Study calculates Kashaev invariants for twice-iterated torus knots.
problem Calculating Kashaev invariants for complex knots.
method Asymptotic analysis and topological interpretations.
result Obtained formulas linking invariants to Chern-Simons and Reidemeister torsion.
A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers.…
New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
Formula for Alexander polynomial of twisted torus knots derived.
problem Calculating Alexander polynomial for a specific class of knots.
method Knot group presentation combined with Fox's calculus.
result Explicit formula for Alexander polynomial of twisted torus knots.
Study shows differences of torus knots do not form L-space knots.
problem Understanding concordance of knots and their relation to L-space knots. method Examined differences of torus knots and their concordance properties.
result Subgroups generated by two positive torus knots contain no nontrivial L-space knots. Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
New families of twisted torus knots found with essential surfaces.
problem Whether all twisted torus knots have essential tori.
method Analyzing sequences of twists on torus knots.
result Found two new families of toroidal twisted torus knots.
New examples of positive twisted torus knots with same surface slope found.
problem Finding new examples of positive twisted torus knots with same surface slope.
method Extending Dean's conditions and using new four-parameter family.
result Four-parameter family of positive twisted torus knots with same surface slope.
The strong slope conjecture helps identify torus knots.
problem Detecting torus knots using colored Jones polynomials.
method Observation and application of the strong slope conjecture.
result An adequate knot with matching polynomial degrees is a (2,q)-torus knot. Knots formed from torus knots are not concordant to L-space knots.
problem Understanding concordance in knots formed from connected sums of torus knots.
method Analyzing properties of knots formed from connected sums of torus knots.
result Knots formed from torus knots are not concordant to L-space knots.
Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.
problem Classifying twisted torus knots with Horadam parameters.
method Using recursive Horadam parameters to generalize Lee's work on Fibonacci parameters.
result Families of twisted torus knots with Horadam parameters are provided.
Classifies Legendrian and transverse torus knots in S3.
problem Classifying Legendrian and transverse torus knots in S3. method Complete coarse classification using Legendrian and transverse properties.
result Complete classification of torus knots in contact structures on S3. Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
Study finds infinite non-fibered twisted torus knots.
problem Identifying non-fibered twisted torus knots.
method Explicit formula for Alexander polynomial, leading coefficients analysis.
result Infinite families of non-fibered twisted torus knots found.
Study on sums of torus knots concordant to alternating knots.
problem Which sums of torus knots are concordant to alternating knots?
method Effective obstructions based on Heegaard Floer homology.
result Described some effective obstructions for sums of two torus knots.
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.
problem Proving the non-triviality of welded knots and ribbon torus-knots.
method By generating examples and determining the fundamental group of the concerned welded knot.
result Non-triviality of welded knots and ribbon torus-knots is demonstrated.
Researchers confirm a relation between knot invariants and provide formulas for torus knots.
problem Confirming a relation between knot invariants and providing formulas.
method Explicit formulas and algorithms for certain ADO-invariants of torus knots obtained from the series invariant of knot complements.
result Explicit formulas and algorithms for certain ADO-invariants of torus knots.
The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relat…
Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.
problem Determining the minimum number of band surgeries to unknot torus knots.
method Used the torsion order of unoriented knot Floer homology.
result Pinch number and unoriented band unknotting number coincide for torus knots.
Characterizes Legendrian knots in lens spaces.
problem Classifying Legendrian knots in lens spaces.
method Splitting lens spaces and using convex Heegaard decomposition.
result All Legendrian torus knots in universally tight lens spaces are classified.
Study shows unknotting number of certain virtual torus knots equals standard torus knot's unknotting number.
problem Determining the unknotting number of virtual torus knots.
method Analyzing virtual knots derived from standard torus knots and counting crossing changes.
result Virtual unknotting number of certain virtual torus knots equals the unknotting number of the corresponding standard torus knot.
We show that any non-minimal bridge decomposition of a torus knot is stabilized and that n-bridge decompositions of a torus knot are unique for any integer n. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…
Complexity class determined for recognizing torus knots.
problem Recognizing specific torus knots and related knots.
method Based on recent work on detecting knottedness.
result Recognition problem is in NP and co-NP.
Determine pairs of torus knots with genus one cobordisms, with exceptions.
problem Identify pairs of torus knots with genus one cobordisms.
method Combine obstructions from Heegaard Floer knot complex with explicit constructions.
result Determine pairs of torus knots with genus one cobordisms, with exceptions.
A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine-Tri…
Classifies exceptional Legendrian torus knots using contact surgery diagrams.
problem Classifying exceptional Legendrian realisations of non-trivial torus knots.
method Contact surgery diagrams and upper bounds on tight contact structures.
result Classification of exceptional Legendrian realisations of torus knots.
Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.
problem Computing Jones polynomial for a specific family of links.
method Computed Jones polynomial for T((p,q),(2,s)) links, focusing on s=2n. result Jones polynomial is trivial if and only if the knot is trivial.
Sharp signature bound for 3-strand torus knots proved.
problem Determining the topological 4-genus of 3-strand torus knots.
method Using McCoy's twisting method and improving upper bounds.
result Signature bound is sharp for 3-strand knots and off by at most 1 for 4- and 6-strand knots.
The twisted torus knots lie on the standard genus 2 Heegaard surface for S3, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored knots.
The study counts SU(2) representations for torus-covering knots.
problem Counting irreducible metabelian SU(2) representations for torus-covering knots.
method Using Fox's p-colorability and knot determinant.
result Similar to classical knots, the number of representations is determined.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
New method detects sliceness of knots in a torus.
problem Detecting sliceness of knots in a torus.
method Group-theoretical combinatorial techniques.
result Constructs many sliceness obstructions.
This paper extends knot invariants using instantons to study torus knot groups.
problem Understanding the topology of knots and their representations.
method Generalization of equivariant singular instanton Floer theory.
result Irreducible singular instanton homology of torus knots for rational holonomy parameters are Z/4-graded abelian groups. Study connects knot polynomials with number theory sums.
problem Alexander polynomials and Dedekind sums of torus knots.
method No specific method mentioned; connects known concepts.
result Established relationship between knot theory and number theory.
Classifies knots in a special 3D space.
problem Classifying knots in a specific geometric space.
method Complete coarse classification of non-loose torus knots.
result Complete classification of knots in S1imesS2. We show that most cabled knots over torus knots in S3 satisfy the AJ-conjecture, namely each (r,s)-cabled knot over each (p,q)-torus knot satisfies the AJ-conjecture if r is not a number between 0 and pqs.
In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.
New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid M(n,m) for (n,m)-torus knot groups and other braid groups. result New Garside structures for (n,m)-torus knot groups and related braid groups are constructed. Triple-crossing number bound for knots and links, especially torus knots.
problem Finding bounds for triple-crossing numbers of knots and links.
method Using the genus of a knot or link, we derive bounds for the triple-crossing number.
result Triple-crossing number of torus knots and many other knots is at least twice their genus.
Study on nonorientable analogue of Milnor's conjecture for torus knots.
problem Prove a conjecture about the nonorientable 4-genus of torus knots.
method Relied on Ozsváth, Stipsicz, and Szabó's lower bound for γ4 and new closed formulas for the signature of torus knots. result Proved the conjecture for many infinite families of torus knots.
Detects torus knots using SL(2,C) representations and instanton Floer homology.
problem Detecting torus knots using SL(2,C) representations.
method Instanton Floer homology and SL(2,C) representations.
result Proves that a version of the A-polynomial detects torus knots.
Proves conjecture about integer sums of torus knot torsions.
problem Integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for torus knots.
method Introduced Verlinde numbers from modular S-matrix, proved integrality through recursion formulas.
result Proven integrality of sums of (g-1)st powers of adjoint Reidemeister torsions for all torus knots and non-negative g.
Formula found for braid index of n-bridge braids.
problem Finding a formula for the braid index of n-bridge braids. method Elementary, effective, self-contained proof.
result Closed form formula for braid index of n-bridge braids. We prove that an iterated torus knot type fails the uniform thickness property (UTP) if and only if all of its iterations are positive cablings, which is precisely when an iterated torus knot type supports the standard contact structure. We also show that all iterated torus knots that fail the UTP support cabling knot …
Maximum Levine-Tristram signature of torus knots follows a reduction formula.
problem Determining the maximum Levine-Tristram signature for torus knots.
method Proved a reduction formula analogous to Gordon-Litherland-Murasugi's classical signature result.
result Maximum Levine-Tristram signature of torus knots satisfies a reduction formula.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
problem Detecting and distinguishing knots using Khovanov homology.
method Examining all prime knots with up to 20 crossings, conjecturing Legendrian simplicity.
result Numerical evidence supports Khovanov homology distinguishing torus and twist knots.