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 M1 is any finite-dimensional integrable type-1 simple highest-weight Uv,1-module whose weights lie in an admissible lattice and Mt=Φt(M1) is its transported module, then for every oriented link L, the corresponding normalized invariants satisfy Iv,tMt(L)=Iv,1M1(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.
Related
- Isomorphic classification of the coordinate algebras of the two-parameter braided quantum groups SL2.
- Ozone groups and centers of two-parameter quantum Heisenberg algebras
- Exotic knot traces and the epsilon-invariant
- A lower bound for the genus of a knot using the Links–Gould invariant
- Acyclic quantum cluster algebras via Hall algebras associated to cluster categories
BibTeX
@misc{fan2026knot,
title = {{The Knot Invariant Associated to Two-Parameter Quantum Algebras II}},
author = {Zhaobing Fan and Tianhao Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02481},
eprint = {2609.02481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02481},
}
RIS
TY - UNPB
TI - The Knot Invariant Associated to Two-Parameter Quantum Algebras II
AU - Zhaobing Fan
AU - Tianhao Zhu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02481
UR - https://arxiv.org/abs/2609.02481
AB - 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=Φ_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.
ER -