On the defect and stability of differential expansion

Abstract

Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in r-th symmetric representation does not change with r, if it is large enough. This fact reflects the non-trivial and previously unknown properties of the differential expansion, and it turns out that from this point of view there are universality classes of knots, characterized by a single integer, which we call defect, and which is in fact related to the power of Alexander polynomial.HOMFLY polynomials are Wilson-loop averages in 3d Chern-Simons theory [1], which in this simplest model depend only on the topology of the Wilson line (knot). Therefore one can separate and study the group-theory properties of observables -and this is a non-trivial and very interesting problem, for a brief summary of results see [2]. From the quantum field theory perspective knot polynomials are direct generalization of conformal blocks, and this relation [3] , which was temporarily postponed because of the insufficient "experimental" material.In this note we describe empirically obtained properties of these expansions for symmetric representations [r] (where r is the length of the single-line Young diagram). It looks like there are different universality classes of such expansions, characterized by a single integer, which we call "defect" δ K . Moreover, these newly observed properties allow to identify 2(δ K + 1) with the power of Alexander polynomial and lead to a peculiar stability property of symmetrically colored HOMFLY for large enough r: what stabilizes is not the polynomial itself, but the set of its coefficients -i.e. something like the "coordinates" g r,j , introduced in [24]. Theoretical analysis of these observations, proofs and extension to non-(anti)symmetric representations are beyond the scope of the present text.

Colored HOMFLY and generalized Mandelbrot set

“…It is already sufficient that too well-developed branches of theory are now connected, and the exchange of ideas and concepts can now take place. Moreover, already the simplest examples we look at below, demonstrate that resultants know a lot about other non-trivial properties of knot invariants -like defects [31] of differential expansions [32][33][34][35] -what supports our belief that the relation to Mandelbrot world is a deep property and not just an amusing observation or a joke. This means that further work is needed to appreciate the real significance of this newly-emerging research direction, which finally extends the still-mysterious Mandelbrot property beyond a single example.…”

“…This is not a surprise, because at this point the roots of all symmetricallycolored H r are described by (6.2). Namely, they are all expressed through the roots Q α of the Alexander polynomial, whose degree is related to the defect δ K of the differential expansion [31]:…”

Checks of integrality properties in topological strings

“…This formalism is successfully developed in [31] and [33] and has already allowed us to find the inclusive Racah matrices for R = [2,2] and even R = [3,1]. In combination with the differential expansion method [142][143][144][145][146][147][148][149][150], this provides extensions to other rectangular representations. Further progress (for other nonrectangular representations) is expected after developing the ∆-technique briefly outlined in [33].…”

Factorization of differential expansion for antiparallel double-braid knots

“…Thus Alexander polynomial Al (m,n) [1] = 1 + mn{q} 2 has degree one, and defect [110] of the differential expansion is zero for the entire family. This means that we can use the conjecture of [105] for the shape of differential expansions for rectangular representations of the defect-zero knots.…”