New York Journal of Mathematics
Volume 29 (2023), 1075-1096


Charles Frohman, Joanna Kania-Bartoszynska, and Thang Le

Skein algebras of three-manifolds at 4th roots of unity

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Published: October 27, 2023.
Keywords: Skein algebras, Kauffman bracket, quantum topology.
Subject [2010]: 57K31.

This paper introduces an algebra structure on the part of the skein module of an arbitrary 3-manifold M spanned by links that represent 0 in H1(M;Z2) when the value of the parameter used in the Kauffman bracket skein relation is equal to
± i. It is proved that if M has no 2-torsion in H1(M;Z) then those algebras,
K± i0(M), are naturally isomorphic to the corresponding algebras when the value of the parameter is ± 1. This implies that the algebra K± i0(M) is the unreduced coordinate ring of the variety of PSL2(C)-characters of π1(M) that lift to SL2(C)-representations.


This material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Author information

Charles Frohman
Department of Mathematics
The University of Iowa
Iowa City, IA 52242, USA


Joanna Kania-Bartoszynska
National Science Foundation
Arlington, VA, 22230, USA


Thang Le
School of Mathematics
Geogia Institute of Technology
686 Cherry St NW, Atlanta, GA 30332, USA