The authors construct positive decompositions of fundamental HOMFLY polynomials in variables φ, φ̄, D for arbitrary knots, not just bipartite ones, and give a criterion to detect when such a decomposition conceals a bipartite realization.
Khovanov polynomials for satellites and asymptotic adjoint polynomials
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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.
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Bipartite expansion beyond biparticity
The authors construct positive decompositions of fundamental HOMFLY polynomials in variables φ, φ̄, D for arbitrary knots, not just bipartite ones, and give a criterion to detect when such a decomposition conceals a bipartite realization.