Journal of Knot Theory and Its Ramifications

In this paper, we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism $A$ in such a category naturally splits as $A=B\circ C$ such that $C\circ B$ is an identity morphism.

We study how the genus, the simplicial volume and the $L^2$-Alexander invariant of Li and Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the $L^2$-Alexander invariant contains strictly more information than the pair (genus, simplicial volume). Along the way, we prove that the $L^2$-Alexander invariant detects the figure-eight knot $4_1$, the twist knot $5_2$ and an infinite family of cables on the figure-eight knot.

In the paper, we study the incoherent nullification number $\bar{n}(L)$ of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

Given a knot $K$, we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\ge 2$ these “generalized knot groups” determine $K$ up to reflection.

The second author has shown that for $n\ge 2$, the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues $S{K}_{a,b}=T_{a,b}\#T_{-a,b}$, $G{K}_{a,b}=T_{a,b}\#T_{a,b}$, constructed as connected sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\ge 2$ and $n$ satisfying a coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(S{K}_{a,b})$ and $G_n(G{K}_{a,b})$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\ge 2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group.

In this paper, we prove that for any odd prime $p$ greater than 3, the modular group representation associated to the $\mathrm{\text{SO}}{(p)}_2$-topological quantum field theory can be defined over the ring of integers of a cyclotomic field. We will provide explicit integral bases. In the last section, we will relate these representations to the Weil representations over finite fields.

It had been proved by Birman and Goldberg that the normal closure of the pure braid group $P_n(D)$ in the pure braid group of the torus $P_n(T)$ is the commutator subgroup $[P_n(T),P_n(T)]$. In this paper, we are going to study the case of full braid groups: i.e. the normal closure of $B_n(D)$ in $B_n(T)$, which turns out to have an interesting geometric description.