arXiv research

This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their "usual" counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar "virt…

This is the first in a series of papers studying w-knotted objects (w-knots, w-braids, w-tangles, etc.), which make a class of knotted objects which is {w}ider but {w}eaker than their usual counterparts. The group of w-braids was studied (as "{w}elded braids") by Fenn-Rimanyi-Rourke and was shown to be isomorphic to th…

Holomorphic quantum modular forms linked to knot volumes.

Develops a calculus for knotted objects, including classical links.

Survey of quantum enhancements in knot theory.

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev, who thought all free knots to be trivial). As it turned …

This paper studies virtual knots using mosaic diagrams.

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. Th…

We prove that for some knot-like objects one can easily recognize non-equivalence w.r.t. all Reidemeister moves by studying some equivalence classes modulo only 2nd Reidemeister moves. There are applications to virtual knots, graph-links and looped graphs.

A polynomial knot is a smooth embedding $κ: \real \to \real^n$ whose components are polynomials. The case $n = 3$ is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological knot. This paper contains basic results for these knots as well as many examples.

Survey on knotoids and braidoids, their theory and applications.

Paper develops knot invariants for long knots in a torus.

Graphs from knot types help identify unique knots.

Graphs represent knot adjacency for n crossings.

In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in 4-dimensional space. When all layers of interpretation are stripped off, what r…

New invariant fully describes finite type invariants of knots in homology 3-spheres.

TQFT signatures linked to trace fields of knots.

Data science enhances knot theory by analyzing invariant relations.

New method uses mosaics to study wild knots.

Parity defined for based matrices, a new example of virtual knot parity.

This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in a compact, accessible manner, and to show how to use it for computational purpo…

We developed an efficient algorithm to factorize knots.

Models of random knots help understand typical knot behavior.

Alternating knots follow a pattern theorem, making them rarer than previously thought.

New knot invariant from braided Hopf algebra.

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classi…

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-t…

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

We define an algebraic/combinatorial object on the front projection $Σ$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashb…

Topological quantum computers use hyperbolic knots for computations.

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocy…

Proposes a more efficient knot selection method for sparse Gaussian processes.

Study convex embeddability in linear and circular orders, applying to knots.

New algebraic construction of knot contact homology using perverse sheaves.

This paper confirms a bound for folded ribbonlength of 2-bridge knots.

A new knot selection method speeds up sparse Gaussian process approximations.

In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. …

The paper connects quivers to knot complements and studies their BPS states and 3d N=2 theories.

New method identifies knots in protein chains using virtual knots.

The paper constructs many knotted and linked objects in higher dimensions.

We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational blow-downs. We show how to detect corks and plugs inside of some exotic manifolds.…

New quantum invariants for planar knotoids improve knot classification.

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-link…

Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of Reidemeister-type moves. In this paper, we introduce two natural extensions of classical knot …

New method detects and compares folding pathways of knotted proteins.

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomolog…

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic…

The paper describes topological properties of arcs and crossings in knot theory.