Algorithms for Computing Some Invariants for Discrete Knots

Given a cubic knot K, there exists a projection of the Euclidean space onto a suitable plane such that p(K) is a knot diagram and it can be described in a discrete way as a cycle permutation. Using this fact, we develop an algorithm for computing some invariants for K: its fundamental group, the genus of its Seifert surface and its Jones polynomial. 3 : p   P 3  3 P  


Introduction
Considering the set consists of the lattice and all the straight lines parallel to the coordinate axis and passing through points in , we say that a knot 3 In [1] it was shown that any classical knot is isotopic to a cubic knot and by [2] we know that there exists a generic projection p of any cubic knot into a suitable plane.If we combine these two results, we have that p(K) is a diagram of K and it can be described in a discrete way as a cyclic permutation of points w w (with some restrictions).This allows us to develop an algorithm for computing the fundamental group of K, the genus of its Seifert surface and its Jones polynomial.

Discrete Knots and Some Invariants
Consider an oriented cubic knot K.In [2] it was proved that we can associate to K a unique sequence of points such that , i j , by a unit edge, n v is likewise joined to 1 by a unit edge, and the numbering of the i 's is compatible with the orientation of K. Henceforth, we will assume that all the coordinates of the points in K are positive.
An advantage of cubic knots is that there exists a canonical generic projection p (for details see [2]).In fact, let , where is the well-known transcendental number.Let P be the plane through the origin in orthogonal to N and consider the orthogonal projection .Then  be its projection into the plane P. Thus K is a polygonal curve contained in P with some self-intersections called inessential vertices or crossings.The crossings are not contained in     , where , list of points at P,  , , , , L w w , n w , an empty set I, the constant numbers A and B given above.for all do . We will determine which crosses "over" the other.Since, the line segments i and k are the image under the projection p of two segments whose endpoints are , k , respectively, we have that these both segments are parallel to two different canonical coordinate vectors.Let y , then we compare the first coordinate of the vectors i i and , i.e., we compare and a k .Thus, k , then we compare the second coordinate of the vectors i y k , and we have the same criteria of the previous case changing a by b.The last case z i = z k is analogous to the previous one.
Let c be a crossing point of the segment over the segment .Consider the vectors Thus we have two possible configurations: If det(M) > 0, we say that c is a positive crossing; If det(M) < 0, then c is a negative crossing.

Algorithm 2. Crossing criteria
Require: The list of indexes of intersection points , , , , , , and the list of points in , where and  , , , , r c  be an oriented discrete knot and 1 2 be its crossings.We will compute the fundamental group K denoted by , using the Wirtinger presentation (see [3,4]).We will start describing the set of generators of   1 K P (see [2]).Suppose that j c is the crossing point of the linear segment .Now we are going to rearrange the crossj c in such a way that 1 2 r .Let i be the segment of We know by Wirtinger presentation that there exists a bijection between the set of segments and the set of generators of , so the set of generators of   a r Again, by the Wirtinger presentation we know that for each .
, then we have the relation given by .

Seifert Surface
Given a knot K there exists an algorithm to construct its Seifert surface via an oriented diagram of it (for details see [3,4]).Roughly speaking, suppose that the corresponding diagram has r crossings, then the crossings are replaced by two disjoint arcs respecting the orientation.At the end, we obtain a collection of s simple closed curves called Seifert curves.We construct a Seifert surface F for K considering each Seifert curve as the boundary of a disk.The disks are connected at each crossing by a twisted band (so we need r bands).The genus of F is 1 2 s r   .The Seifert genus of a knot is the minimal genus possible for a Seifert surface of that knot.
Next, we apply the above algorithm to our case.As in the previous section, , , , r c c c  denotes an oriented discrete knot and are its crossings, where j c is as above.Let , , , In [2] it was defined the bijective map , given by if and ,  .This permutation can be expressed as a product of s disjoint cycles, where each cycle represents a Seifert curve.Hence we can compute the Seifert genus g.  

Jones Polynomial
The Jones polynomial is a very important invariant of an oriented knot K.We compute the Jones polynomial of a cubic knot K using the method described on "The knot atlas website" ( [5]) applied to our case.Let i k be pairs of indexes such that j k crosses over and up to rearrangement, we can assume that   , where the index n + 1 is equal to 1.
For each pair   , s s i k , consider the segments as , , cs and ds such that i s  C cs , i s + 1  C as , k s  C ds and k s + 1  C bs .Now we take the following expressions .Next we compute the writhe number denoted by w, which is equal to the number of positive crossings minus the number of negative crossings.
Finally, the Jones polynomial   J q is equal to , where q denotes a variable, q and w is the writhe number.
Create array Take curve segments , In this case, its fundamental group has 3 generators a1, a2, a3; and relations: a3a2 = a2a1, a1a3 = a3a2, a2a1 = a1a3.Its genus surface is one and its Jones polynomial is   q q q J q     .

Figure Eight Knot
Considering the figure eight knot as a cubic knot, in this case, we have 40 vertices.See Figure 3.
We now compute its fundamental group, its genus surface and its Jones polynomial.Thus, its fundamental group has 4 generators a1, a2, a3, a4; and relations: a2a4 = a1a2, a1a3 = a3a2, a4a2 = a3a4, a3a1 = a1a4.Its genus surface is one and its Jones polynomial is   2 1 1 q q q q J q the indexes s and l satisfy that j s k  g and j l

Algorithm 4 .
Seifert surface Require: The set or indexes  1 , , r  the genus of the knot .0 g  Create a function    where l is an index if and l B

1 .
s p such that l, m, n, p are the labels of the edges around k C   that crossing, starting from the incoming lower edge l and proceeding counterclockwise direction.Example:   , , , l m n p such that m is next to l in counterclockwise direction, n is next to m in counterclockwise direc tion, etc. Left-Handed Trefoil Knot Considering the left-handed trefoil knot as a cubic knot, see Figure 1, where you can see the corresponding vectors v i 's; .1, , 24 i   Now, we apply our program to compute its fundamental group, genus Seifert and Jones polynomial.See Fig- ure 2.
Notice that in the above expressions the order does not matter; for instance, the expressions [as, ds] and [ds, as] are equal.Now, we compute the formal product of all the above expressions to obtain a new expression Q.