arXiv research

New invariant measures complexity of 2-knots in 4D space.

Algorithm computes knot Floer complex for knots of thickness one.

Study algebraic obstructions to knot-like complex realizability.

New complexity measure for shake-slice knots established.

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing cha…

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and …

Study of knots and links in 2-complexes, defining linking numbers and polynomials.

Proves volume conjecture for twist knots using complex analysis.

We show that the genus problem for alternating knots with $n$ crossings has linear time complexity and is in Logspace$(n)$. Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. We show that the genus problem for these knots belongs to $TC^0$ circuit complexity c…

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…

New method generates optical vortices in any knot shape.

The paper constructs exotic knotted surfaces and curves in 4-manifolds.

The paper classifies knot Floer complexes of low width, simplifying knot bases.

Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invari…

New method speeds up knot computations in 3D.

The paper proves a linear diameter bound for hyperbolic knot complexes.

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

Automatic computation speeds up crosscap number calculation for alternating knots.

Investigates ropelength of complex knots and links.

Extends knot invariant to filtered grid complexes.

Computer experiments reveal complex knots that don't simplify.

This paper computes Kakimizu complexes for all 11 crossing prime alternating knots.

Proofs knot homology connected sums using grid complexes.

The paper determines the structure of Kakimizu complexes for genus one hyperbolic knots.

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in ${\sf NP}$ under the same assumption, and t…

New Garside structures found for torus knot groups and related braid groups.

Knots can be constructed and decomposed using Murasugi sums of Seifert surfaces.

Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.

Sharp bounds for Kirby-Thompson invariants of knotted surfaces computed.

We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yieldin…

Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he re…

Study shows crossing numbers for algebraic knots can differ by arbitrarily large amounts.

Formula for satellite operators using knot Floer homology.

New method computes knot Floer homology for satellite knots.

New estimate of semimeander complexity for knots with more than 10 crossings.

Formula calculates knot Floer complexes for specific cable knots.

Verifies knot conjecture for 24-crossing knots.

We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.

We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.

New bounds found for complexity of spun knots.

We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.

Tests using knot Floer homology detect prime knots with high accuracy.

The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with windin…

New subgroup found in knot homology concordance group.

New formula for dual knots using involutions.

Calculates knot $X$-torsion order using spectral sequences.

New methods compute Alexander polynomials for complex knots.