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.
On rectangular HOMFLY for twist knots
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abstract
As a new step in the study of rectangularly-colored knot polynomials, we reformulate the prescription of arXiv:1606.06015 for twist knots in the double-column representations $R=[rr]$ in terms of skew Schur polynomials. These, however, are mysteriously shifted from the standard topological locus, what makes further generalization to arbitrary $R=[r^s]$ not quite straightforward.
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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.