Related papers: A Generalization of Pretzel Links via Spatial Grap…

In this paper we introduce the tied links, i.e. ordinary links provided with some ties between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called…

This paper is the second part of our comprehensive study on the braid index problem of pretzel links. Our ultimate goal is to completely determine the braid indices of all pretzel links, alternating or non alternating. In our approach, we…

The colored $\mathfrak{sl}_{3}$ Jones polynomial $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ are given by a link and an $(n_{1}, n_{2})$-irreducible representation of $\mathfrak{sl}_{3}$. In general, it is hard to calculate $J_{(n_{1},…

In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the…

Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results.…

In these notes we review the calculation of Jones polynomials using a matrix representation of the braid group and Temperley-Lieb algebra. The pseudounitary representation that we consider allows constructing ``states'' from the…

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also…

We give a general procedure that provides, given any particular pretzel link, a braid whose closure is the pretzel link. Moreover, we manage to give a specific braid word in terms of the entries of the pretzel link.

We give an alternate expansion of the colored Jones polynomial of pretzel links which recovers the degree formula in arXiv:1807.00957. As an application, we determine the degrees of the colored Jones polynomials of a new family of 3-tangle…

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As…

New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used…

We use planar 4-valent graphs and a graphical calculus involving such graphs to construct an invariant for balanced-oriented, knotted 4-valent graphs. Our invariant is an extension of the $sl(n)$ polynomial for classical knots and links. We…

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that…

Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.

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…

We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime…

In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore…

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel…

We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices…