Journal of Knot Theory and Its Ramifications

Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In this paper, we discuss the rather complicated extension of Joyce’s observation to several different medial quandles and reduced (one-variable) Alexander modules associated with classical links. The theme is that for links, medial quandles provide stronger invariants than reduced Alexander modules.

The Cabling Conjecture states that surgery on hyperbolic knots in $S^3$ never produces reducible manifolds. In contrast, there do exist hyperbolic knots in some lens spaces with non-prime surgeries. Baker constructed a family of such hyperbolic knots and posed a conjecture that his examples encompass all hyperbolic knots in lens spaces with non-prime surgeries. Using the idea of seiferters we construct a counterexample to this conjecture. In the process of construction, we also derive an obstruction for a small Seifert fibered space to be obtainable by a surgery with a seiferter.

We define an annular concordance invariant and study its properties. When specialized to braids, this invariant gives bounds on band rank. We introduce a modified chain complex to reformulate the invariant. Then, by focusing on a special case, we give a refinement of the transverse invariant $\widehat{𝜃}$. We also study the relationship of this invariant with transverse and braid monodromy properties.

We discuss when homogeneous quasipositive links are positive. We show that a homogeneous diagram of a quasipositive link whose number of Seifert circles is equal to the braid index is a positive diagram, and show that the truth of Murasugi–Przytycki’s conjecture on braid index of alternating links implies that alternating quasipositive links are positive.

We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.

For classical knots, $\mathrm{\Delta }$-moves and crossing changes can both unknot any knot. But many virtual knots cannot be unknotted with these moves. Moreover, there are many virtual knots that can be unknotted by crossing changes but cannot be unknotted by $\mathrm{\Delta }$-moves. We show that the derivatives of the Jones polynomial can detect whether a virtual knot can be unknotted by $\mathrm{\Delta }$-moves.

Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2 and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak’s work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these 12 sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a minimal generating set of oriented Reidemeister-type moves for spatial trivalent graph diagrams contains 10 moves.

Let $UV{B}_n$ and $UV{P}_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands, respectively. We study the groups $UV{B}_n$ and $UV{P}_n$, and our main results are as follows: for $n\ge 5$, we give a complete description, up to conjugation, to all possible homomorphisms from $UV{B}_n$ to the symmetric group $S_n$. For $n\ge 3$, we characterize all possible images of $UV{B}_n$, under a group homomorphism, to any finite group $G$. For $n\ge 5$, we prove that $UV{P}_n$ is a characteristic subgroup of $UV{B}_n$. In addition, we determine the automorphism group of $UV{P}_n$ and we prove that ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ is a subgroup of the outer automorphism group of $UV{B}_n$. Lastly, we show that $UV{B}_n$ and $UV{P}_n$ are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group $W{B}_n$ and we discuss about its automorphism group.

The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent strands $s$ full twists. In this paper, we show that the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(max\{p,q,r\}+|s|r)$ where $p$ and $q$ are positive. Furthermore, if $r\le p-q$, then the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(p+(\left|s\right|-1)r)$.