Defect and degree of the Alexander polynomial

Abstract

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $q^{\pm 2}$ of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of $C$-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

arXiv comment21 pages

References in corpus (33)

• Integrability and Matrix Models
• On Khovanov's categorification of the Jones polynomial
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• sl(3) link homology
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Evolution method and "differential hierarchy" of colored knot polynomials
• Asymptotics of the colored Jones function of a knot
• Superintegrability summary
• On universal knot polynomials
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Towards effective topological field theory for knots
• Differential hierarchy and additional grading of knot polynomials
• On the defect and stability of differential expansion
• Colored HOMFLY polynomial via skein theory
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Superintegrability as the hidden origin of Nekrasov calculus
• The C-polynomial of a knot
• Are there p-adic knot invariants?
• Perspectives of differential expansion
• Extension of KNTZ trick to non-rectangular representations
• Factorization of differential expansion for non-rectangular representations
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Algebra of quantum $\mathcal{C}$-polynomials
• Evolution properties of the knot's defect
• Invariants of knots and links at roots of unity
• Are Maxwell knots integrable?

Cited by in corpus (2)

• Towards the theory of Yangians
• Machine learning of the well known things