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We consider maps from braid groups to groups \Gamma_{n}^{k} and
representations of groups \Gamma_{n}^{k}. These groups considered naturally
when studying Voronoi diagrams on 2-surfaces. By studying dynamics of n points
Voronoi diagrams undergo a flip when four closest points are on the same cirle
(the other ones being outside this circle). Hence, the groups \Gamma_{n}^{4}
have 3(n\choose 4) generators  d_{ijkl} indexed by quadruples
of numbers which are split into two pairs of formally opposite ones.
The three types of relations in \Gamma_{n}^{k} are as follow:
each generator squared equals 1 (each generator is an involution);
every two generators sharing no more than two indices commute, and
(the pentagon relation)
 
d_{ijkl}d_{ijlm}d_{jklm}d_{ijkm}d_{iklm}=1.
The latter relation appeared in various forms in
many papers (Kapranov-Voevodsky), but not in group forms.
Groups \Gamma_{n}^{4} naturally appear when studying spaces of triangulations
of polygons and surfaces.
 
The recoupling theory used earlier by L.H.Kauffman and S.Lins for constructing
invariants of 3-manifolds gives invariants of braids through represenations of
the groups \Gamma_{n}^{4}. The examples caluclated belong to I.M.Nikonov.
 
On the other hand, it was known from cluster algebra theory
that the Ptolemy relation for inscribed quadrilaterals gives rise to
solutions of the Pentagon equation, hence
(Manturov, Fedoseev, Kim, Nikonov,
"Invariants and Pictures", WS,2020)
gives rise to braid invariants.
 
In the present talk we show how this Ptolemy relation satisfies
the Biedenharn-Elliot equation and hence gives rise
to (Viro-Turaev) invariants of 3-manifolds.
 
Hence, the groups \Gamma_{n}^{k} give rise to two new and deep connections
between braids and 3-manifolds, namely, the constructions used for invariants
of the former can be used for the latter and vice versa.
 
Many unsolved problems will be formulated.

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Seminars

on knot theory and related topics

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Chicago -8h

Moscow +0h

Beijing +5h

Seoul +6h

Tokyo +6h

Kim, Seongjeong,

e-mail :ksj19891120@gmail.com