Extend determinant representations beyond rectangular colors and current knot families

Extend the determinantal representations of colored HOMFLY-PT polynomials to non-rectangular representations and to more general knot families.

Background

The paper derives determinant formulas for HOMFLY-PT polynomials of twist and antiparallel double-braid knots when the color is a rectangular representation. These formulas arise by expressing differential-expansion weights as products of paired minors and resumming them with the Cauchy–Binet identity.

The abstract explicitly identifies extending the construction to non-rectangular representations and to broader knot families as unresolved. The later discussion explains that non-rectangular colors involve additional evolution channels that are not captured by the exterior-power mechanism established for rectangular representations.

References

Extensions to non-rectangular representations and to more general knot families remain open problems.

— Determinantal representation of colored HOMFLY for double braids  (2609.09437 - Kononov et al., 8 Sep 2026) in Abstract

Extending the determinant formula requires explicit matrices replacing (\ref{eq:double-matrix-entries}) and a factorization replacing (\ref{eq:Z-trefoil-factorization}); these remain to be constructed for general non-rectangular colors.

— Determinantal representation of colored HOMFLY for double braids  (2609.09437 - Kononov et al., 8 Sep 2026) in Section 5, subsection “Non-rectangular representations”