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On the length conjecture for twist knots

Samuel Panitch, Mauricio Romo

math.GTarXiv:2609.02706

Abstract

Using the newly developed 3d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the 3d quantum trace map behaves naturally with respect to Dehn filling.

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