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.
Study links' concordance using signatures and nullity.
problem Link concordance invariance of Levine-Tristram signatures.
method Determine invariance for complex numbers on the unit circle.
result Signature and nullity give link concordance invariants.
Paper confirms Kashaev's signature conjecture for links.
problem Proving Kashaev's conjecture about link invariants.
method Using Seifert surface definition and diagrammatic approach.
result Established Kashaev's conjecture, providing a new formula for Alexander polynomial.
Study on signatures of positive braids with bounds derived.
problem Understanding signatures of positive braids and their invariants.
method Derived lower bounds for Levine-Tristram signatures, and upper and lower bounds on signature ratios.
result Established bounds on signatures of positive braids, uniformly valid across monoids.
Survey of the Levine-Tristram signature invariant for links.
problem Defining and understanding the Levine-Tristram signature invariant for links.
method Recalling three and four dimensional definitions, listing properties, and providing references.
result Comprehensive overview of the Levine-Tristram signature invariant.
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.
Extends a formula for the homomorphism defect of a signature map to coloured braids.
problem Evaluate the homomorphism defect of a signature map for coloured braids.
method Uses a 4-dimensional interpretation of the signature and new 4D tools like the Maslov index and isotropic functor.
result Generalizes the formula of Gambaudo and Ghys to coloured braids and tangles.
New formulas estimate link signatures near 1.
problem Estimating link signatures close to 1.
method Two approaches: 3D and 4D, using generalized Seifert surfaces and a new extension to the torus.
result New estimates on Levine-Tristram signature near 1.
New proof of link 4-genus bounds and satellite link invariants.
problem Determining the 4-genus of links and its behavior under satellite operations.
method New proof using Levine-Tristram signatures and satellite constructions.
result The 4-genus of a link does not increase under certain satellite operations.
In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable p…
Study rational homology balls using Casson-Gordon invariants to measure complexity.
problem Measuring complexity of rational homology balls.
method Use Casson-Gordon invariants and Levine-Tristram signatures.
result Obtain lower bounds on the number of 1-handles in handle decompositions.
A new method computes link invariants from diagrams.
problem Computing link invariants efficiently.
method Single symmetric matrix from a link diagram.
result Multivariable Alexander polynomial computation.
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.
Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.
problem Understanding knots that divide ribbon knotted surfaces and their properties.
method Defining half ribbon knots, computing half ribbon genus and fusion number, and comparing with Levine-Tristram signatures.
result Computed half ribbon genus and fusion number for various knots, including new computations of doubly slice genus.
We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change when we cross a singular point of C. Applying link invariants as Murasugi's sign…
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.
We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in Cn+1, using Seifert forms of high-dimensional non-spherical links, the Levine--Tristram signatures and the generalized Murasugi--Kawauchi inequality obtained in earlier work …
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 n-shake-slice knots with specified complexity. Extended signatures help distinguish non-concordant links.
problem Distinguishing non-concordant links using signatures.
method Defined and studied an n-variable extension of the Levine-Tristram signature, proving it a concordance invariant on a dense subset of the torus.
result Found an infinite family of 3-component links not concordant to their mirror images, detectable only by the extended signature.
In [BF12] the authors associated to a knot K an invariant n_R(K) which is defined using the Blanchfield form and which gives a lower bound on the unknotting number. In this paper we express n_R(K) in terms of Levine-Tristram signatures and nullities of K. In the proof we also show that the Blanchfield form with real co…
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…
Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.
problem Understanding homotopy ribbon concordance for knots.
method Using Blanchfield pairings and twisted Alexander polynomials.
result Existence of infinite families of knots with same Blanchfield form but not homotopy ribbon concordant.
Given a knot K we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of K. This lower bound subsumes the lower bounds given by the Levine-Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotti…
We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous…
We compute the Heegaard Floer homology of S13(K) (the (+1) surgery on the torus knot Tp,q) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of Tp,q as …
New invariant for knotted tori, similar to classical invariant.
problem Defining a new topological invariant for knotted tori.
method Analogous to Levine-Tristram invariant, using gauge theory for singular connections.
result Invariant matches Echeverria's invariant and Langte Ma's general result.
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further,…
New Arf invariants for colored links determined by linking numbers.
problem Extending Arf invariant to colored links.
method Using generalized Seifert forms to construct quadratic forms and determining Arf invariant.
result New Arf invariants for colored links are determined by linking numbers.
The Upsilon invariant bounds cobordisms between knots and their braid index.
problem Bounding cobordisms between knots and their braid index.
method Using Ozsváth, Stipsicz, and Szabó's Upsilon-invariant.
result Established inductive formulas for the Upsilon invariant of torus knots.
We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings Mm⊂Nm+2, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the c…
New lower bound for knot genus using Links-Gould invariant.
problem Finding a tighter lower bound for knot genus.
method Representation theory of Uqgl(2∣1) to prove degree of Links-Gould polynomial bounds Seifert genus. result The Links-Gould polynomial provides a new lower bound on knot genus, detecting specific knots like Kinoshita-Terasaka and Conway.
To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…
New unoriented algebraic concordance group defined using mock Seifert matrices.
problem Understanding unoriented algebraic concordance of knots in thickened surfaces.
method Introducing mock Seifert matrices and using them to define unoriented algebraic concordance.
result The unoriented algebraic concordance group is abelian and infinitely generated.
Given a link L in the 3-sphere, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply-connected positive-definite smooth 4-manifold; the knot case has been studied extensively in work of Cochran-Harvey-Horn. Such a 4-manifold is necessarily homeomorphic to a (punctured) c…
The paper revisits expected signatures in semimartingale models, providing new formulae and simplifying complexity.
problem Computing expected signatures in semimartingale models.
method Revisits and provides new formulae for computing expected signatures in a general semimartingale setting.
result Log-transform of expected signatures simplifies complexity, leading to signature cumulants.
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.
Deep signature/log-signature FBSDE algorithm improves accuracy and training time.
problem Solving FBSDEs with state and path dependent features.
method Incorporates deep signature/log-signature transformation into RNN model.
result Improves accuracy and training time compared to existing methods.
A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifol…
New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
Introduces flat discrete signatures for financial data analysis.
problem Representing financial data for machine learning without continuous transformation.
method Introduced flat discrete signatures and discrete signatures, generalizing flat discrete signatures.
result Flat discrete signatures can represent quadratic variation relevant in finance.
Method extracts cancer signatures from genome data, reducing noise and variability.
problem Identifying stable cancer signatures from noisy genomic data.
method Applied statistical risk models from finance to cancer genome data, using NMF.
result Extracted signatures have lower variability and improved stability.
This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology …
We define the Analytical signature, the Hodge signature and the de Rham signature for a foliated manifold with boundary with foliation transverse to the boundary. We show that all these signatures coincide and a Hirzebruch formula is valid.
The paper examines the consistency of Lasso regression applied to signature analysis of time series data.
problem Consistency of Lasso regression in signature analysis of time series data.
method The paper studies the consistency of Lasso regression applied to signature analysis of time series data, both theoretically and numerically.
result The Lasso regression is consistent both asymptotically and in finite sample for certain types of time series and processes.
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.
New methods price American options in rough volatility models.
problem Pricing American options under rough volatility.
method Integrating deep-signature and signature-kernel learning into optimal stopping problem solutions.
result Performance comparison in rough Heston and rough Bergomi models.
Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.
problem Computing signatures of torus links and their cores.
method Use Neumann's equivariant signatures and rewrite Hirzebruch's formula for torus links (without cores) in terms of integral points in a parallelogram.
result Rewritten Hirzebruch's formula for torus links with cores using integral points in a parallelogram.
This paper extends the C*-signature to non-Witt spaces using noncommutative geometric methods.
problem Extending the signature to non-Witt spaces with noncommutative geometric methods.
method Noncommutative geometric methods, combinatorial framework, and comparison with analytical signature.
result Constructing the C*-signature on non-Witt spaces.