Coefficients of Catalan states of lattice crossing. II: Applications of \(\Theta_A\)-state expansions (Q6564526)

This paper under review is one of a series of articles devoted to investigating the coefficients of Catalan states of a lattice crossing \(L(m, n)\). When \(m=n\), the Kauffman bracket skein module of a lattice crossing appears naturally in the calculation of the \(n\)-th colored Jones polynomial. Due to the existence of Jones-Wenzl idempotents, the coefficients of states with no returns are of particular interest. In previous work, several new methods have been proposed to study these coefficients, e.g. the plucking polynomial of rooted trees and the \(\Theta_A\)-state expansion. In this paper, by using the \(\Theta_A\)-state expansion, the authors provide a factorization theorem for the plucking polynomial and prove the so-called removable arc theorem and vertical factorization theorem. These results enhance our understanding of the coefficients of Catalan states of lattice crossings.\N\NFor Part I see [\textit{M. K. Dabkowski} and \textit{C. Wu}, ``Coefficients of Catalan States of Lattice Crossing I: $\Theta_{A}$-state Expansion'', Preprint, \url{arXiv:2203.00061}].