arXiv research

New invariant distinguishes singular knots and links.

Defines algebraic structures from singular knot theory.

Study of singular knots connects knot theory with quantum algebra.

Study on singular twisted links and virtual braids, extending knot theory concepts.

New Alexander polynomial for singular knots improves upon existing methods.

The article extends a knot polynomial algorithm for singular knots.

Psybrackets define invariants for complex knots and links.

New moves for singular knots identified and described.

Survey of algebraic structures for singular knots.

Invariants for singular knots and links using non-commutative cocycles.

New invariants for singular knots and links defined using shadow structures.

We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.

Generates moves for oriented singular links using algebraic structures.

Study knot singularities in Bogomolny equation solutions.

Classifies certain 3D knots with specific properties.

Study shows infinite distinct outer metric Lipschitz classes for knots in $S^3$.

Generalizes Jones polynomial to singular knots.

Enhances psyquandle invariants for singular and pseudoknots.

A new invariant for complex knots and graphs using rational tangles.

Proves a conjecture linking knot Floer homology and HOMFLY-PT homology for singular knots.

We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial a…

Knot lattice homology invariant of smooth knot type in rational homology spheres.

The paper introduces new knot invariants using singular instanton gauge theory.

New spectral sequences define knot invariants.

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomial…

Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the ``boundary knot'' that occ…

We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We also introduce singular semiquandles and virtual singular semiquandles which define…

Projections of knotted spheres in high dimensions show complex singularities.

The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.

Positive braids linked to knot invariants and geometric monodromy groups.

Researchers found faithful representations for surface singular braid monoids.

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the sin…

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…

Study surgery obstructions for Seifert fibered homology spheres using knot and Heegaard Floer homology.

Generalizes biquandles to psyquandles for singular and pseudolinks invariants.

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma--Hecke algebras ${\rm Y}_{d,n}(u)$ and the theory of singular braids. The Yokonuma--Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphis…

New polynomial invariant distinguishes singular links.

We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibilit…

This paper extends knot invariants using instantons to study torus knot groups.

Classifies uncolored bonded knots with up to 7 singularity points.

The paper calculates the slicing degree of knots using advanced homology theories.

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$…

This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.

In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism.

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

We show that the location of the first singularity of the Upsilon function of an algebraic knot is determined by the first term of its Puiseux characteristic sequence. In many cases this gives better bounds than the tau invariant on the genus of a cobordism between algebraic knots.

New invariant for knotted tori, similar to classical invariant.