Geometric estimates from spanning surfaces

Abstract

We derive bounds on the length of the meridian and the cusp volume of hyperbolic knots in terms of the topology of essential surfaces spanned by the knot. We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound. As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots with meridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.

A Survey of Hyperbolic Knot Theory

“…Another instance where Theorem 5.3 applies is to knots with a pair of essential spanning surfaces S 1 and S 2 ; in this case the surface S is taken to be the disjoint union of the two spanning surfaces. The following appears in [24].…”

Geometry of alternating links on surfaces

“…Compare Corollary 7.2 to Theorem 7.4. The hypotheses are slightly weaker for the corollary, and indeed, Burton and Kalfagianni's geometric estimates on slope length [12] apply anytime a knot is hyperbolic with two essential spanning surfaces. Thus it applies to hyperbolic adequate knots, for which r(π(K), F ) = 2.…”

“…We will apply the 6-Theorem to bound exceptional fillings. First, we need to bound the lengths of slopes on weakly generalised alternating links, and this can be done by applying results of Burton and Kalfagianni [12]. Burton and Kalfagianni give bounds on slope lengths for hyperbolic knots that admit a pair of essential surfaces.…”

“…If |q| > 5.373(1 − χ(F )/c) then the Dehn filling of X(K) along slope σ yields a hyperbolic manifold.Proof. As in[12], the Euclidean area of a maximal cusp C in a one-cusped 3-manifold X satisfiesarea(C) ≤ (σ) (µ) |i(σ, µ)| .Work of Cao and Meyerhoff[14] implies area(C) ≥ 3.35. Plugging in the bound on (µ) from Theorem 7.1, along with the fact that |i(σ, µ)| = |q|, and solving for (σ), we obtain (σ) ≥ 3.35c 3(c − χ(F )) |q|.…”

State graphs and fibered state surfaces

“…State surfaces are known to be related to knot and link polynomials; see for example [4,7,18]. State surfaces are also related to the hyperbolic geometry of the link complement for hyperbolic links; see for example [1,2,5,7,8,9]. Because of their connections with geometry, topology, and link invariants, these surfaces have recently become objects of much study in knot theory.…”