conditions holds: • For any p ∈ Λ, the microlocal Morse cone defined as in (3.1) has rank 1, namely • The self microlocal hom restricts to a constant sheaf supported on Λ, namely We write Sh s Λ (Y ) ⊂ Sh Λ (Y ) for the subcategory of simple sheaves along Λ. The triangulated structure is lost when we pass to this subcategory.
There is a distinction between a sheaf F ∈ Sh Λ (Y ) being "simple along Λ" or "simple along its micro-support". The two notions are equivalent if SS(F) ∩ T ∞ Y = Λ. However in the definition Sh Λ (Y ), an object F is only required to have its micro-support intersecting T ∞ Y in a subset of Λ. When Λ has multiple connected components, such as in our case the conormal tori of links, the first notion is strictly stronger than the second (defining fewer objects).
In the case that we consider in this paper when Y = X = R 3 or S 3 , and Λ = Λ K the Legendrian conormal tori of a link K, simple sheaves microsupported along Λ admit an easier description. The derivation is parallel to the case studied in [Ga2] when K is a knot. It is because both the micro-support and the simpleness are local properties, and the local pictures for knots and links are no different.
We outline the argument with references in [Ga2]. For simplicity, we consider a sheaf F ∈ M od Λ (X) concentrated at homological degree 0. The micro-support constraints force that F restricted to each component of the link, or the link complement is a local system [Ga2,Lemma 3.1]. In terms of group representations, these local systems are equivalent to Here π K is the link group and Z K i := π 1 (K i ). Conversely, we can reconstruct the sheaf from these local systems by gluing. The gluing data is an element in an extension class, and according to [Ga2,Lemma 3.2,Lemma 3.3], it is equivalent to a collection of linear maps which satisfy the following compatibility conditions. For each 1 ≤ i ≤ r, let m i , i be the meridian and longitude in the peripheral subgroup, then T i satisfies (a) ρ( i )•T i = T i •ρ i (K i ), and (b) m i acts on the image of T i as identity. We conclude that the sheaf F is equivalent to the collection of data (ρ, V, ρ The sheaf F is simple along Λ if and only if cone(T i ) has rank 1 for each 1 ≤ i ≤ r. It is simple along its singular support if and only if cone(T i ) has rank at most 1 for each 1 ≤ i ≤ r. These statements follow from the first of the equivalent definitions of simpleness.
In this paper, we focus on the representation of the link group, namely (ρ, V ). The simpleness imposes strong restrictions to this representation. If cone(T i ) has rank 1, then T i is either injection with a rank 1 cokernel, or T i is surjective with a rank 1 kernel. Consider the first case when T i is injective, then W i can be regarded as a subspace of V of codimension 1. The condition (b) of T i requires m i acting on a codimension one space as identity. In the other cases, including when rank(cone(T i )) = 1 with surjective T i , or when rank(cone(T i )) = 0, the action on m i is entirely trivial. Combining these cases, we see that for F to be simple, it is necessary to require (ρ, V ) satisfying that the action of each meridian m i fixes a subspace of codimension at most 1.

Properties of the augmentation representation
Given the framed cord algebra of a framed link, we have constructed a link group representation for each augmentation of the algebra. In this section, we study the properties of these representations. 4.1. Microlocal simpleness. Let M od Λ K (X) be the abelian category of sheaves on X microsupported along the link conormal Λ K . Let M od s Λ K (X) ⊂ M od Λ K (X) be the full subcategory of simple sheaves. Suppose F ∈ M od s Λ K (X), then j −1 F is a local system on X \ K. The local system is equivalent to a representation ρ : π K → GL(V ). The simpleness of F requires that any meridian acts on V as identity on a subspace of codimension 1 or 0.
Based on this observation, we make the following notion of simpleness.
is microlocally simple if ρ(m) is almost identity for every meridian m ∈ π K . Proposition 4.3. Augmentation representations are microlocally simple.
Proof. Let (ρ , V ) be the augmentation representation associated to an augmentation : Recall that R is the n × n matrix determined by . As a vector space, V is spanned by the column vectors R j , 1 ≤ j ≤ n. To verify that a linear automorphism T ∈ GL(V ) is almost identity, it is sufficient to show that (id V − T )R j is contained in a rank 1 subspace of V for all 1 ≤ j ≤ n.
To prove (ρ , V ) is microlocally simple, we need to verify that every T = ρ (m) with m being a meridian is almost identity. Since any two meridians with coherent orientations belonging to the same component K i ⊂ K are conjugate, it suffices to select a meridian m i for each component K i and prove that ρ (m i ) ±1 are both almost identity. It is easy to see that if ρ (m i ) is almost identity, so is ρ (m i ) −1 . Finally recall that {m t } 1≤t≤n are meridian generators of the link group π K . We observe that the generating set contains at least one meridian for each component.
Following these arguments, the assertion in the proposition reduces to show that for any generating meridian m t , there exists a rank 1 subspace U t ⊂ V such that We prove the desired result.
Remark 4.4. Microlocally simple knot group representations are called "KCH representations" or "unipotent KCH representations" in earlier papers [Ng4,Co,Ga2]. We stop using these names for two reasons. First, the abbreviation "KCH" as an prefix of the representation emphasizes more on contact topology and its relation to sheaf theory, instead of the property of the representation itself. It is more revealing to borrow the notion of simpleness from microlocal sheaf theory. For the second reason, it is a locally property whether a meridian action is diagonalizable or unipotent. When we work with links, the previous naming system fails to generalize in a concise way when meridians of different components act differently. On the other hand, the word "microlocal" has a good implication that one should consider the local behaviors separately for each component of the link. 4.2. Vanishing. Suppose K ⊂ K is a sublink. The sublink group is π K = π 1 (X \K ). Note this fundamental group is taken over the complement of K in X, forgetting that K is a sublink of K. The natural open inclusion j : X \ K → X \ K induces a map on fundamental groups: The composition defines a functor of abelian categories Rep(π K ) → Rep(π K ). If we identify representations of a fundamental group as local systems, the functor coincides the pull back functor j −1 : loc(X \ K ) → loc(X \ K).
In the previous subsection, we proved that augmentations are microlocally simple. It follows from the definition that the group action of a meridian has two possibilities. It either fixes the entire vector space, or defines an invariant subspace of codimension one. In this subsection, we focus on the first case, and give a sufficient condition on augmentations so that the associated augmentation representations fit into this situation.
Definition 4.5. Let K ⊂ X be a link and K ⊂ K be a sublink. A simple link representation ρ : π K → GL(V ) vanishes on K if ρ(m ) = id V for any meridian m of K .
The definition comes from an observation in the sheaf theory. Let E ∈ loc(X \ K) be the local system determined by a link group representation (ρ, V ). Let j : X \ K → X be the open embedding. We consider the underived push forward F = j * E. If (ρ, V ) is vanishes on K , then F is microsupported on the components other than K , i.e.

SS(F)
Proposition 4.6. Let K ⊂ X be a link and K ⊂ K sublink. Suppose an augmentation : Cord(K) → k satisfies either one the following conditions: • for any framed cord γ starting from K , (γ) = 0; or • for any framed cord γ ending on K , (γ) = 0, then the associated augmentation representation (ρ , V ) vanishes on K .
Proof. By Lemma 2.2, we can assume there is an integer s with 1 ≤ s ≤ n, such that the closure of strands {1, · · · , s} is precisely the sublink K .
By definition, the associated augmentation representation vanishes on K if for any meridian m t , 1 ≤ t ≤ s, the action ρ (m t ) is identity. Recall from (2.4) that Therefore it suffices to show that either (p −1 t · p j ) = 0 for any j ∈ {1, · · · , n}, or R t is the zero vector.
Suppose maps all cords starting on K to zero. For any t ∈ {1, · · · , s} and any j ∈ {1, · · · , n}, p −1 t · p j is a cord starting on K . Hence (p −1 t · p j ) = 0 and the assertion follows. Suppose maps all cords ending on K to zero. For any t ∈ {1, · · · , s}, R t consists of augmentations of cords ending on K . Therefore R t is a zero vector as expected.
In each of the cases, we have proven the assertion.
4.3. Separability. In this section, we discuss a sufficient condition for which the augmentation representation is separable. We begin with an example to motivate the notion of separability.
We consider the two-component unlink and Hopf link (Figure 4.1). Each of them has two components, and every component is an unknot. A link is split if it is the union of two sublinks that lie in two disjoint solid balls. In our example, the unlink is split while Hopf link is non-split.
Therefore he augmentation variety of Hopf link is We can similarly compute the augmentation variety of the two-component unlink We observe that the augmentation variety of the unlink is contained in that of Hopf link, corresponding to case (B) when (a 12 ) = (a 21 ) = 0. In this case, all mixed cords are augmented to zero. Though Hopf link is non-split, some augmentations behave as if the framed cord algebra came from a split link. We remark that the idea of sending mixed cords to 0 has been considered in contact geometry such as: [Mi, AENV].
We propose a counterpart of this phenomenon on the sheaf side. The following lemma will show that the direct sum of two simple sheaves which are microsupported along disjoint Legendrians is again simple. Lemma 4.7. Let Y be a manifold, and Λ 1 , Λ 2 ⊂ T ∞ Y two disjoint Legendrian submanifolds. If F 1 ∈ D b,s Λ 1 (Y ), F 2 ∈ D b,s Λ 2 (Y ), then F 1 ⊕ F 2 ∈ D b,s Λ 1 Λ 2 (Y ). Proof. Because Λ 1 , Λ 2 are disjoint, µhom(F 1 , F 2 )| T ∞ Y = µhom(F 2 , F 1 )| T ∞ Y = 0. The simpleness of F i yields µhom (F i Now we present a sufficient condition on augmentations so that the associated augmentation representations split into direct summands. Definition 4.8. Suppose K = K 1 K 2 is the union of two sublinks K 1 , K 2 . A link group representation ρ : π K → GL(V ) is separable with respect to the partition if there exists link group representations ρ i : π K i → GL(V i ) for each i = 1, 2, such that where V i is considered as a π K -representation through the composition Proposition 4.9. Suppose K = K 1 K 2 is the union of two sublinks K 1 , K 2 . If an augmentation : Cord(K) → k maps all mixed cord between K 1 and K 2 to zero, the induced augmentation representation is separable with respect to the partition.
Proof. To simplify the presentation of the proof, we apply Lemma 2.2 and assume there is an integer s with 1 ≤ s ≤ n − 1, such that the closure of strands P 1 := {1, · · · , s} is K 1 , and the closure of strands P 2 := {s + 1, · · · , n} is K 2 .
LetR j , 1 ≤ j ≤ s be the column vectors ofR 1 . We define The sublink group π K 1 acts on V 1 in the following way. For γ ∈ π K 1 , we define ρ 1 (γ)R j = (p −1 α · γ · p j ), for 1 ≤ j ≤ s. The procedure to check that (ρ 1 , V 1 ) is a representation of π K 1 is similar to that in Theorem 2.16. We remark two subtleties in the construction. First, π K 1 is generated by meridians m t , 1 ≤ t ≤ s, which is a subset of meridian generators of π K , but the quotient relations are different from those in π K . Second, we take skein relations in Cord(K 1 ) instead of those in Cord(K).
From now on, we abuse notations and regard (ρ i , V i ), i = 1, 2 as a representation of π K . Taking the direct sum, we get a representation ρ 1 ⊕ ρ 2 : π K → GL(V 1 ⊕ V 2 ).
We prove that (ρ 1 ⊕ ρ 2 , V 1 ⊕ V 2 ) is isomorphic to the augmentation representation (ρ , V ). Recall that V 1 is spanned by {R j } 1≤j≤s , V 2 is spanned by {R j } s+1≤j≤n , and V is spanned by {R j } 1≤j≤n . We define a morphism between vector spaces: It is an isomorphism of vector spaces. By hypothesis, R ij = 0 for both 1 ≤ i ≤ s < j ≤ n and 1 ≤ j ≤ s < i ≤ n. Therefore the intersection of Span{R j } 1≤j≤s and Span{R j } s+1≤j≤n is trivial. Further by definition, linear relations in V 1 are preserved in Span{R j } 1≤j≤s , and similar for V 2 . We conclude the vector space isomorphism. It remains to check that ϕ preserves the group action. Namely, for any v ∈ V 1 ⊕ V 2 and any γ ∈ π K , we want It suffices to verify the equation for any spanning vectorR j and any generating meridian m t . The left hand side of (4.1) becomes, Recall {−} is the component function. If K {t} and K {j} belong to different sublink, then (p −1 t · p j ) = 0 by hypothesis. The equation can be further simplified to ρ (m t ) · ϕ(R j ) = R j .
For the right hand side of (4.1), we compute in two cases.
We complete the proof.
Remark 4.10. In the proof, we define an induced representation for a sublink. It is natural to ask whether this construction is functorial. In general it is not possible, because for a sublink K 1 ⊂ K, one does not have a natural embedding Cord(K 1 ) → Cord(K). The natural morphism between these two framed cord algebras is in the opposite direction: Under the hypothesis of Proposition 4.9, an augmentation : Cord(K) → k factors through this morphism, giving an induced augmentation 1 . The augmentation representation of 1 : Cord(K 1 ) → k is isomorphic to ρ 1 : π K 1 → GL(V 1 ) constructed in the proof. It is similar for 2 : Cord(K 2 ) → k.
Remark 4.11. A natural question is whether the augmentation representation is irreducible. The answer is no. If K is a link, then any example in Proposition 4.9 such that V 1 , V 2 are non-trivial is reducible.