The Knot Invariant Associated to Two-Parameter Quantum Algebras II

Abstract

Fan, Ma, and Xing constructed oriented-tangle invariants from finite-type two-parameter quantum algebras. In this paper, we construct an explicit parameter-transport comparison between two-parameter modules and their corresponding one-parameter modules, and we verify this comparison on every elementary oriented-tangle operator. After extending scalars to a common coefficient field, we prove that if $M_1$ is any finite-dimensional integrable type-1 simple highest-weight $U_{v,1}$-module whose weights lie in an admissible lattice and $M_t={\Phi }_t(M_1)$ is its transported module, then for every oriented link $L$, the corresponding normalized invariants satisfy $I_{v,t}^{M_t}(L)=I_{v,1}^{M_1}(L)$. Consequently, the two invariants assign equal values to exactly the same pairs of oriented links and therefore have the same distinguishing power; this includes the vector representations of the finite classical types A, B, C, and D. Sean Clark's comparison of ordinary and super quantum knot invariants is obtained as a specialization of the same transport principle in which explicit scalar factors are allowed.

29 pages, 4 figures