相关论文: Intrinsic Linking and Knotting in Virtual Spatial …
Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that…
Two geometric spaces are in the same topological class if they are related by certain geometric deformations. We propose machine learning methods that automate learning of topological invariance and apply it in the context of knot theory,…
We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends…
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the…
We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.
We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded…
The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded…
The first and last named authors have demonstrated the existence of knots for which every integral slope is non-characterizing. In this short note, we extend this result in two ways. There exists a knot that shares for every integer n the…
Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves,…
We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes…
Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data…
The unknotting number is the classical invariant of a knot. However, its determination is difficult in general. To obtain the unknotting number from definition one has to investigate all possible diagrams of the knot. We tried to show the…
We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…
We find the minimal number of links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for…
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…
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower…
Piecewise-linear virtual knots are discussed and classified up to edge index six.
Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the…
A $2k$-move is a local deformation adding or removing $2k$ half-twists. We show that if two virtual knots are related by a finite sequence of $2k$-moves, then their $n$-writhes are congruent modulo $k$ for any nonzero integer $n$, and their…
In this paper we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain…