Alexander polynomials appear in two opposite roles in two Kashaev phases of Chern-Simons theory due to co-existing branches in the quasiclassical limit with non-trivial versus vanishing classical actions.
Evolution method and "differential hierarchy" of colored knot polynomials
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials on such evolution parameters is very easy to find. We apply this evolution method to study of the families of knots and links which include the cases with just two parallel and anti-parallel strands in the braid, like the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand links. When the answers were available before, they are immediately reproduced, and an essentially new example is added of the "double braid", which is a combination of parallel and anti-parallel 2-strand braids. This study helps us to reveal with the full clarity and partly investigate a mysterious hierarchical structure of the colored HOMFLY polynomials, at least, in (anti)symmetric representations, which extends the original observation for the figure-eight knot to many (presumably all) knots. We demonstrate that this structure is typically respected by the t-deformation to the superpolynomials.
fields
hep-th 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
In the Kashaev limit the non-homogeneous quantum A-polynomial splits into phases tied to zero action and deformed hyperbolic volume, with a byproduct expectation that the classical A-polynomial at L=1 is proportional to the Alexander polynomial.
citing papers explorer
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Two roles of Alexander in two Kashaev phases
Alexander polynomials appear in two opposite roles in two Kashaev phases of Chern-Simons theory due to co-existing branches in the quasiclassical limit with non-trivial versus vanishing classical actions.
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More on Kashaev limits of the quantum $A$-polynomials
In the Kashaev limit the non-homogeneous quantum A-polynomial splits into phases tied to zero action and deformed hyperbolic volume, with a byproduct expectation that the classical A-polynomial at L=1 is proportional to the Alexander polynomial.