Related papers: The $F$-polynomial invariant for knotoids
In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants…
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are…
The aim of this paper is to introduce a polynomial invariant $f_K(t)$ for virtual knots. We show that $f_K(t)$ can be used to distinguish some virtual knot from its inverse and mirror image. The behavior of $f_K(t)$ under connected sum is…
In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic…
In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is…
A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By…
The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and…
The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…
This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way.…
In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface,…
This paper deals with the study of a new family of knot invariants: the $L^2$-Alexander invariant. A main result is to give a method of computation of the $L^2$-Alexander invariant of a knot complement using any presentation of default 1 of…
In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the…
We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister…
We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to…
F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a…
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…
Although most knots are nonalternating, modern research in knot theory seems to focus on alternating knots. We consider here nonalternating knots and their properties. Specifically, we show certain classes of knots have nontrivial Jones…
We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be…
This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…