Additivity Properties of the Width of Knots

Jennifer Schultens

September 2, 2002

Abstract

The width of knots satisfies w(K_{1}#K_{2}) ≥ max(w(K_{1}), w(K_{2})) for any two
knots K_{1}, K_{2}. In particular, w(K_{1}#K_{2}) ≥ \frac{1}{2}(w(K_{1}) + w(K_{2})).

1 Introduction

The notion of thin position, introduced by D. Gabai [G], has been employed with great success in many geometric constructions. Yet the underlying notion of the width of a knot remains shrouded in mystery. Little is known about the width of specific knots. By stacking a copy of K_{1} in thin position on top of a copy of K_{2} in thin position, it is easily seen that w(K_{1}#K_{2}) ≤ w(K_{1}) + w(K_{2}) − 2. The investigation here is the starting point of a more general investigation into the nature of the interaction between incompressible surfaces in a knot complement and the width of a knot. We here show that the width of a composite knot is bounded below by the maximum of the widths of its summands and therefore also by one half the sum of the widths of its summands.

The question as to how the width of a knot behaves under the operation of connected sum of knots is inspired, in part, by a consideration of the fact that the bridge number satisfies the equation b(K_{1}#K_{2}) = b(K_{1}) + b(K_{2}) − 1, see the paper [S] by H. Schubert, or [Sch2] for a much shorter proof. The shorter proof in [Sch2] crystallized out of an investigation into whether or not thin position arguments clarify the behaviour of bridge number under connected sum. The answer to that question appears to be no. Thin position arguments appear to provide a much more refined analysis than required for the recovery of Schubert’s result. On the other hand, thin position can be viewed as a generalization of bridge position, so it is natural to ask to what extent an analogue of Schubert’s result holds for the width of a knot.

The proofs in [S], [Sch2] do not carry over to this more refined setting. The argument in [Sch2] fails in settings where the swallow follow torus is too convoluted:

The correspondance between tunnel numbers of knots and bridge numbers of knots, strongly irreducible generalized Heegaard splittings and knots in thin position deserve to be investigated more closely. Suffice it to say that this correspondance played a role in the discovery of the simple argument below. This correspondance also leads one to believe that the degeneration of width under connected sum of knots is possible, i.e., that there might be knots K_{1}, K_{2}, such that w(K_{1}#K_{2}) <

Figure 1: Convoluted portion of a swallow follow torus

w(K_{1})+w(K_{2})−2. In the context of this correspondance, it deserves to be pointed out that for small knots (knots whose complements contain no essential closed surfaces), neither width nor tunnel number degenerate under connected sum; i.e., for small knots, width of knots satisfies w(K_{1}#K_{2}) = w(K_{1}) + w(K_{2}) − 2 and tunnel number satisfies t(K_{1}#K_{2}) = t(K_{1}) + t(K_{2}). This is proven in [RS] and [Sch1], respectively.

I wish to thank Yoav Rieck and Marty Scharlemann for helpful conversations. I also wish to thank RIMS, where this work was begun, and Professor Tsuyoshi Kobayashi for inviting me to RIMS. I also wish to thank the MPIM-Bonn where this work was completed. This work was supported in part by the grant NSF-DMS 0203680.

2 Basic Definitions

For standard definitions and results concerning knots, see [BZ], [L] or [R]. For standard definitions and results pertaining to 3-manifolds, see [H] or [J].

Definition 2.1. A \underline{height function} for a 3-manifold M is a Morse function h : M → [0, 1] with exactly two critical points (a minimum and a maximum) in interior(M). Note that the only closed 3-manifold possessing a height function is S^{3}. Let h : S^{3} → [0, 1] be a \underline{height function}. Then the regular level surfaces of h are spheres. Let K ⊂ S^{3} be a knot and let c_{1}, …, c_{n} be the critical values of h|_{K} listed in order so that h(c_{1}) < ⋯ < h(c_{n}). Choose r_{1}, …, r_{n−1} so that c_{i} < r_{i} < c_{i+1} and set R_{i} = h^{−1}(r_{i}). The \underline{width of K relative to h}, denoted by w(K, h), is ∑_{i} |K ∩ R_{i}|. The \underline{width of K}, denoted by w(K), is the minimum of this relative width over all height functions. (We say that K is in \underline{thin position} if it is presented with respect to a height function realizing its width.)

Here R_{i} is a \underline{thin level} if c_{i} is a maximum value for h|_{K} and c_{i+1} is a minimum value for h|_{K}; and R_{i} is a \underline{thick level} if c_{i} is a minimum value for h|_{K} and c_{i+1} is a maximum value for h|_{K}.

Definition 2.2. Let K = K_{1}#K_{2} be a composite knot with decomposing sphere S. Then S^{3} − η(K ∪ S) has two components. Each of these components is a torus. Such a torus is called a \underline{swallow follow torus}. A subannulus of a swallow follow torus T is parallel into ∂η(K_{1}) or ∂η(K_{1}).

In the first case, we call T the swallow follow torus that swallows K_{2} and follows K_{1}. In the second case, we call T the swallow follow torus that swallows K_{1} and follows K_{2}.

We denote the component of S^{3} − T containing K_{1}#K_{2} by int(T) and the other by ext(T). Set C(K_{i}) = S^{3}−η(K_{i}). Then for T the swallow follow torus that swallows K_{2} and follows K_{1}, ext(T) = C(K_{1}). For T the swallow follow torus that swallows K_{1} and follows K_{2}, ext(T) = C(K_{2}).

Definition 2.3. Let K = K_{1}#K_{2} ⊂ S^{3} be a composite knot. Suppose that T is a swallow follow torus for K_{1}#K_{2} and h : S^{3} → [0, 1] is a height function. We may assume that h_{T} is a Morse function. Consider the singular foliation, F_{T}, of T induced by h_{T}. Let σ be a leaf corresponding to a saddle singularity. Then σ consists of two circles, s_{1}, s_{2}, wedged at a point. If either s_{1} or s_{2} is inessential in T, then we call σ an \underline{inessential saddle}. Otherwise, σ is an \underline{essential saddle}.

A collar of σ has three boundary components s̃_{1}, s̃_{2}, s̃_{3}. One of these boundary components, s̃_{1}, is parallel to s_{1}, another, s̃_{2} is parallel to s_{2}. We call σ an \underline{upper saddle} (\underline{lower saddle}) if the third boundary component, s̃_{3}, lies above (below) σ.

Let σ, s̃_{1}, s̃_{2}, s̃_{3} be as above. We may assume that s̃_{1} and s̃_{2} are in the same level surface L of h. Then since L is a sphere, s̃_{1} and s̃_{2} cobound an annulus in L. If a collar of s̃_{1} ∪ s̃_{2} in this annulus is contained in V, then σ is a \underline{nested saddle}.

Figure 2: A nested saddle and an unnested saddle

As for the case of a knot in thin position, we can define thick and thin levels for swallow follow tori. We may do so without introducing a notion of thin position for a swallow follow torus. (This could easily be done, but is not what is required here.)

Definition 2.4. Let c_{1}, …, c_{n} be the values at which h|_{T} has saddle singularities and let r_{1}, …, r_{n−1} be regular values for h|_{T} with c_{i} < r_{i} < c_{i+1}. Set R_{i} = h^{−1}(r_{i}). Then R_{i} is a \underline{thin level for h|_{T}} if c_{i} corresponds to an upper saddle and c_{i+1} is corresponds to a lower saddle; and R_{i} is a \underline{thick level for h|_{T}} if c_{i} corresponds to a lower saddle and c_{i+1} corresponds to an upper saddle. Suppose R is a thin level for h|_{T} and S is a thick level for h|_{T}, and there are no thick or thin levels between R and S. Then the component W of S^{3} − (R ∪ S) cobounded by R and S is called a \underline{slice}. We denote R by ∂_{−}W and S by ∂_{+}W. A component cut off by the highest thick level or by the lowest thick level is also called a \underline{slice}.

We call W an \underline{upper slice} if it contains upper saddles and a \underline{lower slice} if it contains lower saddles.

3 Four lemmata

We here prove a sequence of lemmata providing a solid torus endowed with a height function satisfying the requirements needed in the main construction in the succeeding section.

Lemma 3.1. Let h : S^{3} → [0, 1] be a height function, K_{1}, K_{2} ⊂ S^{3} knots and T a swallow follow torus that swallows K_{2} and follows K_{1}. Then for any level surface L, each component of ∂(L ∩ ext(T)) is either inessential in T or a meridian of C(K_{1}).

Proof: Let F_{T} be as in Definition 3. Let σ be the highest essential saddle in F_{T}. For s̃_{1}, s̃_{2}, L as in Definition 3, let D̂_{1}, D̂_{2} be the (disjoint) disks bounded by s̃_{1}, s̃_{2} in a level surface L̃ just above or just below L. As σ is the highest essential saddle in F_{T}, the portion C of T lying above σ is a planar surface. Furthermore, F_{T} |_{C} contains only inessential saddles. Thus each component of ∂C disjoint from σ bounds a disk in T. It also follows that there is a disk D that can be obtained from C by capping off all components of ∂C disjoint from σ by disks. In particular, σ is an upper saddle.

Now consider D̂_{i}. Any curve in T ∩interior(D̂_{i}) is contained in D and thus bounds a subdisk of D. A standard innermost disk argument then shows that D̂_{i} is isotopic rel ∂D̂_{i} to a disk whose interior is disjoint from T, i.e., lies either entirely in int(T) or entirely in ext(T). Since s̃_{i} is parallel to s_{i}, s̃_{i} is essential in T. Furthermore, since int(T) is knotted, T is incompressible, whence s̃_{i} is essential in the closure of ext(T). This implies that D̂_{i} must be isotopic rel ∂D̂_{i} to a disk whose interior lies entirely in int(T), in particular, s̃_{i} is a meridian of int(T). Thus s̃_{i} is a meridian of C(K_{i}).

Now any leaf of F distinct from s̃_{i} is disjoint from s̃_{i} and hence must either be inessential, or also a meridian of C(K_{i}).

Figure 3: The highest essential saddle

The following definition helps us understand the connectivity of ext(T).

Definition 3.2. Let c_{1}, …, c_{m} be the critical values of h|_{T} corresponding to nested saddles and let C_{i} = (h|_{ext(T)})^{−1}(c_{i}). (If c_{i} corresponds to an upper saddle, then a level surface just below C_{i} has one more component than a level surface just above C_{i}. The situation is reversed if c_{i} corresponds to a lower saddle.)

Consider the component of C_{i} consisting of two surfaces wedged at a point and let C̃_{i} be one of these two surfaces. It is homeomorphic to a component of a level surface just below C_{i} in the case that c_{i} corresponds to an upper saddle and homeomorphic to a component of a level surface just above C_{i} is c_{i} corresponds to a lower saddle.

Define a graph Γ by assigning a vertex to each component of ext(T)−(C̃_{1}∪⋯∪C̃_{m}) and connecting two vertices v_{1}, v_{2} by an edge if and only if the components V_{1}, V_{2} of ext(T)−(C̃_{1} ∪⋯∪C̃_{m}) corresponding to v_{1}, v_{2} meet along C̃_{i} for some i ϵ {1, …, m}.

Lemma 3.3. Γ is a tree.

Proof: Suppose that γ is a cycle in Γ. We may assume that the vertices, v_{1}, …, v_{k} in γ are distinct. We construct a circle α in ext(T) using γ as a model. Since components of ext(T) − (C̃_{1} ∪ ⋯ ∪ C̃_{m}) are path components, each vertex gives rise to an arc α_{i} connecting the components of C̃_{i} corresponding to the edges incident to v_{i}. As C̃_{i} is path connected, the endpoints of the α_{i} can be connected up to form a circle α in ext(T). After a small isotopy, α intersects each component of C̃_{i} − T exactly once. Choose one such component, say C̃_{1}.

By Lemma 3.1, each component of ∂C̃_{i} is either inessential in T or a meridian. Consider the surface C obtained by capping off each component of the former type by a pushoff of the disk bounded in T. Then C is a planar surface in ext(T) all of whose boundary components are merdians. Now consider the surface Ĉ obtained from C by capping off each component of ∂C with a meridian of int(T). Then Ĉ is a 2-sphere in S^{3} that meets the circle α in S^{3} exactly once. But this is impossible. Thus there is no cycle in T.

Lemma 3.4. Suppose M = S^{3} − η(K) for some knot K ⊂ S^{3} and h : M → [0, 1] a height function whose level surfaces are connected. Then there is a sequence of knots K_{0} = K, K_{1}, …, K_{n} such that K_{n} is the unknot, corresponding to a sequence of manifolds M_{0} = C(K_{0}), …, M_{n} = C(K_{n}), such that M_{n} is a solid torus. These manifolds are such that M_{i} is obtained from M_{i−1} by cutting and pasting along a regular level surface L_{i−1} of h_{i−1} in such a way that each component of ∂(M_{i} ∩ L_{i−1}) is reidentified with itself. In particular, there is a sequence of height functions h_{0} : M_{0} → [0, 1] = h : M → [0, 1], h_{1} : M_{1} → [0, 1], …, h_{n} : M_{n} → [0, 1].

Furthermore, if h extends to a height function on S^{3}, then h_{i}, for i = 1, …, n, extends to a height function on S^{3}.

Proof: Let l be a curve on ∂M that goes through each essential saddle of h|_{∂M} and is monotone between essential saddles. If l bounds a disk in M, then the conclusions of the lemma hold.

We prove the lemma by induction on the number of thin levels of h|_{∂M}.

Step 1: Constructing disks for l.

Let W be a slice. We may assume that W is an upper slice. Then l ∩ W consists of an even number (what goes up must come down) of monotone arcs, Γ_{1}^{W}, and arcs containing exactly one critical point, more specifically, a maximum, Γ_{2}^{W}. We consider how l ∩ h^{−1}(r) varies as r traces out the image of the level surfaces comprising W beginning at the top. It begins with endpoints of Γ_{1}^{W}, remains the same until a maximum appears which then splits into two points that can be considered to lie on two monotone arcs ending at the maximum, remains the same until this happens again, … and ends after a finite number of iterations.

Since level surfaces are connected, any pair of arcs in Γ_{1}^{W} can be connected via an arc in ∂_{−}W and continuously varying copies of this arc in the level surfaces of h comprising W piece together to a disk. (The appearance of maxima is no obstruction, as this one point in a level surface may easily be avoided by the connecting arc.) Call such a disk D_{γ_{1},γ′_{1}}, if it is cobounded, together with one arc in ∂_{−}W and one arc in ∂_{+}W by γ_{1} and γ′_{1}.

Similarly, given an arc γ ϵ Γ_{2}^{W}, piece together a disk D_{γ} by beginning with the maximum m of γ and then employing the two monotone subarcs of γ − m as above. Here D_{γ} is cobounded by γ together with an arc in ∂_{+}W.

These disks allow us to prove the lemma under the hypothesis that there are no thin levels:

Step 2: Suppose M = L ∪_{S} U, for L a lower slice and U an upper slice, then the conclusions of the lemma hold.

Set D^{L} = {D_{γ} | γ ⊂ Γ_{2}^{L}} and D^{U} = {D_{γ} | γ ⊂ Γ_{2}^{U}}. Note that in this case Γ_{1}^{L} = Γ_{1}^{U} = ∅. Denote the arcs in ∪_{γ⊂Γ_{2}^{L}}∂D_{γ} ∩S by α and the arcs in ∪_{γ⊂Γ_{2}^{U}} ∂D_{γ} ∩S by β. Denote the components of α by a_{1}, …, a_{k} and the components of β by b_{1}, …, b_{k}.

Construct a third set of arcs γ as follows: Consider S − α. This is a connected surface, hence there is an arc g_{1} ⊂ S −α with ends on the same punctures as b_{1}. Note that if S − (α ∪ g_{1}) is disconnected, then a separating circle is formed by g_{1} along with one of the a_{j}. So b_{1} shares both endpoints with one of the a_{j}. Thus k = 1, i.e, K is the unknot. Thus S − (α ∪ g_{1}) is connected.

Having constructed g_{1}, …, g_{i}, such that S − (α ∪ g_{1} ∪ ⋯ ∪ g_{i}) is connected, there is an arc g_{i+1} ⊂ S − (α ∪ g_{1} ∪ ⋯ ∪ g_{i}) with ends on the same puncture as b_{i+1}. Here S − (α ∪ g_{1} ⋯ ∪ g_{i} ∪ g_{i+1}) is also connected unless a separating circle is formed by g_{1} ⋯∪g_{i} ∪g_{i+1} along with i+1 of the a_{j}. By the same reasoning as in the preceeding paragraph this implies that i + 1 = k. Thus either S − (α ∪ g_{1} ⋯ ∪ g_{i} ∪ g_{i+1}) is connected or i + 1 = k.

Cutting M along S and reidentifying in such a way that b_{i} is identified with g_{i} for i = 1, …, k produces a knot complement M_{1}. Let K_{1} ⊂ S^{3} be the knot of which M_{1} is the complement. (Here K_{1} is obtained from K by inserting a pure braid near the thick level.) Let h_{1} be the height function obtained on M_{1}. Here K_{1}, M_{1}, h_{1} satisfy the hypotheses of the lemma. But now (∪_{γ ⊂ Γ_{2}^{L}}∂D_{γ}) ∩ (∪_{γ ⊂ Γ_{2}^{U}} ∂D_{γ}) = ∅ and S ∩ ((∪_{γ ⊂ Γ_{2}^{L}}∂D_{γ}) ∪ (∪_{γ ⊂ Γ_{2}^{U}} ∂D_{γ})) is an imbedded circle c̃ that meets each point of l ∩ S exactly once. Since S is a connected planar surface, c̃ cuts S into two components S_{1}, S_{2}. Now since each component of ∂S_{i} − c̃ misses l, each such component is inessential in ∂M and hence bounds a disk in ∂M. Thus c̃ bounds a disk D̃.

Hence l bounds the disk D̃ ∪ ((∪_{γ ⊂ Γ_{2}^{L}}D_{γ}) ∪ (∪_{γ ⊂ Γ_{2}^{U}} D_{γ})). Whence the lemma follows if there are no thin levels.

Now suppose that the lemma holds if there are up to n−1 thin levels and suppose M = (L_{1} ∪_{S_{1}} U_{1}) ∪_{F_{1}} ⋯ ∪_{F_{n−1}} (L_{n} ∪_{S_{n}} U_{n}) (the L_{i}’s are meant to be lower slices, the U_{i}’s upper slices). Consider M^{1} = L_{1} ∪_{S_{1}} U_{1}, M^{2} = M − M^{1}. Then ∂M^{1} = ∂_{−}U_{1} ∪ (∂M ∩ M^{1}) and ∂M^{2} = ∂_{−}L_{2} ∪ (∂M ∩ M^{2}) where ∂M ∩ M^{i} consists of annuli. Pair off the components of ∂_{−}U_{1} ∩ ∂M that cobound an annulus in ∂M ∩ M^{2} and those of ∂_{−}L_{2} ∩ ∂M that cobound an annulus in ∂M ∩ M^{1}.

Construct M̂^{1} by adjoining an upper slice along ∂_{−}U_{1} in which paired off components of ∂_{−}U_{1} ∩∂M are connected by an annulus with one non nested essential saddle and one maximum. Similarly, construct M̂^{2}. Construct l_{i} by adjoining to l ∩ M^{i} arcs that run once through the critical point of the saddle in an added annulus and that are monotone otherwise. Then there are no thin levels in M̂^{1} and n − 1 thin levels in M̂^{2}.

Now the inductive hypothesis applies to M̂^{1} and M̂^{2}. After cut and past operations, l_{i} bounds a disk D̂_{i} in M̂^{i}. We may assume that D̂_{i} intersects each wedge in subdisks constructed as in Step 1. Then each arc in D̂_{1} ∩ ∂_{−}U_{1} and D̂_{2} ∩ ∂_{−}L_{2} cobounds a subdisk of D̂_{i} contained in M̂^{i}−M^{i} and these subdisks are disjoint. (Each such subdisk is bounded by a component of ∂D_{1} ∩ ∂_{−}U_{1} or ∂D_{2} ∩ ∂_{−}L_{2} together with a component of l_{i} − l ∩ M^{i}.) Set D_{i} = D̂_{i} ∩ M^{i}. Then D_{i} is a disk in M^{i}.

Reglueing M^{1} and M^{2} along F_{1} may introduce points of intersection of D_{1} ∩ ∂_{−}U_{1} and D_{2} ∩ ∂_{−}L_{2}. The argument in Step 2 shows how to eliminate such points of intersection and create a planar surface D_{1} ∪ D_{2}. Since the cut and paste operations preserve the connectedness of l, and since ∂(D_{1}∪D_{2}) = l, D_{1}∪D_{2} is the disk required.

The final comment follows from the fact that the level surfaces of h are in oneto-one correspondance with the level surfaces of h_{i}. (In particular, h_{i}|_{∂M_{i}} contains no nested saddles.) Indeed, there is a one-to-one correspondance of the boundary components of a level surface of h with the boundary components of a level surface of h_{i}. Thus if h : M → [0, 1] extends to a height function ĥ : S^{3} → [0, 1], then this extension is via a solid torus with an appropriate height function. In this solid torus, regular level surfaces are either meridian disks or boundary parallel disks. Via this same height function on this solid torus, h_{i} : M_{i} → [0, 1] also extends to a height function ĥ_{i} : S^{3} → [0, 1] defined by ĥ_{i}|_{M_{i}} = h_{i} and ĥ_{i}|_{S^{3}−M_{i}} = ĥ|_{S^{3}−M}.

Figure 4: The transformation of a knot exterior as in Lemma 3.4

Lemma 3.5. Let K, K̃ ⊂ S^{3} be knots and let T be a swallow follow torus for K#K̃ that swallows K̃ and follows K. Suppose M = ext(T) and h : M → [0, 1] a height function. Then there is a sequence of knots K_{0} = K, K_{1}, …, K_{n} such that K_{n} is the unknot, corresponding to a sequence of manifolds M_{0} = C(K_{0}), …, M_{n} = C(K_{n}), such that M_{n} is a solid torus (i.e., K_{n} is the unknot). These manifolds are such that M_{i} is obtained from M_{i−1} by cutting and pasting along a regular level surface L_{i−1} of h_{i−1} in such a way that each component of ∂(M_{i} ∩ L_{i−1}) is reidentified with itself. In particular, there is a sequence of height functions h_{0} : M_{0} → [0, 1] = h : M → [0, 1], h_{1} : M_{1} → [0, 1], …, h_{n} : M_{n} → [0, 1].

Furthermore, if h extends to a height function on S^{3}, then h_{i}, for i = 1, …, n, extends to a height function on S^{3}.

Proof: The proof is by induction on the number of nested saddles in F_{T}. If there are no nested saddles, then the lemma follows by Lemma 3.4.

Suppose that there are n nested saddles and that the lemma holds whenever there are at most n − 1 nested saddles. Let v be an endvertex of Γ. Then v corresponds to a component V of ext(T) − (C̃_{1} ∪ ⋯ ∪ C̃_{m}) that meets no other nested saddles. Furthermore, V meets only one component of C̃_{1} ∪ ⋯ ∪ C̃_{m}, say C̃_{1}. We will assume that c_{1} is an upper saddle. The other case follows analogously.

Since C̃_{1} is connected, it follows that V is foliated by connected level surfaces. Alter h|_{V}, maintaining the relative positioning of all level surfaces of h|_{V} to obtain a height function h̄ : ext(T) → [0, 1], so that V is contained between C_{1} and another level surface L of h, where L is chosen so that h|_{ext(T)−V} has no critical values in (h(L), c_{1}).

(Note that here if h|_{ext(T)} extends to S^{3}, h̄|_{ext(T)} may or may not extend to S^{3}. In S^{3}, isotoping V upward could force components corresponding to other vertices of Γ to be isotoped upward as well. This would impede progress all together. For this reason we consider the height function h̄|_{ext(T)} and below the height function h̃|_{ext(T)}. It follows from the reasoning below that these temporarily considered height functions help locate the cut and paste operations required.)

We denote the singular point in C̃_{1} by c. Let a be a monotone arc from c to a minimum of T in V. Since h̄|_{ext(T)−V} has no critical values in (h̄(L), c_{1}), there is a monotone arc b from c to a point just below the level surface h̄^{−1}(h(L)) in ext(T)−V. Consider the height function obtained by modifying h̄|_{ext(T)} along a collar of a ∪ b in T in such a way that a ∪ b becomes a monotone arc and so that no new critical points are introduced. The effect is that c and the minimum are no longer critical points. (A saddle and a minimum have been canceled.) Denote the new height function by h̃|_{ext(T)}. See fig. 4a.

Figure 5: Transforming h to h̃

Consider a subarc e of the collar of a ∪ b in T that lies in the level surface of h̄|_{ext(T)} containing c. As h̄|_{ext(T)} is modified, e is changed into a new arc d (that has exactly one critical point, a minimum). Simultaneously, a disk, D, cobounded by d and an arc, which we may also call e, connecting the endpoints of e, emerges. Now ext(T) with the new height function h̃ satisfies the inductive hypothesis. Thus the conclusions of the lemma hold and there is a solid torus M_{n} with height function h̃_{n} obtained from ext(T) in the required manner.

Note that the procedure in Step 2 of the proof of Lemma 3.4 may be refined for the setting at hand. Specifically, consider the subarc a_{i} lying in the level surface S̃ of h̃|_{ext(T)}. Here a_{i} lies on a component of a level surface S̃ of h̃ but not necessarily of h. See fig. 4b.

Figure 6: A disk for a level surface of h̃ yields a disk for h

Consider the arc δ = S̃∩D. Here δ is a separating arc with ends on one component of ∂S̃. It cuts S̃ into two components that can be identified with components of the level surface h̄^{−1}(h̃(S̃)). It follows from the construction of h̃, that an arc b_{i} lies entirely on one side of δ. See fig. 4c.

To refine the procedure in Step 2 of the proof of Lemma 3.4 we choose g_{i} to lie on the same side of δ as b_{i}. A very similar argument to that in Step 2 of the proof of Lemma 3.4 shows that this can be done. (In the case of a nested saddle that is a lower saddle, we interchange the roles of α and β.)

Thus the cut and paste operation performed on S̃ can be performed so that each point of D∩S̃ is reidentified with itself. This implies that the cut and paste operation on S̃ can be factored into two cut and paste operations, one on each of the components of S̃−D. Thus the corresponding cut and paste operation can be performed for h̄|_{ext(T)} by factoring it into two cut and paste operations each performed on one of the two components of h̄^{−1}(h̃(S̃)).

The components of the level surfaces of h̄|_{ext(T)} are in one-to-one correspondance with the components of the level surfaces of h|_{ext(T)}. Thus all cut and paste operations on level surfaces of h̄|_{ext(T)} can be realized as successions of cut and paste operations on the level surfaces of h|_{ext(T)}.

The final statment follows as before.

4 the main construction

Theorem 4.1. w(K_{1}#K_{2}) ≥ max(w(K_{1}), w(K_{2})).

Figure 7: Intersection pattern of α and β in S̃

Proof: Let h : S^{3} → [0, 1] be a height function that realizes the width of K = K_{1}#K_{2} and let T be a swallow follow torus that swallows K_{1} and follows K_{2}. We wish to construct a height function h̃ : S^{3} → [0, 1] such that w(K_{1}, h̃) = w(K).

By construction, (int(T), K) ≅ (solid torus, K_{1}). We use this fact as follows: Here ext(T) is homeomorphic to C(K_{2}). In order to avoid confusion, briefly consider S^{3} and remove ext(T), then int(T) is homeomorphic to a closed regular neighborhood of K_{2}. We do not consider this imbedding of int(T) in S^{3}. Instead, we consider an abstract copy of int(T). Let D be a meridian disk of int(T), i.e., a properly imbedded disk that meets K once. Then if we attach a solid torus V to ∂(int(T)) such that a longitude of ∂V is identified with ∂D, the resulting 3-manifold is S^{3}, and

Figure 8: Modification of the figure in fig. 1 (int(T) ∪ V, K) ≅ (S^{3}, K_{1}), i.e., K becomes isotopic to K_{1} in this S^{3}.

We now employ h to construct a height function on this latter copy of S^{3}. To do so, we must extend the Morse function h|_{int(T)} to S^{3}. Let V = M_{n} be the solid torus provided by Lemma 3.5. Then V is endowed with a Morse function h_{n} : M_{n} → [0, 1]. We denote the Morse function on int(T) ∪ V given by h|_{int(T)} on int(T) and h_{n} on V by h̃. The identification is such that ∂h_{n}^{−1}(r) matches up with ∂h|_{int(T)}^{−1}(r) for all rϵ[0, 1].

(Since h_{n}^{−1}(r) = h|_{ext(T)}^{−1}(r) for all r, this can be done. In fact, Lemmata 3.4 and 3.5 can be translated into precise instructions on which boundary components to identify. Again, note that this is an abstract identification.)

This provides a Morse function h̃ : S^{3} → [0, 1]. Since its critical points are in one-to-one correspondance with the critical points of h : S^{3} → [0, 1], it is a height function. Note that there is now a one-to-one correspondance between the critical points of h|_{K_{1}#K_{2}} and the critical points of h̃|_{K_{1}}. Thus w(K_{1}) ≤ w(K_{1}#K_{2}).

The proof that w(K_{2}) ≤ w(K_{1}#K_{2}) proceeds analogously.

5 Corollaries

Corollary 5.1. w(K_{1}#K_{2}) ≥ \frac{1}{2}(w(K_{1}) + w(K_{2})).

Proof: Suppose w(K_{1}) ≥ w(K_{2}), then

w(K_{1}#K_{2}) ≥ w(K_{1}) = \frac{1}{2}(2w(K_{1})) ≥ \frac{1}{2}(w(K_{1}) + w(K_{2})).

Similarly in the other case.

Corollary 5.2. w(K_{1}# … #K_{n}) ≥ \frac{1}{n}(w(K_{1}) + ⋯ + w(K_{n})).

Proof: Suppose w(K_{1}) ≥ max(w(K_{2}), …, w(K_{n})), then

w(K_{1}# … #K_{n}) ≥ w(K_{1}) = \frac{1}{n}(n w(K_{1})) ≥ \frac{1}{n}(w(K_{1}) + ⋯ + w(K_{n})).

Similarly in the other cases.

Theorem 5.3. If K is a satellite knot with companion J, then w(J) ≤ w(K).

Proof: The proof is analogous to the one above.

It remains to find examples of knots with widths exhibiting degenerative behaviour with respect to connected sum.

6 A note on computing the width of a knot

The thick and thin levels of a knot in thin position allow for a computation of the width of a knot.

Lemma 6.1. Let S_{i_{1}}, …, S_{i_{k}} be the thick levels of K and F_{j_{1}}, …, F_{j_{k−1}} the thin levels. Set a_{i_{l}} = | K ∩ S_{i_{l}}| and b_{j_{l}} = | K ∩ F_{i_{l}}|. Then

w(K) = 2 ∑_{l=1}^{k} a_{i_{l}}^{2} − 2 ∑_{l=1}^{k−1} b_{j_{l}}^{2}.

Proof: This can be proven by a direct computation and repeated use of the Gauss Summation Formula. The proof is inspired by a comment by Clint McCrory (his comment concerned the case in which thin position is bridge position, and the illustration is a square of dots.) It is illustrated in fig. 6. Each dot represents two points of intersection with a regular level surface between two critical level surfaces. For instance, the dots in fig. 6 represent the case in which the critical values, listed from the highest to the lowest are a maximum, maximum, maximum, maximum, minimum, minimum, maximum, maximum, minimum, minimum, maximum, maximum, maximum, minimum, minimum, minimum, minimum, minimum.

Figure 9: Dark dots indicate squares that are added, white dots indicate overlap squares that are subtracted

References

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[H] 3-manifolds J. Hempel, Annals of Math. Studies 86 (1976), Princeton University Press [J] Lectures on three-manifold Topology W. Jaco, Regional Conference Series in

Mathematics 43 (1981), Amer. Math. Soc.

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[RS] Thin position for a connected sum of small knots, to appear in Algebraic and Geometric Topology

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[Sch1] Additivity of tunnel number for small knots J. Schultens, Comment. Math. Helv. 75 (2000) 3 353–367

[Sch2] Additivity of bridge numbers of knots, J. Schultens, math.GT/0111032, preprint

[Sch3] Heegaard genus of Haken manifolds, J. Schultens, math.GT/0108028, preprint

Jennifer Schultens Department of Mathematics and CS 400 Dowman Dr. Emory University Atlanta, GA 30322, USA jcs@mathcs.emory.edu