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.

6 pages

References in corpus (4)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Equations on knot polynomials and 3d/5d duality

Cited by in corpus (26)

• On universal knot polynomials
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Differential expansion for link polynomials
• Nimble evolution for pretzel Khovanov polynomials
• Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?
• Rectangular superpolynomials for the figure-eight knot
• Are there p-adic knot invariants?
• Perspectives of differential expansion
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• Knot polynomials for twist satellites
• HOMFLY for twist knots and exclusive Racah matrices in representation [333]
• Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Pentad and triangular structures behind the Racah matrices
• Difference of mutant knot invariants and their differential expansion
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Algebra of quantum $\mathcal{C}$-polynomials
• Defect and degree of the Alexander polynomial
• Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed $\mathcal{R}$-matrix approach
• Evolution properties of the knot's defect
• Towards the theory of Yangians
• Matrix model and dimensions at hypercube vertices
• Colored HOMFLY and Generalized Mandelbrot set