Cyclotomic expansions of colored $SU(n)$ invariants of two-strand torus knots and Bailey transforms

Abstract

Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen--Liu--Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots $T(2,2p+1)$ and give a Bailey theoretic realization of the cyclotomic coefficients. For $n\geq2$ and $K_p=T(2,2p+1)$, the colored $SU(n)$ invariants admit an expansion $$ J_N^{SU(n)}(K_p;q) = \sum_{m=0}^{N} \left(\prod_{j=0}^{m-1}\{N-j\}\{N+n+j\}\right) H_m^{(n,p)}(q), $$ where $H_m^{(n,p)}(q)\in\mathbb Z[q^{\pm1}]$ is independent of the color $N$. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin--Zheng formula for $T(2,2p+1)$ then gives a well-poised Bailey kernel. A terminating very-well-poised ${}_6ϕ_5$ summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen--Liu--Zhu $SU(n)$ volume conjecture for $T(2,2p+1)$.