Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
Classifies trees with strictly unimodal q-polynomials.
problem Classifying rooted trees with strictly unimodal q-polynomials.
method Classification based on plucking polynomials and criteria for trapezoidal shapes.
result Generalizes results on strict unimodality of q-binomial coefficients.
Paper proves BGW tau-function can be represented as Q-polynomials.
problem Enumerative geometric interpretations of BGW tau-function.
method Proves BGW tau-functions are hypergeometric tau functions of BKP hierarchy.
result Original BGW tau-function can be represented as a linear combination of Schur Q-polynomials.
Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of Q-polynomial is less than its determinant. We give a refinement of their evaluation.
New formulas for knot polynomial evaluations from covering spaces.
problem Evaluating knot polynomials uniquely from covering spaces.
method Using singular determinants and linking pairings.
result Explicit formulae for Jones and Q-polynomial evaluations. This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; how…
Let G be a signed graph. Let G^ be the graph obtained from G by replacing each edge e by a chain or a sheaf. We first establish a relation between the Q-polynomial of G^[6] and the W-polynomial of G [9]. Two special dual cases are derived from the relation, one of which has been studied in [8]…
Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of such a ring defines new link invariants. In this sense, they produce the well…
The paper studies polynomials and ideals from colored Jones polynomials for links.
problem Understanding the structure of colored Jones polynomials for links.
method Investigates commutative and noncommutative ideals derived from colored Jones polynomials.
result Formulates the link version of the AJ conjecture.
New invariant for rooted trees with connections to knot theory.
problem No specific problem stated, focuses on a new mathematical invariant.
method Description of a new invariant of rooted trees.
result The invariant has connections to knot theory and homological algebra.
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. Researchers found a q-series identity for a specific knot using sl3 representations.
problem Finding a q-series tail for sl3 colored Jones polynomials. method Explicit formulas for the tail of sl3 colored Jones polynomials for (2,2m)-torus links. result An identity of q-series connecting sl3 colored Jones polynomials and Ramanujan false theta function. Three-layer neural networks learn hierarchical polynomial functions efficiently.
problem Learning hierarchical polynomial functions with three-layer neural networks.
method Layerwise gradient descent on square loss, focusing on feature learning.
result Achieves optimal sample complexity for learning hierarchical polynomials.