Related papers: On computational aspects of two classical knot inv…
We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…
In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write…
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
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…
We introduce two new families of polynomial invariants of oriented classical and virtual knots and links defined as decategorfications of the quandle coloring quiver. We provide examples to illustrate the computation of the invariants, show…
We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant.…
We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the…
We enhance the quandle counting invariants of oriented classical and virtual knots and links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a…
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…
We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the…
We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.
The classical knot recognition problem is the problem of determining whether the virtual knot represented by a given diagram is classical. We prove that this problem is in NP, and we give an exponential time algorithm for the problem.
The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which…
This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from…
We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a…
We enhance the psyquandle counting invariant for singular knots and pseudoknots using quivers analogously to quandle coloring quivers. This enables us to extend the in-degree polynomial invariants from quandle coloring quiver theory to the…
We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used…
Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin…
We show that every periodic virtual knot can be realized as the closure of a periodic virtual braid and use this to study the Alexander invariants of periodic virtual knots. If $K$ is a $q$-periodic and almost classical knot, we show that…