A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Polynomially parameterizes knots and spheres, proving analogous results.
For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ\times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove re…
The paper connects knot theory and cluster algebras via dimer face polynomials.
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…
New link polynomials linked to cluster theory.
Developed algorithms to compute three polynomial invariants of veering triangulations.
Quantum polynomials are derived from a specific tribracket structure.
Study improves HOMFLY polynomial coefficients for positive braid links.
Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…
New knot polynomials yield simple results modulo primes.
New knot polynomials reveal patterns and mutations.
The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…
We show that for a twist knot, the A-polynomial can be obtained from recurrences for the summand in Masbaum's formula of the colored Jones polynomial. Our result supports the AJ conjecture due to S.Garoufalidis.
Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots w…
New results on algebraic knots with Brieskorn polynomials.
We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, unde…
It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describi…
Homology handles with trivial Alexander polynomial bound a 3D sphere.
Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.
In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe polynomial is also discussed.
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative re…
We generalize the index polynomial invariant to the case of virtual tangles. Three polynomial invariants result from this generalization; we give a brief overview of their definition and some basic properties.
Computes A-polynomials of knots from Whitehead sister link fillings.
The paper constructs biharmonic maps between spheres using polynomial maps.
Paper extends Cohen's method to compute Jones polynomial for certain braid subfamilies.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
Relative Thom polynomials for maps around boundaries established.
The Penrose-Kauffman polynomial connects knot theory to graph coloring.
We introduce a polynomial invariant of graphs on surfaces, , generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for , analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statisti…
Simplified A-polynomial calculation for twisted knots.
As a generalization of a fundamental result about the Alexander polynomial of links, we give a description of a Torres condition for the twisted Alexander polynomial of links associated to a unimodular representation.
Algorithm learns polynomial transformations of Gaussian distributions.
Study groups with polynomial growth, finding structure and applications.
We give a characterization for the Alexander Polynomials of closed orientable 3-manifolds M with first Betti number 1, as well as some partial results for the characterization problem for M having first Betti number > 1. We first prove an analogue of a theorem of Levine: that the product of an Alexander polynomial of M…
Recently, it has been shown that the Jones polynomial, in [LS19], and the Alexander polynomial, in [NT18], of rational knots can be obtained by specializing -polynomials of cluster variables. At the core of both results are continued fractions, which parameterize rational knots and are used to obtain cluster variabl…
New formulas derived for Jones polynomial of rational links.
Strongly polynomial algorithm for approximate Forster transforms and halfspace learning.
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.
We show that the standard stochastic gradient decent (SGD) algorithm is guaranteed to learn, in polynomial time, a function that is competitive with the best function in the conjugate kernel space of the network, as defined in Daniely, Frostig and Singer. The result holds for log-depth networks from a rich family of ar…
Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.
We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollobás-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ri…
Characterizes values at infinity for real polynomial maps with 2D fibers.
We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.
The study computes trace fields and minimal polynomials for specific knots and links.
It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model…
New bound on Jones polynomial for specific positive links.
This study shows the moment-SOS hierarchy converges in polynomial optimization over product of spheres.