Journal of Knot Theory and Its Ramifications

In this paper, we study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an $S$-equivalent invariant, and give a criterion to a relation between the two pairings. We also observe that the pairings of some knots are equivalent, and that the pairings of other knots are not equivalent.

In this paper, we give a necessary condition for two virtual knots to be related by a finite sequence of twist moves by using the affine index polynomial, which is a Vassiliev invariant of degree $1$. Trapp showed that a numerical Vassiliev invariant of degree $n$ has a polynomial growth of degree $\le n$ on a twist sequence of knots, which can be extended to a twist sequence of virtual knots.

We calculate the growth of the affine index polynomial for a twist sequence of virtual knots and find the difference of the affine index polynomials of two virtual knots, which are related by a twist move. Moreover, we give a lower bound for the number of twist moves needed to transform $K$ to $K^{\prime }$ if $K$ and $K^{\prime }$ are virtual knots related by a finite sequence of twist moves.

We compute the involutive knot invariants for pretzel knots of the form P(−2,m,n) for m and n odd and greater than or equal to 3.

Satoh defined a map from virtual links to ribbon surfaces embedded in $S^4$. Herein, we generalize this map to virtual $m$-links, and use this to construct generalizations of welded and extended welded knots to higher dimensions. This also allows us to construct two new geometric pictures of virtual $m$-links, including 1-links.

We show that under a precise condition on the single variable Alexander polynomial, the limit at one of the Tristram–Levine signature of a link is determined by the linking matrix.

Let $S_g$ be a closed and oriented surface of genus $g\ge 2$. A closed curve $\gamma$ on $S_g$ is said to fill $S_g$ (or simply be filling), if its complement in the surface is a disjoint union of topological discs. It is assumed that the curve $\gamma$ is always in minimal position. To a filling curve, we associate a number $b$, the number of topological discs in its complement. For $b=1$, such a filling curve is called minimally intersecting. We prove that for every $b\ge 1,$ there exists a filling curve ${\gamma }_b$ on $S_g$ whose complement is a disjoint union of $b$ many topological discs. Furthermore, we provide an upper bound on the number of mapping class group orbits of closed curves which fills $S_g$ minimally.

The Biquandle Bracket is a generalization of the Jones Polynomial. In this paper, we outline a Khovanov Homology-style construction which generalizes Khovanov Homology and attempts to categorify the Biquandle Bracket. The Biquandle Bracket is not always recoverable from our construction, so this is not a true categorification. However, this deficiency leads to a new invariant: a canonical biquandle 2-cocycle associated to a biquandle bracket.

Group object quandles are sets with compatible group and quandle operations. Their endomorphisms will themselves be group objects and function composition becomes a third compatible operation, giving the structure of a nearring, a ring-like structure lacking one distributive property. We give an orbit decomposition of group object quandles in terms of cosets of a certain subgroup of the additive group, and describe them as central extensions of this subgroup. We then describe and count the elements of the endomorphism nearring and study its ideal structure.