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…
This paper computes knot invariants using w-knotted objects.
problem Computing knot invariants of w-knotted objects.
method Introduces mathematical and computational tools to solve equations in Aw spaces. result Carries out computations of knot invariants up to a certain degree.
Holomorphic quantum modular forms linked to knot volumes.
problem Understanding algebraic properties of quantum modular forms.
method Analyzing descendant state integrals for specific knots.
result Illustrated algebraic properties for the (-2,3,7)-pretzel knot.
Develops a calculus for knotted objects, including classical links.
problem Characterizing and classifying knotted objects, including classical links.
method Introduces Arrow presentations and w-tree presentations to encode knot diagrams, uses clasper theory for welded knots.
result Characterizes finite type invariants of welded knots and long knots.
Survey of quantum enhancements in knot theory.
problem Classifying knots using quantum invariants.
method Collecting and analyzing various quantum invariants.
result New invariants defined by coloring knots with algebraic objects.
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.
problem Understanding virtual knots through mosaic diagrams.
method Developed moves to preserve knot type and showed all virtual knots can be represented.
result Any virtual knot can be represented as a virtual mosaic.
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:ℜ→ℜn 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.
Classical and welded knot moves are classified and their interrelationships discussed.
problem Classical and welded knot moves and their interrelationships.
method Algebraic classification and topological interpretations of local moves.
result All local moves are unknotting operations for welded knots.
Survey on knotoids and braidoids, their theory and applications.
problem Understanding open-ended knot diagrams and their geometric counterparts.
method Review of fundamental concepts and existing research.
result Exploration of knotoids and braidoids in protein studies.
Paper develops knot invariants for long knots in a torus.
problem Understanding long knots in a torus.
method Uses picture-valued and free group valued invariants.
result Developed powerful and easy to compare knot invariants.
Graphs from knot types help identify unique knots.
problem Identifying knots uniquely.
method Created Reidemeister graphs from knot types and analyzed their properties.
result Graph isomorphism type is a complete knot invariant.
Graphs represent knot adjacency for n crossings.
problem Understanding adjacency relationships between knots.
method Defined a new graph Γn to represent n-adjacency. result Proved several results about the new graph Γn. New invariant fully describes finite type invariants of knots in homology 3-spheres.
problem Constructing a universal finite type invariant for knots in homology 3-spheres.
method Refined construction of a new invariant that is strictly stronger and universal.
result New invariant fully describes the graded space of finite type invariants of knots in homology 3-spheres.
TQFT signatures linked to trace fields of knots.
problem Relationship between TQFT signatures and knot trace fields.
method Analysis of Frobenius algebras and TQFTs at specific roots.
result TQFT signatures equal to trace fields of two-bridge knots.
Data science enhances knot theory by analyzing invariant relations.
problem Understanding the complex relations between knot invariants.
method Topological data analysis applied to knot theory.
result New insights into long-standing conjectures about knots.
New method uses mosaics to study wild knots.
problem Classifying wild knots with infinite knotting behavior.
method Extending knot mosaic theory to represent wild knots with isolated wild points.
result Developed a framework for mosaic tangles and mosaic rigid vertex spatial graphs.
Parity defined for based matrices, a new example of virtual knot parity.
problem Defining parity for a new algebraic structure.
method Introduced parity for based matrices, defined reduced stable parity.
result New example of parity for virtual knots.
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.
problem Computing the prime factorization of knots efficiently.
method Introduced an edge-ideal triangulation to represent knots and developed an algorithm using Regina.
result Our algorithm works well for knots up to 19 crossings and provides new complexity results.
Models of random knots help understand typical knot behavior.
problem Understanding typical knot behavior from a probabilistic viewpoint.
method Presented several randomized models of knots and links, reviewed known results, discussed properties, and explored finite type invariants.
result Asymptotic distribution of knot invariants in random knots studied.
Alternating knots follow a pattern theorem, making them rarer than previously thought.
problem Understanding the scarcity of alternating knots.
method Developed a pattern theorem for alternating knots and used it to prove a conjecture about their rarity.
result Alternating knots are rarer than previously believed.
New knot invariant from braided Hopf algebra.
problem Developing a new knot invariant.
method Non-commutative generalization of knot groups using braided Hopf algebra.
result New quantum character variety as an alternative to skein module.
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.
problem Finite type invariants for ribbon knotted surfaces and their relation to welded string links.
method Developed a theory of finite type invariants for welded string links up to wk-equivalence, studied algebraic structures, and showed characterizations. result Characterizes the information contained by finite type invariants in low degrees for welded string links.
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.
problem The difficulty of calculating quantum invariants of knots.
method Using hyperbolic knots to compute topological quantum computer invariants.
result The hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.
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.
problem Optimizing marginal likelihood for knot selection leads to suboptimal and inefficient placement of knots.
method Uses Bayesian optimization to propose knots one at a time, avoiding multimodal surface issues.
result Improves both accuracy and speed of knot selection compared to current methods.
Study convex embeddability in linear and circular orders, applying to knots.
problem Understanding the quasi-order of convex embeddability in linear and circular orders.
method Combinatorial and descriptive set-theoretic methods applied to arcs and knots.
result Established combinatorial properties and lower bounds for knot complexity.
New algebraic construction of knot contact homology using perverse sheaves.
problem Algebraic construction of knot contact homology.
method Defining a DG category with a braid group action on perverse sheaves.
result The endomorphism algebra of a distinguished object in the category matches fully noncommutative knot DGA.
This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. A new knot selection method speeds up sparse Gaussian process approximations.
problem Efficiently selecting knots for sparse Gaussian processes.
method One-at-a-time Bayesian optimization for knot selection.
result Competitive performance with reduced computational cost.
The elastic trefoil is the twice covered circle, a key finding in knot elasticity.
problem Characterizing the elastic behavior of knotted loops of springy wire.
method Minimizing bending energy and ropelength to penalize self-intersection.
result The elastic trefoil is the twice covered circle, not the round circle.
Study exact Lagrangian cobordisms between Legendrian knots using functorial properties of augmentation categories.
problem Understanding exact Lagrangian cobordisms between Legendrian knots.
method Study the functor between augmentation categories induced by exact Lagrangian cobordisms and establish a long exact sequence.
result Prove the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to exact Lagrangian cobordisms.
The paper connects quivers to knot complements and studies their BPS states and 3d N=2 theories.
problem Understanding the relationship between quivers and knot complements.
method Assigning quivers to knot complements and exploring their physical interpretation.
result Proposed a physical interpretation of quivers in terms of BPS states and 3d N=2 theories.
New method identifies knots in protein chains using virtual knots.
problem Identifying knots in open protein chains.
method Introducing virtual knots to analyze open curves without closure.
result Recovering and extending previous knotting results in proteins.
The paper constructs many knotted and linked objects in higher dimensions.
problem Understanding knotted and linked objects in higher dimensions.
method Using barbell diffeomorphisms to construct examples.
result Infinitely many knotted and linked objects in 4 and 5 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.
problem Classifying and distinguishing planar knotoids with up to five crossings.
method Define biframed planar knotoids and construct new invariants.
result Improved classification of planar knotoids with up to five crossings.
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.
problem Understanding the function of knots in protein folding.
method Topological analysis of protein knotoid distributions and entanglement.
result Reveals unique folding pathway for shallow knotted Carbonic Anhydrases.
The paper describes topological properties of arcs and crossings in knot theory.
problem Understanding the topological nature of arcs and crossings in knot theory.
method Topological description of arcs and crossings as isotopy classes of probes, homotopy classes of diagram elements.
result Sets of arcs and crossings are fundamental for algebraic objects like quandles, partial ternary quasigroups, biquandloids, and crossoids.