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.
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48 results for “signature of knots”
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
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.
Signature functions of knotsmath.GT
Knot signature function defined and conditions for its existence are given.
problem Defining and characterizing the signature function of knots.
method Presentation of necessary and sufficient conditions for a function to be a knot signature function.
result Conditions for a function to be the signature function of a knot are established.
Conditions for integer signatures of high-dimensional knots.
problem Determining signatures of high-dimensional knots with specific Alexander polynomials.
method Necessary and sufficient conditions based on square-free Alexander polynomials.
result Identifies conditions for an integer to be the signature of a knot.
New bounds on knot unknotting number using signature function.
problem Constructing lower bounds on the unknotting number of knots.
method Using the classical knot signature function to construct new bounds.
result New bounds on the unknotting number are twice as strong as previous bounds.
Normal distribution found for 2-bridge knots signatures.
problem Distribution of signatures for 2-bridge knots.
method Calculated average signature, introduced , used limit theorem.
result Distribution of signatures approaches normal as knot complexity increases.
A simplified proof for satellite knot signatures.
problem Proving Litherland's formula for satellite knots.
method Uses linear algebra and basic knot theory.
result A new, elementary proof of the signature formula.
To each unit complex number with positive imaginary part there is defined a Tristram-Levine knot signature function. The set of all such signature functions is linearly independent as a set of functions defined on the set of all knots. The set of averaged signature functions forms a linearly independent set of homomoro…
The paper introduces signatures for virtual knots and applies them to study concordance.
problem Investigating the concordance of virtual knots and their slice genus.
method Defined Tristram-Levine signatures for almost classical knots, used Seifert pairing, and introduced parity projection.
result Established slice obstructions for all virtual knots and determined slice status for almost classical knots.
The paper studies twisted signature invariants of fibered knots and 3-manifolds.
problem Computing twisted signature invariants of fibered knots and 3-manifolds.
method Reduction to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Use of rings of power series to interpret the twisted Milnor pairing and relate it to twisted Blanchfield pairings.
result New twisted generalizations of the Levine-Tristram signature are derived.
We study properties of the signature function of the torus knot . First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.
Lower bound on stable 4-genus of knots using Casson-Gordon signatures.
problem Finding a lower bound on the stable 4-genus of knots.
method Using Casson-Gordon τ-signatures to compute the lower bound.
result A twist knot is torsion in the knot concordance group if and only if it has vanishing stable 4-genus.
The space writhes and signatures of polymer knotscond-mat.soft
Study shows space writhe closely correlates with knot signature in polymers.
problem Understanding the relationship between space writhe and knot signatures in knotted polymers.
method Performed Langevin dynamics simulations of knotted polymers to measure space writhe.
result Space writhe is strongly correlated with knot signature in complex knots.
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.
We define a set of "second-order" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be "first-order signatures". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert …
We give bounds on knot signature, the Ozsvath-Szabo tau invariant, and the Rasmussen s invariant in terms of the Turaev genus of the knot.
The paper calculates knot invariants using Blanchfield forms and obstructs sliceness.
problem Computing and obstructing the sliceness of knots.
method Algorithmic computation of twisted signature invariants using twisted Blanchfield forms and satellite formulas.
result Illustrated algorithm for -torus knots and obstruction of sliceness for certain knots.
We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link …
It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if the…
The paper classifies links based on signature and crossing number properties.
problem Understanding the relationship between signature and crossing number of knots and links.
method Refined theorems and comprehensive classification of links with specific properties.
result Identification of all links where signature + crossing number = 2, closures of positive 3-braids.
A bound on knot unknotting using equivariant signature.
problem Equivariant unknotting of knots.
method Analysis of strongly invertible knots and application of equivariant unknotting moves.
result The equivariant signature provides a lower bound for the equivariant unknotting number.
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.
The paper defines new knot invariants from 2-bridge knots.
problem Defining new invariants for 2-bridge knots.
method Using Gordon-Litherland bilinear forms and essential state surfaces.
result Well-defined invariants and properties explored.
The paper studies twisted signature invariants and their relation to Casson-Gordon invariants.
problem Obstructions to knot concordance using twisted signature invariants.
method Defining a twisted signature function σ_{K,ρ} and proving satellite formulas.
result The twisted signature function σ_{K,ρ} is closely related to Casson-Gordon invariants for appropriate metabelian representations.
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.
Average signature of 2-bridge knots approximates sqrt(2c/π).
problem Estimating the average signature and 4-genus of 2-bridge knots.
method Developed a model for 2-bridge knot diagrams indexed by crossing number, and used it to derive upper bounds for the average 4-genus.
result Upper bound for the average 4-genus of a 2-bridge knot is 9.75c/log c.
Proves a conjecture about knots and their invariants.
problem Relation between Alexander polynomial and signature invariant for two-bridge knots.
method Analyzes two-bridge knots using Fox's conjecture and signature invariant.
result Proves Hirasawa-Murasugi conjecture for two-bridge knots.
New lower bound for doubly slice genus using knot signatures.
problem Finding a lower bound for the doubly slice genus of knots.
method Using the classical signature function to derive a new lower bound.
result Proved that for every nonnegative integer N, there exists a knot with exactly N difference between slice and doubly slice genus.
Explicit relation found between knot torsion and TQFT signatures.
problem Relating Reidemeister torsion of two-bridge knots to TQFT signatures.
method Established an explicit relation using parabolic representations and SU-TQFT.
result Inverse sum of torsions is constant and signatures have similar asymptotic behavior.
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature .
method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree in the quadric.
New complexity measure for shake-slice knots established.
problem Defining and measuring complexity for shake-slice knots.
method Using dualizable patterns and studying knot signatures.
result Existence of -shake-slice knots with specified complexity.
The Dihedral Genus of a Knotmath.GT
The dihedral genus of a knot is related to its signature and minimal surface genus.
problem Determining the dihedral genus of knots and its relation to signatures and minimal surface genus.
method Using dihedral branched covers and properties of Fox colorings, the dihedral genus is linked to the signature of the cover and the Murasugi signature of the knot.
result An infinite family of knots with minimal genus surfaces that realize the four-genus of the knot.
The Kashaev conjecture is proven for classical signatures and Alexander polynomials of links.
problem Proving the Kashaev conjecture for signatures and Alexander polynomials.
method Relating Kashaev's matrix to Gordon-Litherland's work and Kauffman's model.
result Proven Alexander polynomial and classical signature parts of the conjecture for arbitrary links, and full conjecture for definite knots.
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant at takes minimal values on torus links.
The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit circle. The latter are important in deforming U(1) representations of knot group…
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.
This paper computes Jones polynomials and Khovanov homology for weaving knots.
problem Computing Jones polynomials and Khovanov homology for weaving knots.
method Developed recursion relations and used additional results to compute ranks of Khovanov homology.
result Computed ranks of Khovanov homology for knots and provided evidence for normal distribution of ranks.
Ascending numbers are determined for 64 knots with at most n=10 crossings. After proving the theorem about the signature of alternating knot families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings, for which the unknotting number can be confirmed by using th…
New obstructions for knots in high dimensions prevent double sliceness.
problem Preventing double sliceness of knots in high dimensions.
method Developed -signature obstructions for metabelian knot groups.
result Found infinite families of knots not obstructed by previous invariants.
Band surgery affects knot signatures by 0 or 8.
problem Understanding how band surgery impacts knot signatures.
method Using Heegaard Floer d-invariants and L-space properties.
result The absolute value of signature difference is 0 or 8 for quasi-alternating knots related by band surgery.
According to a formula by Gordon and Litherland, the signature of a knot K can be computed as the signature of a Goeritz matrix of K minus a suitable correction term, read off from the diagram of K. In this article, we consider the family of two bridge knots K(p/q) and compute the signature of their Goeritz matrices in…
We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlik…
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge kno…
New invariant measures knot geometry, improving volume-volume inequality.
problem Understanding hyperbolic knots through geometry.
method Introducing natural slope based on cusp geometry, using machine learning for detection.
result Improved inequality linking knot signature, volume, and injectivity radius.
Persistent homology can recognize knotting in curves.
problem Recognizing knotting in curves
method Compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score.
result Systematic differences between knotted and unknotted structures are revealed.
We give an explicit construction of linearly independent families of knots arbitrarily deep in the (n)-solvable filtration of the knot concordance group using the ρ^1-invariant. A difference between previous constructions of infinite rank subgroups in the concordance group and ours is that the deepest infecting knots i…