Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundarystabilized copy of P or the distance d.P / Ä 2genus.Q/.More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either P and Q can be well-separated or P and Q are isotopic or d.P / Ä 2genus.Q/. 57N10; 57M50
Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M , P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or d.K; P / Ä 2 .Q K/. If K is not a 2-bridge knot, then the result holds even if K is removable with respect to Q. As a corollary we show that if a knot in S 3 has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.57M25, 57M27, 57M50
Abstract. Any 2-bridge knot in S 3 has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.
We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2) additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots.1 Technically, it is the quantity that is one less than bridge number which is additive under connected sum. 1 lifts to a Heegaard surface for the resulting 3-manfiold. We can define Heegaard surfaces for 3manifolds with boundary by considering compressionbodies in place of handlebodies (see [26]; this is also explained more below.) The Heegaard genus of a 3-manifold (possibly with boundary) M , denoted g(M ), is the smallest possible genus of a Heegaard surface in M . A closed, orientable 3-manifold M has g(M ) = 0 if and only if M = S 3 . Like bridge number, Heegaard genus is additive under connected sum [11]. There is a vast literature on Heegaard surfaces and their usefulness is now well-established.Tunnel number is a somewhat more recent knot invariant defined by Clark in 1980 [4]. The tunnel number t(K) of a knot K ⊂ S 3 equals the minimal number of arcs which need to be added to the knot so that the exterior is a handlebody. While tunnel number may at first appear unnatural, it is closely connected to Heegaard genus. Indeed, t(K) is one less than the Heegaard genus of the exterior of K. Largely because of this connection, tunnel number has been extensively studied. Unlike bridge number and Heegaard genus, however, tunnel number behaves erratically under connect sum (see, for example, [13,19,20,23]). In this paper we seek to correct this by defining a knot invariant (called "net extent") which is intimately related to Heegaard genus, bridge number, higher genus bridge number and tunnel number. Unlike the latter two invariants, however, it both detects the unknot and is additive under connect sum.The width of a knot was originally defined by Gabai [7] as a tool to prove property R. Various other applications of width quickly emerged [37,38]. Because of its utility it began to be studied it its own right, the main question being its additivity under connect sum 2 [1,25,30]. Eventually, Blair and Tomova [2] proved that there exist families of knots for which Gabai width is not additive. We introduce a second invariant, also called width, applicable to most any knot in many 3-manifolds. It is a slight variation of Gabai width when restricted to spheres transverse to knots in S 3 but is additive under connect sum. Section 6 below considers the relationship between our width and Gabai width. 1.2. The invariants. For our purposes, a (3-manifold, graph) pair (M, T ) consists of a compact orientable 3-manifold M (possibly with boundary) and a properly embedded graph T ⊂ M .Running Assumption: T has no vertices of valence 2 and no com...
We develop the construction suggested by Scharlemann and Thompson in [14] to obtain an infinite family of pairs of knots Kα and K α so that w(K α#K α ) = max{w(Kα), w(K α )}. This is the first known example of a pair of knots such that w(K#K ) < w(K) + w(K ) − 2 and it establishes that the lower bound w(K#K ) ≥ max{w(K), w(K )} obtained in [12] is best possible. Furthermore, the knots Kα provide an example of knots where the number of critical points for the knot in thin position is greater than the number of critical points for the knot in bridge position.
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