Journal of Knot Theory and Its Ramifications

We give a new infinite family of non-fibered 2-bridge knots whose knot groups are bi-orderable.

We study the representations of the braid group in a generalization of the Temperley–Lieb algebra. This algebra was introduced by Kloster and contains the diagrams of the TL algebra with cut strands. We address the question of existence of Markov traces on these representations and make some observations on the possibility of their Baxterizations.

It is known that every knot bounds a singular disk with only clasp singularities, which is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. It is known that the Conway polynomials of knots with clasp number at most two are characterized. In this paper, we focus on the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials, which is called the $\mathrm{\Gamma }$-polynomial. As a result, we characterize the $\mathrm{\Gamma }$-polynomials of knots with clasp number at most two. Moreover, it is known that there exist two homeomorphic classes of clasp disks with two clasp singularities, which are called types $0$ and $1$. We show that there exist infinitely many knots which bound clasp disks of not doubled knots but both types $0$ and $1$, not type $0$ but type $1$, not type $1$ but type $0$, respectively.

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

In this paper, we calculate the Kauffman polynomials $F(K(p,q);a,z)$ of Kanenobu knots $K(p,q)$ with $p,q$ half twists and determine their spans on the variable $a$ completely. As an application, we determine the arc index of infinitely many Kanenobu knots. In particular, we give sharper lower bounds of the arc index of $K(2n,-2n)$ by using canonical cabling algorithm and the 2-cable $\mathrm{\Gamma }$-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

Polyak proved that the set $\{\mathrm{\Omega }1a,\mathrm{\Omega }1b,\mathrm{\Omega }2a,\mathrm{\Omega }3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this paper, we prove that the set of eight directed Polyak moves $\{\mathrm{\Omega }{1a}^{\uparrow },\mathrm{\Omega }{1a}^{\downarrow },\mathrm{\Omega }{1b}^{\uparrow },\mathrm{\Omega }{1b}^{\downarrow },\mathrm{\Omega }{2a}^{\uparrow },\mathrm{\Omega }{2a}^{\downarrow },\mathrm{\Omega }{3a}^{\uparrow },\mathrm{\Omega }{3a}^{\downarrow }\}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least two components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\mathrm{\Omega }3$.

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.

For a link with zero determinants, a $\mathbb{Z}$-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial $\mathbb{Z}$-coloring a $\mathbb{Z}$-colorable link. The minimal coloring number of a $\mathbb{Z}$-colorable link is the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on diagrams of the link. We give sufficient conditions for non-splittable $\mathbb{Z}$-colorable links to have the least minimal coloring number.

In this paper, we give sufficient conditions on group presentations, with two generators and one relator, in order to be the group of a virtual knot diagram. Although those conditions are not enough, we use them to determine, completely, whether or not a Baumslag–Solitar group is the group of a two-bridge virtual knot. Moreover, we present a combinatorial proof of the fact that these groups are not two-bridge classical knot groups.

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to $6$ in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in ${ℝ\mathbb{P}}^3$ and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point.

We introduce the odd index polynomial and the odd arrow polynomial for virtual links which are different from the original index polynomial and arrow polynomial.

Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct an obstruction to cosmetic crossings in classical links.

We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley–Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author’s previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov’s tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov’s conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in $\mathrm{\Sigma }\times [-1,1]$ where $\mathrm{\Sigma }$ is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov’s approach to the Jones polynomial.