2020/10/13 by Bavier, Brandon, Kalfagianni, Efstratia
#57K10 #57K14 #57K31 #57K32 #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2010.06559
We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link.