Thin position for knots and 3–manifolds: a unified approach

Abstract

We unify the notions of thin position for knots and for 3-manifolds and survey recent work concerning these notions. 57M27, 57M25; 57N10

A note on the nearly additivity of knot width

“…In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover (abbreviate it as DBC, and call the method the H-S method ) of (S 3 , k) (see section 3) and they proved that for 2-bridge knots and 3-bridge knots in thin position DBC inherits thin manifold decomposition (note that a knot in a thin position may not induce a thin manifold decomposition by the H-S method in general, see [8] and [6].) Indeed, if k is a non-prime 3bridge knot, then k = k 1 #k 2 for 2-bridge knots k 1 and k 2 , and thin position of k is the sum of k 1 and k 2 by stacking a thin position of one of the Date: 13, Jan, 2010.…”

Dehn surgery on complicated fibered knots in the 3-sphere

“…Step 1: First place K in thin position with respect to the function j. For details on this procedure, see for example [9]. Let ∆ r be the graph of intersection between T r and F (= F 0 ) contained in T r .…”

Width complexes for knots and 3-manifolds

“…See for instance [Howards et al 2007]. In particular, width is well-defined on equivalence classes of representatives and hence on vertices.…”

“…Casson and C. Gordon [1986] have exhibited 3-manifolds that possess strongly irreducible Heegaard splittings of arbitrarily high genus. (See also [Kobayashi 1990;1992].) Such Heegaard splittings correspond to local minima in the width complexes of these 3-manifolds.…”