Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
arXiv:2205.12238 · doi:10.1140/epjc/s10052-022-10851-7
Abstract
For a peculiar family of double braid knots there is a remarkable factorization formula for the coefficients of the differential (cyclotomic) expansion (DE), which nowadays is widely used to construct the exclusive Racah matrices and in arbitrary representations. The origins of the factorization remain obscure and the special role of double braids remains a mystery. In an attempt to broaden the perspective, we extend the family of double braids to antiparallel triple pretzels, which are obtained by the defect-preserving deformation from the trefoil and all have defect zero. It turns out that factorization of DE coefficients is violated quite strongly, still remains described by an elegant formula, at least for all symmetric representations.
References in corpus (40)
- HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
- The colored Jones function is q-holonomic
- Superpolynomials for toric knots from evolution induced by cut-and-join operators
- Colored HOMFLY polynomials of knots presented as double fat diagrams
- Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
- Evolution method and "differential hierarchy" of colored knot polynomials
- Colored HOMFLY polynomials for the pretzel knots and links
- Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
- Asymptotics of the colored Jones function of a knot
- Superintegrability summary
- A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
- Differential hierarchy and additional grading of knot polynomials
- On colored HOMFLY polynomials for twist knots
- Tabulating knot polynomials for arborescent knots
- Superintegrability of Kontsevich matrix model
- On the defect and stability of differential expansion
- Colored HOMFLY polynomial via skein theory
- Differential expansion and rectangular HOMFLY for the figure eight knot
- The colored HOMFLYPT function is -holonomic
- Factorization of differential expansion for antiparallel double-braid knots
- On rectangular HOMFLY for twist knots
- Differential expansion for link polynomials
- The C-polynomial of a knot
- Eigenvalue hypothesis for multi-strand braids
- Rectangular superpolynomials for the figure-eight knot
- Perspectives of differential expansion
- Generalized Q-functions for GKM
- On Factorization of Generalized Macdonald Polynomials
- Extension of KNTZ trick to non-rectangular representations
- Tensorial generalization of characters
- On exclusive Racah matrices for rectangular representations
- Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
- KNTZ trick from arborescent calculus and the structure of differential expansion
- Chern-Simons perturbative series revisited
- Pentad and triangular structures behind the Racah matrices
- Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
- Algebra of quantum -polynomials
- Evolution properties of the knot's defect
- The non-commutative -polynomial of twist knots
- Overview of knot invariants at roots of unity