Journal of Knot Theory and Its Ramifications

We define the affine index polynomial of a flat virtual knot in a similar way as the case of a virtual knot, and show that it is described by the affine index polynomial of any overlying virtual knot. Let K be a virtual knot, and F the underlying flat virtual knot of K. Then we have necessary conditions for the invariant of F about invertibility and amphicheirality of K and F. As applications of the invariant, we raise examples such as (1) F is non-invertible, and (2) K is non-amphicheiral. We also give an alternative proof of a fact that Hrencecin and Kauffman's flat virtual knots are mutually distinct, which is originally proved by Im, Lee and Son.

We show that deleting an edge of a 3-cycle in an intrinsically knotted graph gives an intrinsically linked graph.

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms of linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple q-series for the tail of the knot 8₅. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.

We classify all groups which can occur as the orientation preserving topological symmetry group of some embedding of a Möbius ladder graph in S³.

In [A type D structure in Khovanov homology, preprint (2013), arXiv:1304.0463; A type A structure in Khovanov homology, preprint (2013), arXiv:1304.0465] the author constructed a package which describes how to decompose the Khovanov homology of a link ℒ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for ℒ is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the types A and D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to I. Petkova's decategorification of bordered Floer homology, [The decategorification of bordered Heegaard-Floer homology, preprint (2012), arXiv:1212.4529v1], and shows how they recover the Jones polynomial. We also use the decategorifications to compare this approach to tangle decompositions with M. Khovanov's from [A functor-valued invariant of tangles, Algebr. Geom. Topol.2 (2002) 665–741].

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space. Like torus knots and Berge knots, many known L-space knots admit a Seifert fibered L-space surgery. We give a concrete example of a hyperbolic L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.