arXiv research

Study polynomial trace identities in $SL(2,\IC)$ using quaternion algebras.

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and $n-2$ polynomial functions defined recursively.

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chroma…

The study computes trace fields and minimal polynomials for specific knots and links.

We show that the Alexander-Conway polynomial Delta is obtainable via a particular one-variable reduction of each two-variable Links-Gould invariant LG^{m,1}, where m is a positive integer. Thus there exist infinitely many two-variable generalisations of Delta. This result is not obvious since in the reduction, the repr…

The paper explores orthogeodesics on hyperbolic surfaces and their integer traces.

Let $M$ be a closed connected smooth manifold and $G=\textmd{Diff}_0(M)$ denote the connected component of the diffeomorphism group of $M$ containing the identity. The natural action of $G$ on $M$ induces the trace homomorphism on homology. We show that the image of trace homomorphism is annihilated by the subalgebra o…

Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple …

We construct a new inductive basis of the Birman-Murakami-Wenzl algebra. Using it, we provide a new proof of the existence of the Markov trace on the BMW algebras affording the two-variable Kauffman polynomial. We prove also that all the transverse Markov traces on the BMW algebras are determined by the self-linking nu…

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…

Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In th…

Defines new algebras for virtual link invariants, matching known polynomials.

We prove that the trace of the logarithmic term of the Toeplitz kernel on a contact manifold is a contact invariant, generalizing K. Hirachi's invariant for the Szego kernel on a CR manifold. When the base manifold is the three-sphere, this vanishes identically.

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit…

We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial a…

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…

Paper explores constructing knots with identical traces using specific operations.

Spectral sequence connects knot homologies via algebraic geometry.

Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)

To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral mo…

We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant alpha of triangles in the complex hyperbolic plane. The main result of the p…

In this paper we study properties of the Markov trace ${\rm tr}_d$ and the specialized trace ${\rm tr}_{d,D}$ on the Yokonuma-Hecke algebras, such as behaviour under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\rm tr}_{d,…

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional …

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore,…

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize …

We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on $T^*M$ is made into a space of (full) symbols of operators acting on forms on $M$. This gives rise to the composition of symbols, which is a deformation of the (``super'')commutative multiplication of forms. The …

We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schroedinger operators and Schroedinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda_{N+1} in terms of the lower spectrum, bou…

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.

Extends quantum trace map to SL3(C) for 3D surfaces.

We construct link invariants using the $D_{2n}$ subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories inv…

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.

We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a $q$-state Potts model defined as a planar graph with weighted edges that corresponds to the knot. For any integer $q$, we cast…

Non-injectivity proven for trace map on character varieties.

Quantum trace map defines invariants for knots and links, confirming a length conjecture.

A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformations. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometri…

The paper connects curvature data to polynomial coefficients in gluing formulas.

Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we define a numerical index $\operatorname{index}_g(D)$, in terms of a parametrix for $D$ and …

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maxim…

Given a planar curve singularity, we prove a conjecture of Oblomkov-Shende, relating the geometry of its Hilbert scheme of points to the HOMFLY polynomial of the associated algebraic link. More generally, we prove an extension of this conjecture, due to Diaconescu-Hua-Soibelman, relating stable pair invariants on the c…

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

Unified formula for higher traces of linear maps on finite-dimensional normed spaces.

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…