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Statistics on some classes of knot shadows

The present paper is concerned with the enumeration of the state diagrams for some classes of knot shadows endowed with the usual connected sum operation. We focus on shadows that are recursively generated by knot shadows with up to 3 crossings, and for which the enumeration problem is solved with the help of generating polynomials.


Introduction
Let a mathematical knot be identified with its regular projection onto the sphere S 2 . The corresponding representation, called shadow, is a planar quadrivalent diagram without the usual under/over information at every crossing [4,7,9,15]. We can split each crossing of the diagram in one of two ways as shown in Figure 1.  By state is meant one of the obtained diagram with each crossing being split either by a type 0 or a type 1 split. A state can be seen as a collection of disjoint non-intersecting closed curves called circles. For a state S, we let |S| denote the number of its circles. Then, for a knot diagram K with m crossings, we define the following statistics by summing over all states S: where σ(m, k) count the occurrence of the states with p circles, with σ(m, 0) = 0 for all m.
For the sake of simplicity, we call the state-sum formula (1) the generating polynomial. In fact, it is a simplified approach to the so-called Kauffman bracket polynomial [5,13]. We intentionally omit the split variables indicating the over-and under-crossing structure since the summation is calculated with respect to the shadow diagram. The generating polynomial is only intended as a tool at enumerating the state diagrams, and no attempt is made here to investigate its topological property. Moreover, we have the following simplified rule which is then iteratively applied to all crossings in the diagram: In this paper, we mainly focus on the distribution of the number σ(m, k) defined in (1) in terms of generating polynomial for some particular classes of knot shadows.
We organize the paper as follows. In section 2, we construct a recursive definition of some classes of knot shadows and define the associated closure operation. In section 2, we established the generating polynomial for the knots introduced in section 3. Then in section 4, we establish the generating polynomial for the closure of the same knots.

Background
Throughout this paper, unless explicitly stated otherwise, the generic term "knot (diagram)" refers to a shadow drawn on the sphere S 2 . The simplest mathematical knot is the unknot which is a closed loop with no crossings in it. We say that two knots are the same, if one can be continuously deformed to the other so long as no new crossings are introduced and no crossings are removed. Such deformation is called a planar isotopy. A practical illustration would be to consider the corresponding diagram as a "highly deformable rubber" as suggested by Collins [1, p. 12]. To set up our framework, we introduce the following deformation which preserve as well the crossings configuration.
Definition 1 (Denton and Doyle [4]). When we have a loop on the outside edge of the diagram, we can redraw this loop around the other side of the diagram by pulling the entire loop around across the far side of the sphere without affecting the constraints on any of the already existing crossings (see Figure 2). The move is called a type 0 move on the sphere, denoted 0S 2 .
We can then define an equivalence relation on the set of knot shadows such that two knots lie in the same equivalence class if they have the same number of crossings, and if one can be transformed to the other by a finite sequence of 0S 2 moves (modulo planar isotopy). Restricting ourself to the shadows of up to 3 crossings, we give in Table 1 all the possible combination of knot under the 0S 2 move for each given number of crossings [2, p. 14]. We shall refer to this set of knots as elementary knots. For our arguments, we next associate these shadow diagrams with the following operations. Definition 2. The connected sum of two knots K and K ′ , denoted by K#K ′ , is the knot obtained by removing a small arc from each knot and then connecting the four endpoints by two new arcs in such a way that no new crossings are introduced [11].
Analogously, the disconnected sum or disjoint union of two knots K and K ′ , denoted by K ⊔ K ′ , is the diagram obtained by placing the two diagrams inside two non-intersecting domains on the sphere [14, p. 15].
Example 3. Consider the following connected sum: We point out that the connected sum # and the disconnected sum ⊔ are both associative and commutative [20, p. 61]. Moreover, for any knot K, we have K#U = K, where U denote the unknot.
Our framework will make extensive use of the following special notation: Notation 4. Let K be a knot, and let n be a nonnegative integer.
1. K n := K#K# · · · #K n copies with K 0 = U. We say that the knot K n is generated by K, and the knot K is the generator of K n .
In this paper, we establish the generating polynomials of the knots that are generated by the elementary knots. To begin with, we pay a special attention to the following series of knots.
S 3,4 (trefoils) , . Definition 5 (Twist loop). A twist loop is a knot obtained by twisting the unknot. We refer to a twist loop of n half twists as n-twist loop [17,18]. We let T n denote an n-twist loop, with T n := = . (3) Definition 6 (Link). An n-link is a knot which consists of n + 1 linear interlocking circles. We let L n denote an n-link, with The knot L 1 is also referred to as Hopf link.
Definition 7 (Twist link). We construct an n-twist link is by interlocking n series of 1-twist loops, starting from the unknot. We let W n denote an n-twist link, with where the generator is obtained by twisting the Hopf link.
Definition 8 (Hitch knot). Ashley [3, #50, p. 14] describes the half hitch as "tied with one end of a rope being passed around an object and secured to its own standing part with a single hitch", see Figure 4. We define a n-hitch knot, denoted by H n , as the shadow obtained by joining together the two loose ends of a thread of n half hitches. The corresponding connected sum is given by The knot H 1 is known as trefoil.
Definition 9 (Overhand knot). The overhand knot is a knot obtained by making a loop in a piece of cord and pulling the end through it. For instance, we see in Figure 5 a single and a two series of overhand knot. If as previously we join together the two loose ends of a n series of overhand knot, then we call the projected shadow an n-overhand knot, and we shall refer to such knot as O n . We have In the present representation, we also refer to the knot O 1 as trefoil.
Besides, we define as well the closure or the closed connected sum of a knot as the connected sum with itself as shown in Figure 6. The closure of the unknot, which is a disjoint union of two closed loops, is therefore the simplest of all the closure of knots. We let K denote a closure of the knot K.  Let us then present the n-foil knot, the n-chain link, the n-twisted bracelet, the n-ringbolt hitching, and the n-sinnet of square knotting which are respectively the closure of the n-twist loop, the n-link, the n-twist link, the n-hitch knot and the n-overhand knot. In what follows, we give the formal definition of these knots, and give the corresponding shadow diagrams.
Definition 10 (Foil knot). An n-foil [18] is a knot obtained by winding n times around a circle in the interior of the torus, and 2 times around its axis of rotational symmetry [1, p. 107]. Definition 11 (Chain link). An n-chain link consists of n unknotted circles embedded in S 3 , linked together in a closed chain [12]. Definition 12 (Twist bracelet). An n-twist bracelet (or a twisted n-chain link [12]) consists of n twisted link intertwined together in a closed chain [16]. Figure 9: n-twist bracelets, n = 0, 1, 2, 12.
Definition 13 (Ringbolt hitching). By n-ringbolt hitching, we mean a series of n half hitches that form a ridge around a ring or loop. Definition 14 (Sinnet of square knotting). Ashley [3, #2906, p. 471] defines a chain sinnet as a knot which are made of one or more strands that are formed into successive loops, which are tucked though each other. Here, we borrow the term n-sinnet of square knotting to describe a closed chain of n overhand knot, see Figure 11. Remark 15. The pairs of knots shown in Figure 12 are equivalent under the 0S 2 move. The 0S 2 move does not remove nor create a crossing, therefore a complete split leads to the same state diagram. We then expect that knots belonging to same equivalence class have equals generating polynomials.

The generating polynomial for the connected sum
The present section is devoted to computing the generating polynomials of the previously introduced knots.

Preliminaries
Regarding the knot operations # and ⊔, we have the following immediate results: A first extension is as follows.
Proposition 16. For an arbitrary knot K and the unknot U, the following property holds Proof. The initial diagram is accompanied with the unknot and so are each resulting diagram after a series of splits.
Corollary 17. Let n be a nonnegative integer, and let K be an arbitrary knot. Then We can generalize Proposition 16 as follows Proposition 18. For two arbitrary knots K and K ′ , the following equality holds Proof. Let us first compute the states of K. Each of these states is accompanied with the diagram of K ′ . Hence we have where i runs over all the states of K ′ , and k i is the corresponding number of circles. Finally by Corollary 17 we obtain and we conclude by noting that Proposition 19. For two arbitrary knots K and K ′ , the following equality holds Proof. If we first compute the states of the knot K ′ , then for each of these states, there exists exactly one circle which is connected to K ′ . In terms of polynomial, it means where, as previously, i runs over all the states of K, and k i is the corresponding number of circles. The result immediately follows from formula (9).
Corollary 20. Let K, K ′ and K ′′ be three knots. Then We will make extensive use of the following particular case Corollary 21. For an arbitrary knot K and a nonnegative integer n, we have Remark 22. Formula (13) suggests that in order to compute the generating polynomial for the knot K n , we simply have to compute that of the generator K. This formula also means that the generating polynomials associated with knots generated by elementary knots that lie in the same equivalent class are exactly the same, regardless of the choice of the arcs at which the connected sum is performed.
Theorem 23. The generating function for the sequence K n (x) n≥0 with respect to the occurrence of the generator K (marked by y) and the state diagrams (marked by x) is Proof. We write K(x; y) := n≥0 K n (x)y n , and the result follows from formula (13).
The results in subsection 3.2 -3.6 are all then immediate application of Corollary 21 and Theorem 23. For each of the concerned knots, we give the generating polynomial and the associated generating function. If available, we also give the A-records and the short definition from the On-Line Encyclopedia of Integer Sequences (OEIS) [19].

Twist loop
Let T n (x) := k≥0 t (n, k) x k denote the generating polynomial for the n-twist loop.
Theorem 24. The generating polynomial for the n-twist loop is given by the recurrence relation and is expressed by the closed form formula Proof. See Figure 13, then apply Corollary 21. Corollary 25. The generating function for the sequence T n (x) n≥0 is given by Combining expressions (14) and (15), we obtain the following recurrence relation: The values for the array (t(n, k)) n≥0, k≥0 are given in Table 2 for small value of n and k, with n ≥ k − 1. The result is a horizontal-shifted Pascal's triangle [19, A007318] Table 2: Values of t(n, k) for 0 ≤ n ≤ 10 and 0 ≤ k ≤ 11.
Here, it is immediate that for all nonnegative integer i, (T i ) n = T in . Let us then introduce the following calculations to complete our results.

Link
Let L n (x) := k≥0 ℓ(n, k)x k denote the generating polynomial for the n-link.
Theorem 27. The generating polynomial for the n-link is given by the recurrence relation and is expressed by the closed form formula Figure 14: The states of the 1-link, L 1 (x) = 2x 2 + 2x.
Corollary 28. The generating function for the sequence L n (x) n≥0 is given by .
The expression of the polynomial L n (x) suggests that Therefore, the corresponding polynomial coefficients satisfy the following recurrence relation: Therefore we have Table 5 giving the numbers ℓ(n, k), for 0 ≤ n ≤ 9 and 0 ≤ k ≤ 10.
Remark 29. We have L n (x) = 2 n T n (x). We interpret the factor 2 n as follows: at each pair of crossings making up a "link part", if we split one crossing in either type 0 or 1, then the resulting knot has equals generating with a twist loop. For example, we take n = 7. Letting the splits series be identified by a 0-1's string, we obtain some of the possible splits: 0 512 4608 18432 43008 64512 64512 43008 18432 4608 512 Table 5: Values of ℓ(n, k) for 0 ≤ n ≤ 9 and 0 ≤ k ≤ 10.

Twist link
Let W n (x) := k≥0 w(n, k)x k denote the generating polynomial for the n-twist link.
Theorem 30. The generating polynomial for the n-twist link is given by the recurrence relation and is expressed by the closed form formula Proof. Note that W 1 = L 1 #T 1 . Then by Proposition 19 we have We conclude by Corollary 21.
Corollary 31. The generating function for the sequence W n (x) n≥0 is given by .

Hitch knot
Let H n (x) := k≥0 h(n, k)x k denote the generating polynomial for the n-hitch knot.
Theorem 32. The generating polynomial for the n-hitch knot is given by the recurrence relation and is expressed by the closed form formula Proof. See Figure 15.
Corollary 35. The generating function for the sequence O n (x) n≥0 is given by .
The results are as expected since the knots H 1 and O 1 are equivalent under the OS 2 move. Thus we obtain the same statistics as the hitch knot (subsection 3.5).
4 The generating polynomial for the closure 4

.1 Preliminaries
Let K be a knot diagram. The state diagrams of the closure of K can be illustrated as in Figure 17. Therefore, there exist two polynomials α K , β K ∈ N[x] such that we can write the generating polynomial for the knot K as Conversely, if we "cancel" the connection, then we have the additional identity Let us refer to α K (x) and β K (x) as the components of the polynomial K(x). We can then extend (39) as follows.
Proposition 36. Let K, K ′ be two knot diagrams, and let K(x), K ′ (x) be respectively their generating polynomials. If α K (x) and β K (x) are the components of the polynomial K(x), then we obtain We should note that if K ′ = U, then we get back to the definition of the closure (40). Indeed we have Proof of Proposition 36. We first need to compute the generating polynomial for the knot K. Referring to Figure 17, the "largest double circle" becomes the closure K ′ , while the "largest single circle" is then connected with K ′ . See Figure 18 for the illustration. Next, we recover the same components α K (x) and β K (x), and taking into account the previously mentioned substitution, we write Corollary 37. Let K(x) be the generating polynomial for an arbitrary knot K, and let α K (x) and β K (x) be its components. For a nonnegative integer n, we have Proof. We write K n (x) = (K#K n−1 )(x), then apply Proposition 36. We have By (13) and (40) we obtain Corollary 38. Let K(x; y) := n≥0 K n (x)y n and K(x; y) := n≥0 K n (x)y n respectively denote the generating function for the sequence K n (x) n≥0 and K n (x) n≥0 . Then we have The key to establishing the generating polynomial for the closure of a generated knot is then to identify the components of that of the generator. The next results are direct application of Corollaries 37 and 38.

Foil knot
Let T n (x) := k≥0 f (n, k)x k denote the generating polynomial for the n-foil knot.
Theorem 39. The generating polynomial for the n-foil knot is given by the recurrence relation and is expressed by the closed form formula Proof. By Figure 19, we have α T 1 (x) = 1 and β T 1 (x) = 1. Next apply Corollary 37. Corollary 40. The generating function for the sequence T n (x) n≥0 is given by By (41) and (42), the coefficients f (n, k) must satisfy the following recurrence relation: Hence we have Table 8 for 0 ≤ n ≤ 12 and 0 ≤ k ≤ 12. Also, referring back to (17), we obtain the following sequences: • f (n, 1) = n, the nonnegative integers [ • if k ≥ 3, then we obtain the usual Pascal's triangle [19, A007318].  Table 8: Values of f (n, k) for 0 ≤ n ≤ 10 and 0 ≤ k ≤ 10.
Remark 41. From Table 2 and Table 8 we have from Table 5 and Table 8 we read and finally from Table 7 and Table 8 we read Remark 42. Following Remark 26, we have the corresponding results on the 2n-foil knot and the 3n-foil knot, namely (T 2 ) n (x) = (T 2n )(x) and (T 3 ) n (x) = (T 3n )(x).

Chain Link
Let L n (x) := k≥0 c(n, k)x k denote the generating polynomial for the n-chain link.
Theorem 43. The generating polynomial for the n-chain link is given by the recurrence relation and is expressed by the closed form formula Proof. By Figure 20 we get α L 1 (x) = x + 2 and β L 1 (x) = 1.

Twist bracelet
Let W n (x) := k≥0 b(n, k)x k denote the generating polynomial for the n-twist bracelet.
Theorem 46. The generating polynomial for the n-twist bracelet is given by the recurrence relation and is expressed by the closed form formula Proof. We have α W 1 (x) = x + 2 and β W 1 (x) = 2x + 3, see Figure 21. Corollary 47. The generating function for the sequence W n (x) n≥0 is given by .

Ringbolt hitching
Let H n (x) := k≥0 r(n, k)x k denote the generating polynomial for the n-ringbolt hitching.
Remark 51. Reading Table 6 and Table 13 yields

Sinnet of square knotting
Let O n (x) := k≥0 s(n, k)x k denote the generating polynomial for the n-sinnet of square knotting.
Theorem 52. The generating polynomial for the n-sinnet of square knotting is given by the recurrence relation and is expressed by the closed form formula Proof. Again, the usual routine allows us to give both expressions of O n (x). By Figure 23, we have α O 1 (x) = x 2 + 3x + 3 and β O 1 (x) = 1. We conclude by Corollary 37.
Remark 54. From Table 2 and Table 14 we have and from Table 8 and Table 14 we have

Twist knot
A twist knot is a knot obtained by repeatedly twisting a closed loop, and then linking the ends together [10, p. 57]. We let T n denote a n-twist knot, i.e., a twist knot of n half twists [17]. Examples of twist knots are given in Figure 24. Theorem 55. The generating polynomial for the n-twist knot is given by the relation and has the following closed form Proof. The n-twist knot can be decomposed into the closure of the connected sum of the Hopf link and the n-twist loop, i.e., T n = L 1 #T n , see Figure 25. Applying Proposition 36, we have where α L 1 (x) = x + 2 and β L 1 (x) = 1 are the components of L 1 (x). The closed form immediately follows.
Remark 57. From Table 2 and Table 15 we have and from Table 7, Table 8 and Table 15 we read

The alternative closures
We first introduce the following notation.
Notation 58. Let Then we have Moreover, writing K = U#K allows us to assume that the elementary knot K can be decomposed in a way that one might disconnect some knot factors that are not taken into consideration when connecting with a copy of the actual knot. For the sake of clarity, let us use the asterisk sign * to indicate that none of the arcs of the concerned knot are involved when generating K n and K n . For example, K = T 1 #L * 1 means that the connected sum is performed along some section of T 1 , and L 1 might be disconnected. We distinguish the following excluding cases: • if we cannot disconnect an elementary knot, then K n ∈ T n , T 2n , T 3n , L n , W n , H n , O n ; , O * 1 }, then K n (x) = U 2 #K n (x) and K(x; y) = xK(x; y), see Figure 26; • if K ∈ S 2,2 and K = T 1 #T * 1 , then K n (x) = T n #T n (x), see Figure 27 (a); Figure 27 (b); T n #T 2n (x), if K = T 1 #T * 2 or K = T 1 #T * 1 #T * 1 , see Figure 27 (c); • if K ∈ S 3,3 , then K n (x) = L n #T n (x), if K = L 1 #T * 1 , see Figure 27 (d); T n #L n (x), if K = T 1 #L * 1 , see Figure 27 (e).