arXiv research

Analog of Kauffman bracket for non-orientable knots in thickened surface.

This paper calculates the non-orientable 4-genus for knots with 10 crossings.

J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…

Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.

Study on Legendrian knots and their non-orientable Lagrangian fillings.

New method finds non-orientable knotted surfaces in 4D.

New invariant distinguishes singular knots and links.

In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus $0$. We show that any Legendrian knot has a non-or…

Study non-orientable 4-genus for 11-crossing non-alternating knots.

Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.

Two complete knot invariants from diagrams, finite or infinite.

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

This study simplifies verification of invariants in oriented virtual knots.

Researchers refine the non-orientable $4$-genus of torus knots using Batson's surfaces.

Study on Whitehead doubles and their sliceness properties.

Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.

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

The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conject…

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

New rational band moves simplify knot classification.

Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.

Study on factorizations of knot polynomials for up to 12 crossings.

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of …

Study knots with large $\mathrm{SL}_2(\mathbb{C})$ character varieties.

Knots in 3-manifolds are equivalent if isotopic, except in special cases.

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev inva…

Ribbonness proven for slice knots, solving an old question.

Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.

In [Jo14] and [Jo18] Vaughan Jones introduced a construction which yields oriented knots and links from elements of the oriented Thompson group $\vec{F}$. In this paper we prove, by analogy with Alexander's classical theorem establishing that every knot or link can be represented as a closed braid, that given an orient…

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

Study extends knot genus results to two-component alternating links.

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 non-isomorphic families of such structures over non-abelian groups. We show that the set of c…

This paper extends danceability concept to twisted virtual knots.

Extends knot polynomial to knotted 4-valent graphs.

New invariant distinguishes non-orientable surfaces.

It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the spatial arc and the projection. In a separated paper, the knotting probability of an arc diagram is defined as an invariant under isomorphic …

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

Totally geodesic surfaces found in knots and links.

New knot polynomials distinguish knot orientations without using knot groups.

Study on quandle coloring quivers for (p, 2)-torus knots and links.

We introduce \textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle $2$-cocycle invariants…

Proof of orientable 3-manifolds parallelizability using knot theory.

The paper proves a theorem about shared Dehn surgeries between knots.

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…

Unified framework for 3D and 4D manifold and knot theory.

New knot invariants using biquandle fares.

We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in $Y$-oriented spatial trivalent graph diagrams representing $S^1$-oriented handlebody-k…