Journal of Knot Theory and Its Ramifications

In this paper, we give a combinatorial description of the concordance invariant $𝜀$ defined by Hom, prove some properties of this invariant using grid homology techniques. We compute the value of $𝜀$ for $(p,q)$ torus knots and prove that $𝜀({𝔾}_{+})=1$ if ${𝔾}_{+}$ is a grid diagram for a positive braid. Furthermore, we show how $𝜀$ behaves under $(p,q)$-cabling of negative torus knots.

We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of “simpler” moves not increasing the complexity of the diagram along the way.

In this work, we compute the representation of the mapping class group of the sphere with $4$ punctures arising from the non-semi-simple TQFT [C. Blanchet, F. Costantino, N. Geer and B. Patureau, Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, Adv. Math. 301 (2016) 1–78]. We show that it is faithful. Lastly, we compare quantum representations of punctured spheres in general with those of braid groups.

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

Let $R^f=\mathbb{Z}[A^{\pm 1}]$ be the algebra of Laurent polynomials in the variable $A$ and let $R^a=\mathbb{Z}[A^{\pm 1},z_1,z_2,\dots ]$ be the algebra of Laurent polynomials in the variable $A$ and standard polynomials in the variables $z_1,z_2,\dots .$ For $n\ge 1$ we denote by ${\mathrm{\text{VB}}}_n$ the virtual braid group on $n$ strands. We define two towers of algebras ${\{{\mathrm{\text{VTL}}}_n(R^f)\}}_{n=1}^{\infty }$ and ${\{{\mathrm{\text{ATL}}}_n(R^a)\}}_{n=1}^{\infty }$ in terms of diagrams. For each $n\ge 1$ we determine presentations for both, ${\mathrm{\text{VTL}}}_n(R^f)$ and ${\mathrm{\text{ATL}}}_n(R^a)$. We determine sequences of homomorphisms ${\{{\rho }_n^f:R^f[{\mathrm{\text{VB}}}_n]\to {\mathrm{\text{VTL}}}_n(R^f)\}}_{n=1}^{\infty }$ and ${\{{\rho }_n^a:R^a[{\mathrm{\text{VB}}}_n]\to {\mathrm{\text{ATL}}}_n(R^a)\}}_{n=1}^{\infty }$, we determine Markov traces ${\{T_n^{\prime f}:{\mathrm{\text{VTL}}}_n(R^f)\to R^f\}}_{n=1}^{\infty }$ and ${\{T_n^{\prime a}:{\mathrm{\text{ATL}}}_n(R^a)\to R^a\}}_{n=1}^{\infty }$, and we show that the invariants for virtual links obtained from these Markov traces are the $f$-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each $n\ge 1,$ the standard Temperley–Lieb algebra ${\mathrm{\text{TL}}}_n$ embeds into both, ${\mathrm{\text{VTL}}}_n(R^f)$ and ${\mathrm{\text{ATL}}}_n(R^a)$, and that the restrictions to ${\{{\mathrm{\text{TL}}}_n\}}_{n=1}^{\infty }$ of the two Markov traces coincide.

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups).

Associated to an embedded surface in the three-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, whereform we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs. The main ingredients in the proof of the completeness include a generalization of the Kneser conjecture for three-manifolds with boundary proved here, and extensions of Waldhausen’s theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated.