Connecting essential triangulations I: via 2-3 and 0-2 moves
arxiv(2024)
摘要
Suppose that M is a compact, connected three-manifold with boundary. We
show that if the universal cover has infinitely many boundary components then
M has an ideal triangulation which is essential: no edge can be homotoped
into the boundary. Under the same hypotheses, we show that the set of essential
triangulations of M is connected via 2-3, 3-2, 0-2, and 2-0 moves.
The above results are special cases of our general theory. We introduce
L-essential triangulations: boundary components of the universal cover
receive labels and no edge has the same label at both ends. As an application,
under mild conditions on a representation, we construct an ideal triangulation
for which a solution to Thurston's gluing equations recovers the given
representation.
Our results also imply that such triangulations are connected via 2-3, 3-2,
0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that
Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of
essential triangulation.
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