Jones polynomials:

3        1     4     1 0 1 -1
5        2     7     1 0 1 -1 1 -1
7        3    10     1 0 1 -1 1 -1 1 -1
9        4    13     1 0 1 -1 1 -1 1 -1 1 -1
11       5     16     1 0 1 -1 1 -1 1 -1 1 -1 1 -1
13        6     19     1 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
15        7     22     1 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
17        8     25     1 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
19        9     28     1 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1

Symmetry group: D1

Symmetry type: chiral, invertible.

Signature: 2k

Unknotting number: k



The concept of new knot tables based on knot families can be naturally extended to links. Working with the program LinKnot we succeeded to extend new knot tables to all KLs with n
9 crossings. Tables given in the Appendix A contain generating KLs for every family of KLs, the Conway symbol of the family with conditions for parameters, the number of components, the Alexander polynomial given by general formula, general formulas for the number of different projections (only for alternating KLs), general formulas for unknotting (unlinking) numbers and signatures, data about period(s) of KLs, achirality, and unlinking gap.

Because the complete concept of new KL tables is based on the notion of generating KLs and families originating from them, one of the possible future aims can be a search for new KL invariants that will be the invariants of families, and not only of particular KLs. In the new invariants, to the Reidemeister moves we need to add a change of the length of bigon-chains, i.e., n-moves. By increasing (decreasing) the length of bigon chains in Conway symbols of KLs we obtain KLs belonging to the same family. This transformation makes a transition from one family member to another possible. Unfortunately, this works only with minimal canonical Conway symbols. This is the way that we have obtained KL tables given at the end of the book, that consist only of generating KLs, families derived from them, and parametric data about families (i.e., the KL properties and invariants in a general form). If we are able to recognize the family to which a certain alternating KL belongs, then the Alexander polynomial will itself be sufficient for the recognition of particular alternating KLs, because in every family of alternating KLs there are no KLs with the same Alexander polynomials. Moreover, if we are using the Conway notation for alternating KLs, the minimal Dowker code obtained directly from the Conway symbols by the LinKnot function MinDowAltKL (webMathematica MinDowAltKL) is sufficient for comparing alternating KLs given by their Conway symbols. This function gives minimal Dowker code for all alternating KLs, except for those derived from basic polyhedra permitting flypes. In the case of such basic polyhedra, the minimization is correct with the regard to a fixed basic polyhedron, but not with the regard to all its transforms obtained from it by flypes.

From the results obtained, we strongly believe that all properties of KLs belonging to particular families are well-ordered, and that is possible to extend the particular results and write them in a general form. This works for Alexander polynomials, Jones polynomials, symmetry properties, signatures, unknotting numbers, braid family representatives, and Dowker codes.

One of the main and the most intriguing open questions is Bernhard-Jablan Conjecture. For all KLs that we tested, the unlinking numbers u(L) obtained according to it coincide with the exact unlinking numbers determined using other methods.

The next interesting question is a possibility of establishing connections between coefficients of polynomial invariants and other KL invariants and understanding the topological meaning of certain coefficients. The most interesting conclusion indicated by the results obtained is that unknotting (unlinking) number and signature are linear combinations of numbers participating in the Conway symbol of a KL family.



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