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Topics in Representation Theory
Topics in Representation Theory
The cabling power of HOMFLY knot invariants
The cabling power of HOMFLY knot invariants
Organizers
Speaker
Time
Friday, November 6, 2026 1:00 PM - 2:30 PM
Venue
A3-2a-302
Online
Zoom 361 038 6975
(BIMSA)
Abstract
Replacing a knot by a connected m-stranded cable in its tubular neighborhood allows us to construct, from a single knot invariant, such as the Jones or HOMFLY polynomial, an infinite sequence of new satellite invariants. To measure how many of these invariants are genuinely independent, we expand the knot polynomials into power series and consider the coefficients of degree-n monomials. The Melvin-Morton theorem shows that for the Jones polynomial cabling produces ⌊n/2⌋ independent invariants in degree n.
In this talk, I will present an exact, piecewise quadratic formula for the dimension of the space of knot invariants generated by all cablings of the two-variable HOMFLY polynomial, for which previously only estimates were known.
While the problem can be formulated in entirely elementary terms, its solution brings together a wide variety of techniques combination of techniques, including the Hopf algebra of chord diagrams, the Kontsevich integral, representation theory of gl(N), Vogel's universal Lie algebra, and a linear algebra problem solved using Hankel determinants and period integrals.
In this talk, I will present an exact, piecewise quadratic formula for the dimension of the space of knot invariants generated by all cablings of the two-variable HOMFLY polynomial, for which previously only estimates were known.
While the problem can be formulated in entirely elementary terms, its solution brings together a wide variety of techniques combination of techniques, including the Hopf algebra of chord diagrams, the Kontsevich integral, representation theory of gl(N), Vogel's universal Lie algebra, and a linear algebra problem solved using Hankel determinants and period integrals.