arXiv research

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are dif…

We prove the existence of symmetric critical torus knots for O'Hara's knot energy family $E_α$, $α\in (2,3)$ using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least two smooth $E_α$-critical knots, which supports experimental observations using numerical …

Stability of knots at low regularity, and symmetric critical knots for Möbius energy.

Table of symmetric diagrams for knots up to 10 crossings.

Study shows not all ribbon knots can be symmetric unions.

A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a…

Study of symmetric unions of knots with new inequality and epimorphism results.

The article enumerates doubly symmetric diagrams for knots up to 18 crossings.

This paper investigates symmetric ribbon numbers of low-complexity knots.

New knots found with same determinant but no symmetric relation.

Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.

Two-bridge ribbon knots have symmetric union presentations.

Tanaka shows amphichiral symmetric unions of the unknot are trivial.

We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their kno…

A short proof for a theorem about composite knots.

The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.

Characterizes knot groups and symmetric quandles of surface-links.

Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.

We present the results of Axel Seeliger's tabulation of symmetric union presentations for ribbon knots with crossing numbers 11 and 12 and exhibit possible examples for ribbon knots which are not representable as symmetric unions. In addition, we give a complete atlas of band diagrams for prime ribbon knots with 11 and…

We give an alternative proof of that a critical knot of a Morse-Bott function $f: S^3 \rightarrow \mathbb{R}$ is a graph knot where the critical set of $f$ is a link in $S^3$. Our proof inducts on the number of index-1 critical knots of $f$.

Study identifies prime strongly positive amphicheiral knots with double symmetry.

We prove that all 2-bridge ribbon knots are symmetric unions.

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of un…

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $…

Symmetric elastic knots are found for certain classes with dihedral symmetry.

Extended symmetric union with multiple tangle regions and Alexander polynomial properties.

Symmetric critical points lead to symmetry breaking in neural networks.

This review connects knot invariants to quiver representations.

Study equivariant 4-genus of knots in symmetric 4-manifolds.

New findings on knot operations challenge a long-standing conjecture.

Extended symmetric unions extend properties of Alexander polynomials.

Extends knot invariant computation to symmetrically colored sl_N.

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of index 1. Then we show infinitely many mutually distinct surface-knots that has an em…

The paper defines new knot genera and finds bounds for stabilization distances.

The Palais-Smale condition is proven for various knot energies.

The study counts critical points in knot cobordisms using abelian and metacyclic invariants.

A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…

Study on distinguishing mutant knots using specific representations.

We study the ribbon discs that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number is introduced and compared with the classical ribbon number. We show that the gap between these numbers can be arbitrarily large by constructing an infinite family of ribbon knots with …

Given a knot $K$ parametrized by $r: [0,2π] \to \mathbb{R}^3$, we can define the electric potential on its complement by $Φ(x) = \int_0^{2π} \frac{|r'(t)|}{|x - r(t)|}dt$. Physicists and knot theorists want to understand the critical points of the potential and their behavior. The tunneling number $t(K)$ of a knot is t…

In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of exa…

The twisting number of a ribbon knot is at least as large as its doubly slice genus.

Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.

We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the Möbius energy. For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based…

The paper defines and calculates an upper bound for the equivariant crossing number of two-bridge knots.

Study uncovers complex critical points in tensor decomposition.

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The n-th subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridio…

The paper defines conditions for good involutions in generalized Alexander quandles.