Thin position for knots, links, and graphs in 3–manifolds

Abstract

We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the ideas of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in 3-manifolds. The invariants in one family are similar to bridge number and the invariants in the other family are similar to Gabai's width for knots in the 3-sphere. The invariants in both families detect the unknot and are additive under connected sum and trivalent vertex sum.

Equivariant Heegaard genus of reducible 3-manifolds

“…In [36], Taylor and Tomova defined a set of operations on elements of H(M, T) and used them to define a partial order called thins to and denoted J → H. A minimal element in the partial order is said to be locally thin. The operations involved in the definition of the partial order are all versions of the traditional "destabilisation" and "weak reduction" of Heegaard splittings of 3-manifolds and "unperturbing" of bridge surfaces for links.…”

“…We do not need the precise definitions of the operations in this text, but we do need the following information. For our purposes, we group the operations into four categories (deferring to [36] (IV) untelescoping.…”

“…Proof. We note that in [36] saying that T is irreducible, by definition, means that (M,T) does not contain a once-punctured sphere. The existence of H given J is [36, theorem 6•17].…”

“…The requirement that x ω (S) ≥ 0 for S ∂M helps with some of our calculations. The other requirements help with our use of the material from [36,37]. We note that (W, ∅) is a nice orbifold.…”

“…In this section, we explain the minor adaptions needed to make it work for 3-orbifolds. Definition 4.1 (Taylor-Tomova [36]). Let M be a compact, orientable 3-manifold and T ⊂ M a spatial graph.…”