Journal of Knot Theory and Its Ramifications

Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.

Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica implementation. We discuss other instances of successful “gliding ideas” in the literature — sometimes in disguise — such as the Drinfel’d double construction, Enriquez’s work on quantization of Lie bialgebras, and Audoux and Meilhan’s classification of welded homotopy links.

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary.

We show which crossingless connections between $2(m+n)$ outer boundary points of an annulus can be realized as Kauffman states of the B-type lattice crossing $L_{\mathrm{\text{B}}}(m,n)$. Furthermore, we give a closed-form formula for the number of realizable B-type Catalan states, and we find coefficients of those obtained as Kauffman states of $L_{\mathrm{\text{B}}}(m,1)$ and $L_{\mathrm{\text{B}}}(m,2)$.

Let $r$ be an odd integer, $r\ge 3$. Then the petal number of the torus knot of type $(r,r+2)$ is equal to $2r+3$.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.