arXiv subjects

Ross Elliot

Publications and source records attributed to Ross Elliot.

3 recordsLinked to original sources

arXiv2015

Exceptional knot homology

The goal of this article is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called "hyperpolynomials" that address the "problem of negative coefficients" often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau-Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.

math.QA↗

arXiv2015

Refined composite invariants of torus knots via DAHA

We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries of the DAHA-superpolynomials of torus knots. The latter are conjecturally related to the HOMFLY-PT homology; such a connection is a challenge in the theory of the annulus. At the end, we construct two DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E and closely related to the exceptional Deligne-Gross series of root systems; this theme is of experimental nature.

math.QA↗

arXiv2009

Alexander Polynomials of Periodic Knots: A Homological Proof and Twisted Extension

In 1971, Kunio Murasugi proved a necessary condition on a knot's Alexander polynomial for that knot to be periodic of prime power order. In this paper I present an alternate proof of Murasugi's condition which is subsequently used to extend his result to the twisted Alexander polynomial.

math.AT↗