Journal of Knot Theory and Its Ramifications

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

In this paper, we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy. Given a knot with unknown quasipositivity, we use these predictions to identify braid representatives that are likely to be quasipositive, which we then subject to further testing to verify. Using these techniques, we identify 84 new quasipositive 11 and 12-crossing knots. Furthermore, we show that neural networks are also able to predict and help compute the slice genus and Ozsváth-Szabó $\tau$-invariant of knots.

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot $K$, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of $K$.

A flat virtual link is a finite collection of oriented closed curves $𝔏$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersection points of curves in the virtual homotopy class of $(L_1,L_2)$ equals to the number of terms of a generalization of the Anderson–Mattes–Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol’d [Lagrange and Legendre cobordisms. I, Funkt. Anal. Prilozhen. 14(3) (1980) 1–13, 96.] — consisting of big Legendrian submanifolds between two smaller ones. We are interested in such cobordisms which fit with generating functions, and wonder which structures and obstructions come with this notion. As a central result, we show that the classes of Legendrian concordances with respect to the generating function equipment can be given a group structure. To this construction, we add one of a homotopy with respect to generating functions.

The multivariate Alexander module of a link $L$ has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of $L$.

Region crossing change (RCC) is an operation on link diagrams in which all crossings incident to a selected region are changed. Two diagrams are called RCC-equivalent if one can be transformed to the other via a sequence of RCCs. RCC is an unknotting operation but not an unlinking operation. A set of regions of a diagram is called ineffective if RCCs at every region in that set have no net effect on the crossings of the diagram. The main result of this paper is a construction of the complete collection of ineffective sets for any link diagram. This involves a combination of linear algebraic and diagrammatic techniques including a generalization of checkerboard shading called tricoloring. Using this construction of ineffective sets, we provide sharp upper bounds on the maximum number of RCCs needed to transform between RCC-equivalent knot diagrams and reduced 2- and 3-component link diagrams with fixed underlying projections.

We prove that the knots $13n_{592}$ and $15n_{41,127}$ both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.

The goal of this paper is to tabulate all prime links in the thickened torus $T\times I$ having diagrams having crossing number 5. First, we construct a table of prime projections of links on the torus $T$ having exactly 5 crossings. To this end, we enumerate abstract quadrivalent graphs of special type and consider all possible embeddings of the graphs into the torus $T$ in order to construct prime projections. Then, we prove that all obtained projections are inequivalent. Second, we use the list of prime projections to construct a table of diagrams of prime links in the torus $T$. In order to prove that all those links are inequivalent, we use two modifications of the Kauffman bracket. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed tables.

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let $T(p,q)$ be any torus knot with $p$ even and $q$ odd. The difference between these two invariants on $T(p,q)$ is at least $\frac{k}{2}$, where $p=qk+a$ and $0<a<q$ and $k\ge 0$. Hence, the difference between the two invariants on torus knots $T(p,q)$ grows arbitrarily large for any fixed odd $q$, as $p$ ranges over values of a fixed congruence class modulo $q$. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot $T(p,q)$ is $\frac{1}{2}(p-1)(q-1)$, and Kronheimer and Mrowka later proved that the orientable four genus of $T(p,q)$ is also this same value.