It was previously shown by the first author that every knot in [Formula: see text] is ambient isotopic to one component of a twocomponent, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot [Formula: see text] to be the minimum volume of any link [Formula: see text] in a natural class of alternating, hyperbolic links such that [Formula: see text] is ambient isotopic to a component of [Formula: see text]. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.
Guts, Volume and skein modules of 3manifolds
We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3manifolds. 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 3manifold is a thickened surface, this Kauffman bracket function leads to a Jonestype polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2sided 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.
 Award ID(s):
 2004155
 Publication Date:
 NSFPAR ID:
 10233623
 Journal Name:
 Fundamenta Mathematicae
 ISSN:
 17306329
 Sponsoring Org:
 National Science Foundation
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