Related papers: Parity and Relative Parity in Knot Theory
This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.
We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some…
Quantum walks provide a natural framework to approach graph problems with quantum computers, exhibiting speedups over their classical counterparts for tasks such as the search for marked nodes or the prediction of missing links.…
We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the…
Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all…
We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability…
We introduce the non-self OU sequence and the OU number for link diagrams. Using these, we give a lower bound for the number of necessary Reidemeister moves of type III between two diagrams of the same link.
This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We…
Virtual knot theory has experienced a lot of nice features that did not appear in classical knot theory, e.g., parity and picture-valued invariants. In the present paper we use virtual knot theory effects to construct new representations of…
We fix the notion of parity complex by a judicious selection from among the axioms originally considered by Street. We show that parity complexes so defined, together with the morphisms of parity complexes defined by Verity, form a category…
Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an…
Based on a recently introduced by the author notion of {\em parity}, in the present paper we construct a sequence of invariants (indexed by natural numbers $m$) of long virtual knots, valued in certain simply-defined group ${\tilde G}_{m}$…
In [8], K. Kaur, S. Kamada et al. posed a problem of finding a virtual knot, if exists, with an unknotting index (n,m), where (n,m) is a pair of non-negative integers. In this paper, we address this question by providing infinite families…
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a…
We examine spaces of connected tri-/univalent graphs subject to local relations which are motivated by the theory of Vassiliev invariants. It is shown that the behaviour of ladder-like subgraphs is strongly related to the parity of the…
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 suggest a categorification procedure for the SO(2N) one-variable specialization of the two-variable Kauffman polynomial. The construction has many similarities with the HOMFLYPT categorification: a planar graph formula for the polynomial…
A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $\Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface. AC links provide a useful window for…
A linking pairing is a symetric bilinear pairing lambda: GxG --> Q/Z on a finite abelian group. The set of isomorphism classes of linking pairings is a non-cancellative monoid E under orthogonal sum, which is infinitely generated and…
We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard,…