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.
Study proves obstructions to equivariantly slice strongly negative amphichiral knots.
problem Proving obstructions for equivariantly slice strongly negative amphichiral knots.
method Using determinant, Spinc-structures, Donaldson's theorem, and Heegaard Floer correction terms.
result 8 out of 16 strongly negative amphichiral knots with 12 or fewer crossings are not equivariantly slice.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
Study on equivariant Q-sliceness for strongly invertible knots.
problem Understanding Q-sliceness for strongly invertible knots.
method Constructive and obstructive approaches using Fox-Milnor condition and equivariant concordance.
result Klein amphichiral knots are equivariant Q-slice in a single Q-homology 4-ball.
New knots found that are 4-genus minimal.
problem Finding knots with minimal 4-genus.
method Constructing infinitely many amphichiral knots with specific properties.
result Knots with 4-genus minimal for each g>0. Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
We give a simple obstruction for a knot to be amphichiral, in terms of the homology of the 2-fold branched cover. We work with unoriented knots, and so obstruct both positive and negative amphichirality.
Tanaka shows amphichiral symmetric unions of the unknot are trivial.
problem Understanding amphichiral symmetric unions and their Jones polynomials.
method Analyzing the Jones polynomial of amphichiral symmetric unions of the unknot and generalizing to other knots.
result Amphichiral symmetric unions of any knot with one twist region are trivial.
Research classifies knots based on sliceness and amphichirality.
problem Classifying odd-stranded Turk's head knots based on sliceness and amphichirality.
method Constructing commuting pairs of ambient involutions and analyzing the equivariant Fox-Milnor square condition.
result Established a sharp parity dichotomy for equivariant rational sliceness and Klein amphichirality of odd-stranded Turk's head knots.
The paper classifies involutions on S^4, proving linearities under certain conditions.
problem Classifying involutions on S^4 with specific fixed-point sets.
method Combining surgery theory, Schoenflies theorem, and equivariant topology.
result Linear involutions on S^4 with 1-dimensional fixed-point sets are proven.
Study negative band numbers in braids and links.
problem Characterizing braids with negative band numbers.
method Using Birman, Ko, and Lee's left-canonical form.
result Characterize strongly and almost strongly quasipositive braids.
In 1997 Cochran-Orr-Teichner introduced a natural filtration, called the n-solvable filtration, of the smooth knot concordance group, C. Its terms {F_n} are indexed by half integers. We show that each associated graded abelian group G_n=F_n/F_{n.5}, n>1, contains infinite linearly independent sets of elements of order …
Survey of Turk's head knots and links properties.
problem Characterize Turk's head knots and links.
method Discussion of various results in the mathematical literature.
result Turk's head links are alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime.
Study on Legendrian and transverse realizations of negative torus knots.
problem Classification of transverse and Legendrian realizations of negative torus knots.
method Analysis of contact structures, knot Floer homology, Legendrian surgeries.
result Classification of strongly non-loose transverse and Legendrian realizations.
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p,q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein mod…
We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Sei…
The paper explores symmetric representations of links and conditions for amphichirality.
problem Investigating symmetric representations of links and conditions for amphichirality.
method Using antipodally self-dual and antipodally symmetric maps, the authors provide sufficient combinatorial conditions for amphichirality.
result A link is amphichiral if its self-dual pairing is not one of 6 specific ones.
Survey on embedding 3-manifolds in definite 4-manifolds, focusing on Donaldson's theorem.
problem Understanding embeddings of 3-manifolds in definite 4-manifolds.
method Utilizes Donaldson's diagonalization theorem and combinatorics of integral lattices.
result New result on embedding amphichiral lens spaces in negative-definite manifolds.
Classifies symmetries of knots using group actions and orthogonal representation theory.
problem Classifying symmetries of knots in 3-sphere.
method Using geometrization and orthogonal representation theory, constructing examples, and distinguishing symmetries.
result Cyclic and dihedral families of symmetries of prime knots and composite knots.
Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
Defines knot signature invariant using G-signature theorem.
problem No specific problem stated; focuses on knot theory.
method Uses G-signature theorem to define knot invariant.
result Defines an invariant for strongly invertible knots.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
Study identifies prime strongly positive amphicheiral knots with double symmetry.
problem Characterizing prime strongly positive amphicheiral knots with specific symmetries.
method Examined knots up to 16 crossings, identified prime knots with double symmetry, and presented almost doubly symmetric diagrams.
result Found the first prime strongly positive amphicheiral knot not slice.
New findings on knot genera using advanced techniques.
problem Understanding the 4-genus of knots, especially strongly invertible and periodic ones.
method Innovative concordance group invariants, Donaldson's theorem, and g-signature.
result Many new examples showing the equivariant 4-genus is larger than the 4-genus.
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion wh…
Two knots with unique surgery properties.
problem Characterizing strongly invertible L-space knots.
method Examined surgeries and knot properties.
result Found knots whose surgeries are never Khovanov thin.
According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as φ(z)φ(−z) for some φ(z)∈Z[z]. Moreover, a 2012 example due to Ermotti, Hongler and Weber shows that this is not true for general amphiche…
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
New findings on knot operations challenge a long-standing conjecture.
problem Understanding equivariant unknotting numbers of strongly invertible knots.
method Study of symmetric crossing change operations for strongly invertible knots.
result The equivariant unknotting number is not additive under connected sum.
Links can be transformed into many others using a specific operation.
problem Understanding the relationship between strongly quasipositive links and their concordance.
method Used a satellite operation with a slice knot to transform links.
result Strongly quasipositive links can be transformed into infinitely many other links.
Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
The paper calculates the equivariant genus for a specific type of knot.
problem Calculating the equivariant genus of marked strongly invertible knots associated with 2-bridge knots.
method Analyzing invariant Seifert surfaces for marked strongly invertible knots.
result The paper completely determines the equivariant genus for every marked strongly invertible knot with K a 2-bridge knot. Proves a mod 4 version of Kawauchi's conjecture for amphicheiral knots.
problem Proves a mod 4 version of Kawauchi's conjecture for amphicheiral knots.
method Used finite type invariants and Claude Fable 5 to produce the proof.
result Proves a mod 4 version of Kawauchi's original conjecture for amphicheiral knots.
We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.
Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
Paper finds first infinite family of hyperbolic knots with specific properties.
problem Identifying new hyperbolic knots with specific properties.
method Examined SnapPy census and used knot theory to find new infinite family.
result First infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two.
A knot k is called ``strongly (n-1)-trivial.'' if there exists a projection of k, such that one can choose n crossings of the projection with the property that making the crossing changes corresponding to any of the 2n−1 nontrivial combinations of the selected crossings turns the original knot into the unknot. We …
We show that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a c…
A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility a…
New bounds on knot unknotting numbers using involutive homology.
problem Bounding the unknotting number of strongly invertible knots.
method Using involutive Bar-Natan homology to establish bounds.
result Identified knots with strict inequality between standard and equivariant unknotting numbers.
Specialized knot theory theorems for strongly involutive links.
problem Classical Alexander and Markov theorems for links.
method Equivariant closure map for strongly involutive links.
result Surjective equivariant closure map up to equivalence of strongly involutive links.
We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.
The paper defines and calculates an upper bound for the equivariant crossing number of two-bridge knots.
problem Finding the minimum number of crossings in symmetric diagrams for two-bridge knots.
method Defining and calculating c2(K) for two-bridge knots by restricting diagrams to two types. result An algorithm to determine c2(K) for any two-bridge knot and results up to 14 crossings. Study on knots, genera, and algebraic concordance groups.
problem Understanding the equivariant slice genus of strongly invertible knots.
method Using the Blanchfield form to establish lower bounds and formulate an equivariant algebraic concordance group.
result The equivariant slice genus of an equivariant connected sum of a genus one strongly invertible slice knot is at least n/4.
Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism Φ from equivariant concordance group to a new equivariant algebraic concordance group, proving it lifts known homomorphisms and provides new obstructions. result Obtains a new obstruction to equivariant sliceness and novel lower bounds on equivariant slice genus.
In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of algebraically slice knots contains an infinitely generated free subgroup that is genera…