arXiv research

Proves minimal crossing diagrams for specific spatial graphs.

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams o…

Minimal grid diagrams for 12-crossing prime knots identified.

Minimal crossing virtual links have minimal supporting genus.

Study minimizes crossing points of up to 12 curves on a genus 2 surface.

Minimal grid diagrams found for 13-crossing prime knots.

Enumerates knots up to five crossings and describes moves between them.

Minimal crossing number found in arithmetic curve systems.

We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing …

In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By intro…

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classic…

This paper disproves a conjecture about knot projections under specific homotopy conditions.

Given a 2-crossing minimal chart $Γ$, a minimal chart with two crossings, set $α=\min\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$, and $β=\max\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$. In this paper we study the structure of a neighbourhood of $Γ_α\cupΓ_β$, and p…

The paper improves bounds on knot crossings and tabulates minimal diagrams.

Minimal grid diagrams found for 13-crossing prime knots with 13 arc index.

Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.

New methods find minimal crossing numbers for surfaces in $S^4$.

In this paper, we give definitions of three kinds of minimal charts, and we investigate properties of minimal charts and establish fundamental theorems characterizing minimal charts. To classify charts with two or three crossings we use the fundamental theorems. In the future paper, we give an numeration of the charts …

This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label $m$ of a chart $Γ$ we denote by $Γ_m$ the union of all the edges of label $m$ and their vertices. For a minimal chart $Γ$ with exactly two crossings, we can show that the two crossings are contained in $Γ_α\capΓ_β$ f…

In this paper I give estimates for the minimal crossing number, leading to a short proof that the crossing number is additive for torus links. These estimates are applied to several classes of links. Finally, I prove a part of a conjecture relating the HOMFLY polynomial and the Kauffman polynomial.

For every odd integer $c\ge 21$, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number $c$. The link has two components, and consists of an unknot and a knot which is $(-)$-amphicheiral with odd minimal crossing number. We call the latter knot a {\it Stoimenow knot}. W…

Positive braids minimize knot untangling steps.

Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.

We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.

We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$. We tackle this problem by the so-called…

Cross-entropy loss linked to metric learning, outperforming complex pairwise losses.

The study finds lower bounds for the warping degree of a knot projection.

Quantum machine learning uses quantum cross entropy to minimize loss, but measurement loss affects this process.

We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing…

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 is irreducibly odd, then it is minimal with respect to the number of classical …

PAC-Bayesian bounds for MLPs with cross entropy loss validated.

Connected sum affects crossing numbers of flat virtual knots.

Study shows surprising cobordism distances between certain torus knots.

New method detects changes by maximizing cross-entropy, outperforming existing techniques.

A graph G is called "minimalizable" if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called "strongly minimalizable". In this article, it is explained how minimalizability of a gr…

Optimal curves minimize crossings on surfaces.

In this work we establish the tightest lower bound up-to-date for the minimal crossing number of a satellite knot based on the minimal crossing number of the companion used to build the satellite. If $M$ is the wrapping number of the pattern knot, we essentially show that $c(Sat(P,C))>\frac{M^2}{2}c(C)$. The existence …

We study the minimal crossing number $c(K_{1}\# K_{2})$ of composite knots $K_{1}\# K_{2}$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve $θ_{K_{1},K_{2}}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n…

It is known that the maximal homological degree of the Khovanov homology of a knot gives a lower bound of the minimal positive crossing number of the knot. In this paper, we show that the maximal homological degree of the Khovanov homology of a cabling of a knot gives a lower bound of the minimal positive crossing numb…

The paper studies 4-charts with three crossings and their equivalence to a specific knot.

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.

It is shown that the diameter of the boundary slope set is bounded from above by the twice of the minimal crossing number for a Montesinos knot.

Cryptocurrency market becomes more cross-correlated over time.

Listed 19,513 prime knots with arc index 12-16.

2-twist trefoil has 6 crossings, proving non-trivial knotted surface.

The study calculates the average genus of rational knots and links.