arXiv research

New minimal link diagrams found, including torus links and homogeneous ones.

Proves minimal crossing diagrams for specific spatial graphs.

Minimal grid diagrams for 12-crossing prime knots identified.

In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof al…

The paper explores minimal coloring numbers for $\mathbb{Z}$-colorable links.

Study on knot diagrams showing bridge number can differ from crossing number.

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …

Study finds minimal coloring numbers for torus links using rack colorings.

This paper finds all prime alternating knots with minimal warping degree two.

Minimal moves for surfaces in 4D identified.

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

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 …

The paper simplifies knot and link diagrams with triple-crossings.

Minimal sets of moves for rotational Reidemeister diagrams are identified.

Minimal grid diagrams found for 13-crossing prime knots.

We prove that a virtual link diagrams satisfying two conditions on the Khovanov homology is minimal, that is, there is no virtual diagram representing the same link with smaller number of crossings. This approach works for both classical and virtual links

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

Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result a…

This paper constructs explicit trisection diagrams for elliptic surfaces.

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

Proves Yoshikawa eighth move's independence and finds minimal band moves.

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

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

We characterize planar diagrams which may be divided into n arc embeddings in terms of their chord diagrams, generalizing a result of Taniyama for the case n = 2. Two algorithms are provided, one which finds a minimal arc embedding (in quadradic time in the number of crossings), and one which constructs a minimal subdi…

Given a knot diagram $D$, we construct a semi-threading circle for it which can be an axis of $D$ as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles for minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this n…

Study essential diagrams of knots in $S_{g} imes S^{1}$ and their relation to virtual knots.

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms and knots.

A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link diagram on an oriented 2-dimensional surface.) We prove that if a minimal genus …

A link diagram is said to be lune-free if, when viewed as a 4-regular plane graph it does not have multiple edges between any pair of nodes. We prove that any colored link diagram is equivalent to a colored lune-free diagram with the same number of colors. Thus any colored link diagram with a minimum number of colors (…

A Heegaard diagram for a 3-manifold M is a closed, oriented surface S together with a pair (X, Y) of compact 1-manifolds in S whose components serve as attaching curves for the 2-handles of the two sides of a Heegaard splitting for M. The diagram is positive if X and Y can be oriented so that the intersection number <X…

Proves generalized meander conjectures for knots and spatial graphs.

Proves a theorem for surface links, simplifying diagrams of links.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

Two non-diffeomorphic minimal genus trisections found for a 4-manifold.

For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link.…

Venn diagrams are a graphical way to represent a set system. Each of the n sets is represented by a simple closed curve. The n curves subdivide the plane into 2^n open connected regions, each of which represents the intersection of its containing curves' sets. For example, two overlapping circles can divide the plane i…

Study of knots with at most one Hopf crossing number.

The paper explores parity in knotoids and virtual knots, proving a conjecture and introducing a new polynomial.

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus o…

Polynomials derived from Heegaard diagrams for 3-manifolds.

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots…

Kernelized Taylor diagram visualizes data populations with fewer assumptions.

Characterizes knots with warping sum ≤ 3 and provides some knots with warping sum 4.

Links with minimum tunnel number have one less component than their number of parts.

Study minimum ribbonlength of immersed flat knots and links.

In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a mi…

For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the nu…