Study on r−shake slice knots and proves 0-shake slice knots are slice.
problem Understanding and characterizing r−shake slice knots. method Exploring the relation to corks and proving slice properties.
result Proves 0-shake slice knots are slice.
Polynomially parameterizes knots and spheres, proving analogous results.
problem Parameterizing knots and spheres using polynomials.
method Analogous to classical knots, parameterized long 2-knots and certain classes of knotted spheres.
result Polynomial parameterizations for knotted spheres constructed.
The paper proves tight fibered knots are minimal in a specific knot order.
problem Understanding the minimality of tight fibered knots.
method Proved ribbon concordance forms a partial order and used it to show tight fibered knots are minimal.
result All tight fibered knots are minimal in the ribbon concordance order.
Proves certain knots are slice without shaking.
problem Identifying slice knots without using traditional methods.
method Direct proof for 0−shake slice knots. result Proves 0−shake slice knots are slice. Two algorithms use normal surfaces to detect unknots and prove knots.
problem Detecting and proving the unknot and knottedness of links.
method Normal surface theory algorithms and split-link algorithm.
result Figure-eight knot is proven to be knotted.
Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
Proves cosmetic crossing conjecture for certain knots.
problem Cosmetic crossing conjecture for specific knot types.
method Proof for knots with non-trivial Alexander polynomial; additional assumptions for trivial Alexander polynomial.
result Proves conjecture for specified knot types.
New knot concept extends welded knots, simplifying classification.
problem Classifying welded knots and their complements.
method Introducing 'wen knots', proving subset relationships, characterizing complements.
result Extended welded knots can be fully characterized by the parity of wens.
We prove new results about unknotting fibered positive knots and braids.
problem Proving the unknotting number equals genus for fibered positive knots and braids.
method Analyzing positive braid diagrams and fibered positive knots, proving new constraints and conjectures.
result We found fibered positive knots that cannot be unknotted optimally, contradicting Stoimenow's conjecture.
New knot graphs show most are not Gromov hyperbolic, with special cases.
problem Characterizing Gromov hyperbolicity in knot graphs.
method Defining knot graphs and proving non-hyperbolicity.
result Most knot graphs are not Gromov hyperbolic, with exceptions.
Study proves nontrivial knots can't undergo cosmetic surgeries.
problem Cosmetic surgeries on nontrivial knots.
method Calculated finite type invariant of order 3.
result Nontrivial knots do not admit chirally cosmetic surgeries.
Proves freely 2-periodic knots have two canonical components in their character variety.
problem Identifying freely 2-periodic knots in character varieties.
method Analyzes SL(2, C) character variety and hyperbolic torsion polynomial.
result Character variety has two canonical components for freely 2-periodic 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.
Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.
problem Understanding the structure and properties of transverse knots and their neighborhoods.
method Proves unique standard neighborhoods and structure theorems for non-loose Legendrian knots through destabilization results.
result Finds a manifold with infinite tight contact structures, up to contactomorphism, without Giroux torsion.
The study calculates braid indices for two-bridge knots and proves inequalities.
problem Calculating the braid index of two-bridge knots.
method Proved inequalities and provided average braid index for knots of a given crossing number.
result Average braid index for two-bridge knots with a given crossing number.
New framework distinguishes knots via neighborhood invariants.
problem Distinguishing knots and knotoids.
method Study of knotoid spectra and neighborhood invariants.
result Neighborhood invariants can distinguish knots of higher Gordian distance.
Positive knots are minimal in a specific knot ordering.
problem Understanding the minimal structure of positive knots in a specific knot ordering.
method Analyzing ribbon concordance and using properties of rational Alexander modules.
result Positive knots are minimal in the ribbon concordance ordering.
We prove all knots can be transformed into a trefoil using special diagrams.
problem Transforming any knot into a trefoil using magic tricks.
method Introducing knotholder diagrams to encode transformations.
result All knots can be transformed into a trefoil.
This paper proves hyperbolicity of virtual knot compositions.
problem Proving hyperbolicity of virtual knot compositions.
method Exploring the composition of hyperbolic virtual knots.
result Strong lower bounds on the volume of compositions.
Paper proves knots satisfy a conjecture using Jones polynomial.
problem Proving infinite families of knots satisfy the Cosmetic Surgery Conjecture.
method Computed Jones polynomial and invariants for two knot families.
result Two infinite families of knots satisfy the Purely Cosmetic Surgery Conjecture.
The abstract proves every knot type can be parametrized by smooth functions and studies limit knot types.
problem Understanding all knot types and their parametrizations.
method Proving every knot type can be represented by smooth functions and studying limit knot types.
result Every knot type can be parametrized by smooth functions and limit knot types exist.
Study proves 0-surgery characterizes infinitely many knots.
problem Characterizing infinitely many knots using 0-surgery.
method Classification of knots with specific knot Floer homology properties.
result Proves 0-surgery characterizes infinitely many knots, including new types.
New strict inequalities for knot crossing numbers proved.
problem Establishing strict inequalities between different crossing numbers of knots.
method Proving and generalizing inequalities between n-crossing numbers for various knots. result Optimal strict inequality c9(K)≤c3(K)−2 for many knots, with optimality proven. We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let Q be a spun knot of a virtual 1-knot K by our method. The embedding type Q in S4 depends only on K. Furthermore we prove the following: The submanifo…
Proves special alternating knots cannot be decomposed as non-trivial band sums.
problem Understanding the structure and properties of special alternating knots.
method Analyzes concordance properties and uses band sums to restrict knot decompositions.
result Conjecture that special alternating knots are ribbon concordance minimal is verified in many cases.
The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.
Proves bounds on ribbonlength for various knot types.
problem Finding bounds on the infimal folded ribbonlength of knot types.
method Applied techniques to multi-twist Möbius bands and knot constructions.
result Established bounds for (2,q) torus knots and twist knots. Study proves bridge and braid indices match for twist positive knots.
problem Determining when bridge and braid indices are equal for knots.
method Used knot Floer torsion order to prove for all twist positive knots.
result Bridge and braid indices coincide for all twist positive knots.
Proves rational slopes characterize knot 5_2, except for integers.
problem Characterizing slopes for knot 5_2.
method Proves all rational slopes except integers characterize 5_2, classifies Dehn surgeries, studies almost L-spaces.
result Rational slopes characterize knot 5_2 except for positive integers.
Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.
problem Embedding knots into fractals to compare complexity.
method Proved all knots can be embedded into Menger Sponge, and Pretzel knots into Sierpinski Tetrahedron. Compared iterations needed for given knots.
result Comparison of fractal complexity through knot embedding.
This paper shows that only finitely many knots can be ribbon concordant to any given knot.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Using knot Floer homology and bordered Heegaard Floer homology, the paper proves an inequality relating the knot Floer homology of a satellite knot and its companion.
result Every knot in S3 has only finitely many fibered predecessors under ribbon concordance. New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show similar results for Legendrian knots and prove a "folk" result concerning loose transv…
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic …
Proves volume conjecture for twist knots using complex analysis.
problem Volume conjecture for twist knots.
method Equivalence relation, complex analysis, analytic continuation, function of several complex variables.
result Proves volume conjecture for twist 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.
Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …
Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.
problem Detecting and classifying links based on knot Floer homology ranks.
method Analyzes knot Floer homology ranks to identify and classify links.
result Classifies links of the second smallest knot Floer homology.
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.
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization …
We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove …
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.
We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by term…
Every negative amphichiral knot is rationally slice.
problem Proving every negative amphichiral knot is rationally slice.
method Systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior.
result Every negative amphichiral link is rationally slice.
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rac…
Proves a special knot type is slice.
problem Characterizing slice knots.
method Proof by contradiction and algebraic topology.
result 0-shake slice knots are indeed slice.