Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections

Christian Blanchet (1) , Nathan Geer , Bertrand Patureau-Mirand (2) , Nicolai Reshetikhin

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group $U_qsl_2$ at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin-Turaev ribbon type functor for tangles with quandle representations. This theory applies to the unrestricted quantum group $U_qsl_2$ and produces an invariant of links with a gauge class of quandle representations.

Domains: Geometric Topology [math.GT]

Christian Blanchet, Nathan Geer, Bertrand Patureau-Mirand, Nicolai Reshetikhin. Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections. Selecta Mathematica (New Series), 2020, 26 (2), pp.19. ⟨10.1007/s00029-020-0545-0⟩. ⟨hal-01954883⟩