An algorithm determines knot colorability and determinants from petal projections.
problem Determining knot colorability and determinants from petal projections.
method Algorithm based on petal projections and permutations.
result Determinants of all prime knots with crossing number less than 10 computed.
Determinant modulo 8 classifies virtual knots based on polynomial coefficients.
problem Classifying virtual knots using determinant modulo 8.
method Introduced a determinant for checkerboard colorable virtual knots and proved its classification by the coefficient of z2 in the ascending polynomial. result Determinant modulo 8 classifies virtual knots based on polynomial coefficients.
The paper counts specific knot representations for twist spins.
problem Counting specific SL(2,C) representations of twist spins. method Comparing knot groups of twist spins and their 1-knot determinants.
result The number of representations is determined by the 1-knot determinant.
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.
Formula found for knot determinant in 3-braid weaving.
problem Determining the determinant of a specific type of knot.
method Developed a formula for the determinant of the twisted generalized hybrid weaving knot.
result Proved a conjecture about the knot's determinant.
Study Gram determinants in knot theory, focusing on a Möbius band determinant.
problem Closed formula for the Gram determinant of type (Mb)1. method Survey of Gram determinants, focusing on a Möbius band determinant.
result Speculation on closed formula for (Mb)1 Gram determinant. 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.
The paper discusses knot colorings and their invariants using Goeritz matrices.
problem Distinguishing knots using coloring methods.
method Elementary approach to equivalence between coloring and Goeritz matrices.
result Computing knot determinant and nullity of pretzel knots.
The paper classifies symmetries and determines exterior types of knotted handlebodies.
problem Classifying symmetries and determining exterior types of knotted handlebodies.
method Analysis of hyperbolic knots with non-integral toroidal Dehn surgeries.
result Determines symmetries and exterior types of knotted genus two handlebodies.
Knot groups of hyperbolic 2-bridge knots are uniquely identified by their finite quotients.
problem Identifying knots based on their group structures.
method Proving hyperbolic 2-bridge knots are uniquely determined by their profinite completions.
result Hyperbolic 2-bridge knots are uniquely identified by their finite quotients.
We consider the relations ≥ and ≥p on the collection of all knots, where k≥k′ (respectively, k≥pk′) if there exists an epimorphism πk→πk′ of knot groups (respectively, preserving peripheral systems). When k is a torus knot, the relations coincide and k′ must also be a torus knot; we dete…
Study determines left-orderable properties of knot covers.
problem Determining left-orderability of knot covers.
method Analyzing 2-bridge knots, including double-twist knots, using cyclic branched covers.
result Identifies specific integers for left-orderability of knot covers.
The paper distinguishes non-equivalent branched twist spins using knot determinants.
problem Distinguishing non-equivalent branched twist spins.
method Presented a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. Calculated the first elementary ideals and obtained the condition of the knot determinants by substituting -1 for the indeterminate.
result A sufficient condition to distinguish non-equivalent, non-trivial branched twist spins using knot determinants.
New examples show transverse knots are determined by their branched covers.
problem Transverse knots and their isotopy classes.
method Constructing and analyzing non-isotopic transverse knots with contactomorphic cyclic branched covers.
result Transverse isotopy classes of many transverse knots are determined by the contactomorphism type of their cyclic branched covers.
New number bounds knot complexity, including unknotting and crosscap numbers.
problem Bounding knot complexity and understanding knot types.
method Introducing an unknotting-type number to estimate crosscap number.
result Determines set of knots with crosscap number at most two.
In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the h…
For any given number of crossings c, there exists a formula to determine the number of 2-bridge knots of c crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and …
Generalized knot groups Gn(K) were introduced independently by Kelly (1991) and Wada (1992). We prove that G2(K) determines the unoriented knot type and sketch a proof of the same for Gn(K) for n>2.
Cyclic covers of knots uniquely determine the original knot.
problem Determining knots from their branched covers.
method Examining n-fold cyclic branched covers of alternating prime knots.
result The n-fold cyclic cover of an alternating prime knot uniquely determines the original knot.
Criteria found for knots not determined by their double branched covers.
problem Determining knots from their double branched covers.
method Criteria based on tunnel number one knots and their double branched covers.
result Provided criteria for knots not determined by their double branched covers.
The paper finds bounds and specific tile numbers for knot mosaics.
problem Determining the minimum number of tiles needed to represent knots.
method Analyzing the relationship between tile number and mosaic number of knots.
result Strict bounds and specific tile numbers for various knots are determined.
This paper studies how knots combine using Alexander Polynomials.
problem How knots combine and their determinants behave.
method Basic knot theory, Alexander Polynomials, and composition techniques.
result Generalized solution for knot determinants in compositions.
Knot invariants from XC-structures on Sweedler algebra are trivially determined.
problem Defining and characterizing knot invariants from XC-structures.
method Examining XC-structures on the Sweedler algebra and their relation to knot invariants.
result Knot invariants from XC-structures on Sweedler algebra are completely determined by the framing of the knot.
Jones polynomial bounds and crossing numbers of knots.
problem Bounding the degree of colored Jones polynomial in terms of crossing number.
method Sharpness of bounds determined for adequate knots; application to satellite knots.
result Sharpness of bounds for adequate knots; determination of crossing numbers for specific knots.
This paper finds all prime knots with mosaic number 6 and their minimal space-efficient mosaics.
problem Finding minimal space-efficient mosaics for prime knots with a specific mosaic number.
method Examined prime knots with mosaic number 6, determined their minimal space-efficient mosaics, and calculated their tile numbers.
result A complete list of prime knots with mosaic number 6 and their minimal space-efficient mosaics were found.
Study shows crossing numbers of cable knots are larger than previously thought.
problem Determining the crossing numbers of cable knots.
method Using colored Jones knot polynomials and degree analysis.
result Crossing numbers of (p,q)-cables of adequate knots are larger than q2c. 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.
Paper refines generating function for 2-bridge knot groups.
problem Determining the number of epimorphisms between 2-bridge knot groups.
method Refined generating function considering genus and crossing number.
result Improved formula for epimorphisms between 2-bridge knot groups.
We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of another knot, for different coprime integers m and n.
We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete re…
Study determines Δ-unknotting numbers for two-bridge knots.
problem Determining the minimum number of Δ-moves to simplify two-bridge knots. method Examined two-bridge knots of specific types and calculated their Δ-unknotting numbers. result Identified two-bridge knots with Δ-unknotting number equal to one. New formulas for knot polynomial evaluations from covering spaces.
problem Evaluating knot polynomials uniquely from covering spaces.
method Using singular determinants and linking pairings.
result Explicit formulae for Jones and Q-polynomial evaluations. Paper determines 2-adjacent knots up to 12 crossings.
problem Identifying 2-adjacent knots with up to 12 crossings.
method Used Heegaard Floer d-invariants and Alexander polynomial to obstruct 2-adjacency.
result Proved conjectures about 2-adjacent knots by Ito and Kato.
The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.
problem Identifying knots with nontrivial Alexander polynomials and improving knot classification.
method Elimination of infinite families of knots and use of determinants to improve unknotting number data.
result Elimination of infinite families of knots with nontrivial Alexander polynomials and improvement of unknotting number data.
Study of fundamental groups of knotted solenoid complements in 3D sphere.
problem Determining fundamental groups of knotted solenoid complements.
method Using canonical sequence of knot groups and embedding up to mirror reflection.
result Fundamental groups of knotted solenoid complements are solely determined by a sequence of knot groups and embedding up to mirror reflection.
We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.
Study character varieties of even pretzel knots, determining their mSL(2,C)-representations.
problem Determine character varieties of even classical pretzel knots.
method Compute irreducible mSL(2,C)-representations of even classical pretzel knots. result Clarify the steps to compute the A-polynomial of even classical pretzel knots.
Study on knot and link positivities, supporting conjectures and determining specific cases.
problem Understanding positivities of knots and links and their relation to the Bennequin inequality.
method Analyzing various positivities, providing evidence for conjectures, and determining specific cases.
result Determined strong quasipositivity and quasipositivity for knots up to 12 crossings (with exceptions).
Study determines Turaev genus and dealternating number of torus knots up to a small error.
problem Measuring how close torus knots are to alternating links.
method Analyzes Turaev genus and dealternating number of torus knots with up to 6 strands.
result Exact or near-exact values for Turaev genus and bounds for dealternating number of torus knots.
Problems of knot classification and manifold properties are shown to be NP-complete.
problem Determining knot non-triviality and Thurston norm in 3-manifolds.
method NP-completeness proofs for specific problems in knot theory and 3-manifold topology.
result Problems of knot classification and manifold properties are proven to be NP-complete.
We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the correspond…
We determine the rational Khovanov bigraded homology groups of all Kanenobu knots. Also, we determine the crossing number for all Kanenobu knots K(p,q) with pq>0 or ∣pq∣≤max{∣p∣,∣q∣}. In the case where pq<0 and ∣pq∣>max{∣p∣,∣q∣}, we conjecture that the crossing number is ∣p∣+∣q∣+8.
Knots in circle bundles are uniquely identified by their complements.
problem Determining knots in circle bundles based on their complements.
method Analyzing the complements of knots in orientable circle bundles over surfaces.
result Knots in circle bundles are determined by their complements.
New knots found with same determinant but no symmetric relation.
problem Determining if knots with the same determinant are symmetrically related.
method Constructing a family of knots with the same determinant but no symmetric relation.
result No two knots in the family are symmetrically related.
New invariants for Turaev genus one links identified.
problem Defining invariants for Turaev genus one links.
method Signature determination from Kauffman states and coefficients of Jones polynomial.
result Signature of Turaev genus one knots determined by specific components and crossings.
Study Khovanov homology of positive links and L-space knots, finding vanishing conditions.
problem Understanding Khovanov homology of positive links and L-space knots.
method Analyzing Khovanov homology groups in specific gradings and extending results to (p,q)-cables and Heegaard Floer L-space knots.
result Khovanov homology of positive links and L-space knots vanishes under certain conditions.
Legendrian knots in tight 3-sphere are uniquely determined by their exteriors.
problem Determining Legendrian knots in tight contact structures.
method Contactomorphism type of exterior, Thurston-Bennequin invariant formula.
result Not all Legendrian links in tight 3-sphere are uniquely determined by their exteriors.
The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surface bounded by the knot. We determine the crosscap numbers of torus knots.