Bridge and pants complexities of knots

Abstract

We modify an approach of Johnson to define the distance of a bridge splitting of a knot K in a 3-manifold M using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of (M, K). We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of K. Mathematics Subject Classification 57M25, 57M27, 57M50 (primary). 44 ALEXANDER ZUPAN Theorem 1.1. Let K be a knot in a closed, orientable 3-manifold M, where M has no S 2 × S 1 summands, let Σ be a splitting surface for (M, K), and let Σ h c be an (h, c)-stabilization of Σ. Then the limits lim h,c→∞ B(Σ) and lim h,c→∞ B P (Σ) exist. Moreover, they do not depend on Σ and thus define invariants B(M, K) and B P (M, K) of the pair (M, K).The pants complex P (Σ) of a surface Σ is itself an interesting object of study; for instance, Brock has shown that it is quasi-isometric to the Teichmüller space T (Σ) equipped with the Weil-Petersson metric [2], and Souto has revealed other surprising connections between pants distance and the geometry of certain 3-manifolds [23]. We exploit a theorem of Guéritaud and Futer [3] to prove the following theorem.Theorem 1.2. Suppose that K is a hyperbolic 2-bridge knot, Σ is a (0, 2)-splitting surface for K, v 3 is the volume of a regular ideal tetrahedron, and v 8 is the volume of a regular ideal octahedron. ThenThe paper proceeds in the following manner: in Section 2, we provide background information and include a proof of the analog of the Reidemeister-Singer theorem for bridge splittings in arbitrary manifolds. In Section 3, we introduce the curve, dual-curve, and pants complexes, and in Section 4 we define the distance of a bridge splitting and prove several basic facts about this distance. In Section 5, we use the distance of the previous section to define bridge and pants complexity, and prove the main theorem. In Section 6, we demonstrate several properties of these new invariants and, in Section 7, we define the concept of a critical splitting to provide explicit calculations in Section 8. Finally, in Section 9, we discuss connections between pants distance and hyperbolic volume, and in Section 10 we include several interesting open questions.

Bridge trisections of knotted surfaces in 4-manifolds

“…The other operation increases the number of arcs in τ i : Any bridge splitting of (Y, K) obtained by connected summing with the standard (0, 2)-splitting of the unknot in S 3 is said to be obtained by elementary perturbation, and any splitting obtained by a finite number of elementary perturbations is called a perturbation. As with Heegaard splittings and trisections, it is known that if two bridge splittings of (Y, K) have the same underlying Heegaard splitting, then there is a bridge splitting that is isotopic to perturbations of each of the original splittings, called a common perturbation [Hay98,Zup13], and thus any two bridge splittings of (Y, K) become isotopic after stabilizations and perturbations. The purpose of this section is to define perturbations for generalized bridge trisections and lay out steps toward a proof of a corresponding uniqueness theorem in this setting.…”

Bridge spectra of iterated torus knots

“…If Σ 2 is an elementary perturbation of Σ, then Σ 2 is a (g, b + 1)-surface. Given any two bridge surfaces Σ 1 and Σ 2 for (M, J), there is a third bridge surface Σ * which can be obtained from either Σ i by elementary perturbations and stabilizations [26].…”

Bridge trisections of knotted surfaces in 𝑆⁴

“…In Theorem 2.2 of [32], it is shown that any two bridge splittings for a link L in S 3 have a common perturbation. Although Theorem 2.2 is stated for B = S 3 , the verbatim proof suffices in the case that ∂B, ∂T = ∅, and so we do not include it here.…”

“…Theorem 7.1. [32] Suppose that Σ 1 and Σ 2 are bridge splittings for a tangle T in a punctured 3-sphere B. Then there is a surface Σ * which is a perturbation of both surfaces.…”