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COMPUTATION OF THE KAUFFMAN BRACKET SKELETON

In this paper, we present an invariant called the Kauffman bracket skeleton, which is a simplification of the generalized Kauffman bracket polynomial in two variables. The idea is to take into account only the order and values of coefficients and disregard the degrees of one of the variables. However, the proposed simplification is more compact, and at the same time is not weaker than the original generalized Kauffman bracket polynomial in the sense of, for example, tabulation of prime knots and links in the thickened torus up to complexity 4 inclusively. In order to confirm this fact, we present tables of the proposed invariant for the tabulated prime knots and links in the thickened torus. Also, we construct an algorithm to compute the proposed simplification. The algorithm does not require to work with degrees of the disregarded variable and uses a symmetry of the Kauffman bracket presented by rows of the alternating Pascal’s triangle. Finally, we give a remark on an interpretation of the proposed invariant and algorithm in the case of classical knots and links.


Introduction
One of the main problems of the knot theory is to distinguish the objects under study. This approach involves the problem to construct and compute knot invariants and see if some of them are helpful in the considered particular situation. For example, the generalized Kauffman bracket polynomial turned out to be enough to distinguish all prime knots [1] and links [2] in the thickened torus having diagrams with at most 4 crossings.
In this paper, we present an invariant called the Kauffman bracket skeleton, which is a simplification of the generalized Kauffman bracket polynomial in two (a and x) variables. The idea is to take into account only the order and values of coefficients and disregard the degrees of the variable a. However, the proposed simplification is more compact, and at the same time is not weaker than the original generalized Kauffman bracket polynomial in the sense of, for example, tabulation of prime knots and links in the thickened torus up to complexity 4 inclusively. Also, we construct an algorithm to compute the proposed simplification in an easier way than the original generalized Kauffman bracket polynomial. Indeed, the algorithm does not require to work with degrees of the variable a and uses a symmetry of the Kauffman bracket presented by rows of the alternating Pascal's triangle.
The paper is organized as follows. Section 1 gives the necessary definitions and proves that the Kauffman bracket skeleton is invariant. The proposed algorithm to compute the Kauffman bracket skeleton is constructed in Section 2. A computational example is presented in Section 3, where we also give a remark on an interpretation of the proposed invariant and algorithm in the case of classical knots and links. In addition, in order to show that the proposed simplification is not weaker than the original generalized Kauffman bracket polynomial in the sense of tabulation of prime knots and links in the thickened torus up to complexity 4 inclusively, we present tables of the proposed invariant for the tabulated prime knots and links in the thickened torus.

Knot and Link in the Thickened Torus
Consider a two-dimensional torus T = S 1 × S 1 and an interval I = [0, 1]. By a thickened torus we mean a 3-dimensional manifold homeomorphic to the direct product T × I.
A smooth embedding of m circles in the Int(T × I) is called a link in T × I having m components and denoted by L ⊂ T × I. An one-component link in the thickened torus is called a knot in T × I and denoted by K ⊂ T × I.
As in the classical case, knots and links in the thickened torus can be given by their diagrams. A diagram D ⊂ T of a knot K ⊂ T × I or link L ⊂ T × I is defined by analogy with the diagram of the classical knot except that the knot (or link) is projected into the torus T instead of the plane. We represent the torus T as a square with identified opposite sides. An example of a knot diagram on torus is given in Fig. 5 on the left.

Generalized Kauffman Bracket Polynomial
Let us recall the definition of the generalized Kauffman bracket polynomial [1]. In contrast to the usual Kauffman bracket polynomial of classical knots [4] (see also [3] for the original version called the Jones polynomial), the generalized version [1] takes into account types of curves on the torus (trivial, i.e. bounded a 2-disk, and nontrivial). The papers [5] and [6] give other generalizations of the Kauffman bracket polynomial.
Let D be a diagram of a knot or link in the thickened torus. Endow each angle of each crossing of D with a marker A or B according to the rule given in the center of Fig. 1. Each state s of the diagram D is defined by a combination of ways to smooth each crossing of D such as to join together either two angles endowed with a marker A, or two angles endowed with a marker B, see Fig. 1 on the left and right, respectively. Obviously, if the diagram D has n crossings, then there exist exactly 2 n states of D. where ε(i) is a sign of the i-th crossing of D defined by the rules given in Fig. 2. Note that the writhe of an oriented link diagram is the sum of signs of only those crossings of D that are self-intersections of the components. The exact formula of the generalized Kauffman bracket polynomial [1] is as follows: where is the generalized Kauffman bracket [1]. Here α(s) and β(s) are the numbers of markers A and B in the given state s, while γ(s), δ(s) are the numbers of trivial and nontrivial curves in the torus obtained by smoothing of all crossings according to the state s, and w(D) is the writhe of D. The sum is taken over all 2 n states of D.
Recall that the generalized Kauffman bracket polynomial X (a, x) D is invariant under 1) isotopy (i.e., under all three Reidemeister moves Ω 1 -Ω 3 given in Fig. 3), 2) orientation preserving automorphisms of the torus, 3) orientation reversing automorphism of the torus (up to change of variables a n → a −n ), 4) simultaneous switching of all crossings (up to change of variables a n → a −n ). In most applications (for example, in tabulation), the orientation of knots and links is not of interest. Therefore, above we define the generalized Kauffman bracket polynomial (1) to be an invariant of nonoriented knots and links, i.e. an orientation is used as an auxiliary tool only. Indeed, the generalized Kauffman bracket a, x D is invariant under the same transformations (1) -(4) as the generalized Kauffman bracket polynomial X (a, x) D with the exclusion of the first Reidemeister move Ω 1 . The invariance under Ω 1 is achieved by the factor (−a) −3w(D) taking into account the writhe w(D) determined with the use of the orientation.

Kauffman Bracket Skeleton
Let us define the Kauffman bracket skeleton as a simplification of the generalized Let t m be a tuple composed of nonzero coefficients b jm of the polynomial P m (a), which are ordered in increasing order of j. Note that transition from P m (a) to t m erases information on degrees of the variable a and remains information on order and values of the coefficients b jm . For example, Laurent polynomials P m 1 (a) = a −10 + 5a 10 and P m 2 (a) = 1 + 5a 2 correspond to the same tuple t m 1 = t m 2 = (1, 5).
The formal sum is called the Kauffman bracket skeleton, and tuples t m are called coefficients of the Kauffman bracket skeleton. is completely defined by the Kauffman bracket, which is invariant under these transformations.
Each Ω 1 leads to multiplication of the Kauffman bracket by (−a) ±3 . Therefore, the Kauffman bracket skeleton is multiplied by −1.
Each simultaneous switching of all crossings leads to change of variables a n → a −n in the Kauffman bracket, while degrees of x remains the same. Therefore, the Kauffman bracket skeleton is replaced by the inverted one. The same is true for orientation reversing automorphism of the torus.
The Kauffman bracket skeleton considered up to inversion and multiplication by −1 is an invariant of knots and links in the thickened torus.

Alternating Pascal's Triangle
Recall that Pascal's Triangle is a triangle array of integer numbers constructed as follows.
In the 0-th row (i.e., the topmost row), there is the unique nonzero entry, which is equal to 1 and is situated on the axis of symmetry of Pascal's triangle, while all others entries of the 0-th row are equal to 0.
Let i ∈ {1, 2, ...} be a number of a row, j ∈ {0, 1, 2, ...} be an order number (from left to right) of a nonzero entry in a row, and e i,j be a nonzero entry having the j-th order number in the i-th row. Each entry e i,j is computed as a sum of two entries of the (i−1)-th row, which are situated on the left and right of the desired entry e i,j : e i,j = e i−1,j−1 + e i−1,j+1 , under the assumption that empty entries are equal to 0. In particular, the first and the last number of each row is equal to 1, which is obtained as a sum of 0 and 1. As an example, the first 5 rows of Pascal's triangle are given in Fig. 4(a).
One of possible interpretations of Pascal's triangle states that Pascal's triangle determines binomial coefficients which arise in binomial expansion. Indeed, if a binomial of the form (x + y) is raised to a positive integer power i, then the binomial coefficients e i,j of the expansion coincide with the entries e i,j of the i-th row of Pascal's triangle taking in the same order.
In this paper, we use an alternating version of Pascal's triangle that corresponds to the case when both x and y are negative. Namely, according to (2), we consider In this case, in contrast to Pascal's triangle, each entire of the i-th row is endowed with a sign obtained by the rule (−1) i , where i ∈ {0, 1, 2, ...} is the number of the row. As an example, the first 5 rows of the alternating Pascal's triangle are given in Fig. 4(b).

Algorithm to Compute the Kauffman Bracket Skeleton
Let D be a diagram of a knot or link in the thickened torus having n crossings. In order to compute the Kauffman bracket skeleton S D without computation of the generalized Kauffman bracket polynomial X (a, x) D , we propose the following algorithm. Note that an algorithm to compute the generalized Kauffman bracket polynomial X (a, x) D can be presented by STEPS 1-4 and additional steps on computation of the writhe and sum (1). STEP 1. Enumerate crossings of D. STEP 2. Endow each angle of each crossing of D with a marker A or B according to the rule given in the center of Fig. 1.
STEP 3. Determine all 2 n possible states of the diagram D, where each state is defined by a combination of ways to smooth each crossing of D such as to join together either two angles endowed with a marker A, or two angles endowed with a marker B, see Fig. 1 on the left and right, respectively. STEP 4. For each state s, determine the numbers α(s), γ(s) and δ(s) of markers A in the code of the state s, trivial and nontrivial curves in the torus obtained by smoothing of all crossings of the diagram D according to the state s, respectively. STEP 5. Let {F ℓ (x)} be an infinite sequence of the polynomials F ℓ (x), where ℓ is an ordered number of the polynomial in the sequence {F ℓ (x)} and x is a variable. At present, consider all the polynomials F ℓ (x) to be zero. Further, for each state s, set k = α(s), i.e. k ∈ {0, 1, 2, ..., n}, and redetermine the polynomials F ℓ (x) as follows.
1. If γ(s) = 0, then add the term x δ(s) to the polynomial F k (x).
3. If γ(s) = 2, then add the term x δ(s) to the polynomials F k−2 (x) and F k+2 (x), and add the term 2 · x δ(s) to the polynomial F k (x).

Computational Example
We use the algorithm proposed in the previous section in order to compute the Kauffman bracket skeleton S 3 2 for the knot diagram 3 2 given in Fig. 5 on the left. The results of STEPS 1 -3 are given in Fig. 5, while the results of STEPS 4 -5 are given in Tables 1 and 2 Table 1 Results of STEPS 4 -5 of the algorithm for the knot diagram 3 2

Numbers for state
The terms added to Therefore, we obtain the values of nonzero polynomials in the sequence {F ℓ (x)} given in Table 2.
It turned out that the proposed simplification is not weaker than the original generalized Kauffman bracket polynomial in the sense of tabulation of prime knots and links in the thickened torus up to complexity 4. Namely, the Kauffman bracket skeleton, as well as the original generalized Kauffman bracket polynomial, turned out to be enough to distinguish all 21 prime knots [1] and 26 prime links [2] in the thickened torus having diagrams with at most 4 crossings, see Tables 3 and 4. Note the pair of equivalent links (4 10 and 4 11 ) given in the paper [2]. In order to transform 4 10 to 4 11 , it is enough to slide the horizontal component in the diagram of 4 10 out the top of the square in the bottom, then rotate the square by π, and switch all the crossings, see Fig. 6. Obviously, the Kauffman bracket skeletons of these links are inverted (i.e., equivalent) to each other, see Table 4.  Table 3 The Kauffman bracket skeletons of prime knots in the thickened torus given in [1]