Journal of Knot Theory and Its Ramifications

We prove that the Gaussian parity on free two-dimensional knots is universal.

In this paper, we introduce the Goeritz matrix for a pseudo-link whose entries lie in the Laurent polynomial ring $\mathbb{Z}[u^{-1},u]$, which generalizes the Goeritz matrix for a classical link. We show that the determinant of a modified Goeritz matrix gives a Laurent polynomial invariant for pseudo-links in one variable u with integer coefficients. We also introduce the notions of the signature, determinant, and nullity of pseudo-links. Further, we discuss some properties of the invariants and compute the polynomials and those numerical invariants for several pseudo-knot families.

We extend the classical coloring rule to compute the tangle (fraction) number of a virtual rational tangle. We use the contemporary argument used by Kauffman [Virtual knot theory, Eur. J. Comb. 20(7) (1999) 662–690] and show that tangle (fraction) number is a complete invariant. The main purpose of this paper is to provide an alternate yet elementary proof of John Conway’s classification theorem for the virtual rational tangles.

In this paper, we study the concordance and cobordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.

Previous work [C. Shonkwiler, New computations of the superbridge index, J. Knot Theory Ramifications 29(14) (2020) 2050096] used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. This work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.

The purpose of this paper is to present a new periodicity criterion. For that purpose, we study the HOMFLY-PT polynomial and the Kauffman polynomial of cables of periodic links. Furthermore, we exhibit a couple of examples for which our criterion is stronger than many previously known criteria, among which is the Khovanov homology criterion of Borodzik and the second author.

In this paper, we compute the twisted Alexander invariant of the braid group associated with the Tong–Yang–Ma representation.

Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye’s paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing bars. By adjusting the normalized arrow polynomial of virtual links, we generalize it to twisted links. We show that it is an invariant for twisted link. Finally, we figure out three characteristics of the normalized arrow polynomial of a checkerboard colorable twisted link, which is a tool of detecting checkerboard colorability of a twisted link. The latter two characteristics are the same as in the case of checkerboard colorable virtual link diagram.