Khovanov polynomials for satellites and asymptotic adjoint polynomials

TL;DR

This paper explicitly computes Khovanov polynomials for satellite knots, revealing a deformed linear combination structure and dependence on the pattern and companion invariants, extending quantum group insights.

Contribution

It extends quantum group decomposition to Khovanov polynomials for satellite knots, showing a new structure and dependence on pattern and companion invariants.

Findings

Khovanov polynomials for satellites can be expressed as deformed linear combinations of pattern and companion invariants.

The satellite polynomial varies smoothly with the pattern, with a jump at a critical point related to the s-invariant.

The approach extends quantum group properties to Khovanov polynomials, previously known for HOMFLY polynomials.

Abstract

We compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial "smoothly" depends on the pattern but for the "jump" at one critical point defined by the s-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and tau-invariant for the same kind of satellites.