arXiv research

Jones polynomials have infinitely many roots of unity as zeros.

Continuity of roots of hyperbolic polynomials with smooth coefficients.

Polynomials' roots count tied to surface umbilics.

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…

Study on unimodality of plucking polynomial with delay function.

Categorifies colored Jones polynomial at roots of unity.

In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree.

The paper explores how polynomial roots and operator eigenvalues change with parameters.

We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada polynomials are dense in the complex plane.

The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.

For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.

It is proven that the volume of an infinitesimally flexible polyhedron in $R^3$ is a multiple root of its volume polynomial.

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…

Bounds on knot polynomials for Lie superalgebras of type I.

Continuity of polynomial roots shown for varying coefficients.

Study on quantum invariants of twist knots at specific roots of unity.

The "color" in the colored Jones polynomial is an integer parameter. In this paper, a periodic pattern of the values of the colored Jones polynomial at the second and the third roots of unity is found. If we substitute -1 to the colored Jones polynomial, the value is alternately 1 or the determinant of the given link. …

Study on quantum invariants of twist knots at specific roots of unity.

Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.

We introduce certain polynomials, so-called H.Weyl and H.Minkowski polynomials, which have a geometric origin. The location of roots of these polynomials is studied.

Study shows link polynomial evaluations from Heegaard Floer theory.

Study on Jones polynomials and their roots in the unit circle and complex plane.

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…

We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the $h$-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of p…

We show that the nonzero roots of the torsion polynomials associated to the infinite cyclic covers of a given compact, connected, orientable 3-manifold M are contained in a compact part of the complex plane a priori determined by M. This result is applied to prove that when M is closed, it dominates at most finitely ma…

We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruen…

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $Ω$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in …

We relate the jumps of the signature function of a link to the roots of its first nonzero higher Alexander polynomial.

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polyno…

Researchers map the fundamental group of polynomial strata to a braid group.

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for appro…

New link homologies categorify Jones polynomial at odd prime powers.

Covariance pooling is a feature pooling method with good classification accuracy. Because covariance features consist of second-order statistics, the scale of the feature elements are varied. Therefore, normalizing covariance features using a matrix square root affects the performance improvement. When pooling methods …

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $ωq$ where $ω$ is a complex root of unity, and under the substitution $q \to q^α$ where $α…

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

Unified quantum invariants via intersections of embedded Lagrangians.

Study Type $C$ skein modules using $Sp(2n)$ webs and construct transparent elements.

From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a generalization of the property H_r = (H_1)^r for special polynomials at q=1. We conjecture …

When using Traizet's regeneration technique to construct minimal surfaces, the simplest nontrivial configurations are given as the roots of polynomials that satisfy a hypergeometric differential equation. We exhibit examples of simple minimal surfaces exhibiting the same behavior.

This work improves polynomial approximations for functions with asymmetric behavior.

Researchers lift knot coloring polynomial to Habiro ring.

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$Σ$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove …

In this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture.

The paper finds formulas for a specific invariant order 7.

We find braid positive presentations for most L-space knots, except one, and explore related knot properties.

Study Alexander polynomials of special alternating links and generalize Fox's conjecture.

The topology of $SU(3)$-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to $-1$ are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate t…

New formula simplifies interior polynomial calculation.