Wirtinger systems of generators of knot groups

WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS arXiv:1705.03108v1 [math.GT] 8 May 2017 R. BLAIR, A. KJUCHUKOVA, R. VELAZQUEZ, P. VILLANUEVA Abstract. We define the Wirtinger number of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality can be viewed as establishing a weak version of Cappell and Shaneson’s Meridional Rank Conjecture, and suggests a new approach to this conjecture. Our result also leads to a combinatorial technique for obtaining strong upper bounds on bridge numbers. This technique has so far allowed us to add the bridge numbers of approximately 50,000 prime knots of up to 14 crossings to the knot table. As another application, we use the Wirtinger number to show there exists a universal constant C with the property that the hyperbolic volume of a prime alternating link L is bounded below by C times the bridge number of L. 1. Introduction This work was inspired by an old problem, 1.11 on the Kirby List [12]: Question 1.1. Is every knot whose group is generated by 2 meridians actually an 2-bridge knot? Same for n meridians and n-bridge knots. The above question has become known as the Meridional Rank Conjecture, and has been answered in the affirmative for many classes of links. The case of n = 2 was settled in 1989 by Boileau and Zimmermann [4]. The conjecture has also been shown to hold for generalized Montesinos links [3], [14], torus links [15], iterated cables [9], links of meridional rank 3 whose double branched covers are graph manifolds [2] and knots whose exteriors are graph manifolds [1]. There are no known counter-examples, and the general case remains open. The present work was inspired by the following simple observation about the conjecture. Denote the bridge number and meridional rank of a link L by β(L) and µ(L), respectively. Let us recall the classical argument which establishes β(L) ≥ µ(L). Assume β(L) = m. Then, L admits a diagram with exactly m local maxima x1 , ..., xm (with respect to some axis in the plane). The Wirtinger generators corresponding to the m arcs containing the xi are then easily seen to generate the group of the link complement, by applying the Wirtinger relations in this diagram successively at crossings of decreasing height. What is obvious yet intriguing about this argument is that it does not directly compare the bridge number to the number of meridional generators in a presentation of the link group in which arbitrary valid relations are allowed. Rather, only very particular, diagrammatic, relations are considered. This motivates studying the intermediate link invariant which arises by, intuitively speaking, considering only presentations with the property that the generators are meridional elements and the relations are Wirtinger relations that can simultaneously be realized in a diagram. To formalize this notion, we introduce the combinatorial tool of coloring a link diagram according to the following set of rules. Recall that if L is a link in R3 and p : R3 → R2 is the standard projection map given by p(x, y, z) = (x, y), then p(L) is a link projection if p|L is a regular projection. Hence a link projection is a finite four-valent graph in the plane, and we refer to the vertices of this graph as crossings. A link diagram is a knot projection together with labels at each crossing that indicate which strand goes over and which goes under. By standard convention, these labels take the form of deleting parts of the under-arc at every crossing, and thus we think of a link diagram as a disjoint union of closed arcs, or strands, in the plane, together with instructions for how to connect these strands to form a union of simple closed curves in R3 . MSC codes: 57M25, 57M27, 57M05. The first, third and fourth authors were partially supported by NSF grant DMS-1247679. 1 2 WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS Figure 1. The only knot diagram in which a strand is adjacent to itself. s1 s2 s1 s2 s1 s2 Figure 2. Two coloring moves on the knot 817 , corresponding to the shaded crossings. The coloring process terminates at this stage. More generally, this diagram can not be colored using only two seeds. 817 is a three-bridge knot. Let D be a diagram of a link L with n crossings. Denote by s(D) the set of strands s1 , s2 ,..., sn and let v(D) denote the set of crossings c1 , c2 ,..., cn . Two strands si and sj of D are adjacent if si and sj are the under-strands of some crossing in D. There exists a unique knot diagram up to planar isotopy for which there exists a strand si of D for which si is adjacent to itself, see Figure 1. In all cases we consider, adjacent arcs are understood to be distinct. We call D k-partially colored if we have specified a subset A of the strands of D and a function f : A → {1, 2, . . . , k}. We refer to this partial coloring by the tuple (A, f ). Given k-partial colorings (A1 , f1 ) and (A2 , f2 ) of D, we say (A2 , f2 ) is the result of a coloring move on (A1 , f1 ) if (1) A1 ⊂ A2 and A2 \ A1 = {sj } for some strand sj in D; (2) f2 |A1 = f1 ; (3) sj is adjacent to si at some crossing c ∈ v(D), and si ∈ A1 ; (4) the over-strand sk at c is an element of A1 ; (5) f1 (si ) = f2 (sj ). Denote the above coloring move from one k-partially colored diagram to another by (A1 , f1 ) → (A2 , f2 ). See Figure 2.1 The move (A1 , f1 ) → (A2 , f2 ) captures the fact that if the Wirtinger generators corresponding to si and sk belong to the subgroup of π1 (S 3 − L, x0 ) generated by some set of meridians of L, then, by applying the Wirtinger relation at c, sj is seen to belong to this subgroup as well. We say D is k-meridionally colorable if there exists a k-partial coloring (A0 , f0 ) = ({si1 , si2 , . . . , sik }, f0 (sij ) = j) and a sequence of c(D) − k coloring moves (A0 , f0 ) → (A1 , f1 ) → . . . → (Ac(D)−k , fc(D)−k ), where c(D) denotes the crossing number of D. In particular, Ac(D)−k = s(D), that is, at the end of the coloring process every strand is assigned a color. By design, the set {si1 , si2 , . . . , sik } corresponds to meridional elements that generate the link group via iterated application of the Wirtinger relations in D, so we refer to it as a Wirtinger generating system, and we call its elements seed strands. The minimum value of k such that D admits a Wirtinger generating system with k elements (equivalently, D is k-meridionally colorable) is the Wirtinger number of D, denoted ω(D). Of course, this number will depend on the choice of diagram, but it can be used to define an invariant of L. Definition 1.2. Let L ⊂ S 3 be a link. The Wirtinger number of L, denoted ω(L), is the minimal value of ω(D) over all diagrams D of L. 1Thanks to Patricia Cahn for creating Figure 2. WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS 3 It is easy to see that ω(L) has the property µ(L) ≤ ω(L) ≤ β(L). The first inequality follows from the fact that a Wirtinger generating system is by definition a meridional generating set. The second one is implied by the classical argument relating bridge number to meridional rank. Our main result is to show that this inequality is in fact an equality. Theorem 1.3 (Main Theorem). Let L ⊂ S 3 be a link. The Wirtinger number and the bridge number of L are equal. An immediate consequence of this result is that it provides a novel approach to computing bridge numbers of links. Although the question of finding the minimum of ω(D) over all diagrams D of a given link L is subtle, calculating ω(D) itself is algorithmic. This has allowed us to implement the calculation of ω(D) in Python. In the Appendix, we outline our algorithm for computing ω(D) from a Gauss code for D. The algorithm runs extremely fast in practice. We have used this computational approach to complete the tabulation of bridge number for all prime knots up to 12 crossings and the vast majority of knots with crossing number 13 and 14, thereby adding the bridge numbers of approximately 50, 000 knots to the knot table. Due to the fact that we can not assume that ω(L) is always realized in a minimal diagram of L – although this turns out to be the case for all prime knots of 12 crossings or less – tabulating these bridge numbers requires proving that the upper bounds provided by the Wirtinger number are sharp. Our argument to this effect is presented in Section 3, together with a discussion of our findings relating ω(L) to ω(D) where D is a minimal diagram of L. In the second place, the Wirtinger number allows us to relate β(L) to other diagrammatic link invariants, such as the twist number. Recall that in the sphere of projection containing the link diagram, a twist region is either a maximal collection of bigons in the knot projection stacked end to end or a neighborhood of a crossing which is not contained in any bigon. The integer t(D) denotes the number of twist regions of D. Lackenby [13] showed that if a hyperbolic link has a prime alternating diagram D, then the hyperbolic volume of that link is bounded above and below by linear functions of t(D). We can elevate t(D) to a link invariant by declaring that t(L) be equal to the minimum of t(D) over all diagrams D of L. We obtain: Corollary 1.4. Given a link L, β(L) ≤ 2t(L). Corollary 1.4 has an immediate application to the the study of hyperbolic volumes of links. Closed 3manifolds and link complements with a complete hyperbolic structure can be assigned a well-defined hyperbolic volume. The Heegaard genus of a closed 3-manifold M , denoted g(M ), is the minimum genus of any Heegaard surface for that manifold. Due to a theorem of Jorgensen and Thurston, there exists a constant C such that if M is a closed hyperbolic 3-manifold, then Cg(M ) ≤ vol(M ), where vol(M ) denotes the hyperbolic volume of M . Bridge number can be regarded as the analogue of Heegaard genus in the world of links. Recently, it was shown that there does not exist a C such that for any hyperbolic link Cβ(L) ≤ vol(L) where vol(L) is the hyperbolic volume of the complement of L [5], [8]. It is a challenging open question to establish for what classes of links the analogue of Jorgensen and Thurston’s theorem holds. As a consequence of Corollary 1.4 and the main result from [13], we prove the following analogue of Jorgensen and Thurston’s theorem for prime alternating hyperbolic links: Theorem 1.5. There exists a universal constant C with the property that every prime alternating hyperbolic link L satisfies the inequality Cβ(L) < vol(L). 2. Proof of the main theorem Let D be a k-meridionally colorable diagram of some link L. Our proof strategy will be to construct from D a Morse embedding of L into R3 with exactly k local maxima. This will be carried out in two steps. First, we will study the process of extending a partial coloring of D across the entire diagram. The purpose is to extract geometric information about the link L from the sequence of coloring moves. Secondly, we will use the information obtained to construct the desired embedding. It will prove useful to record the order in which strands are colored, as follows. Definition 2.1. Suppose D is link diagram with crossing number c(D). Assume D can be k-meridionally colored by starting with a Wirtinger generating system {si1 , si2 , . . . , sik } and performing coloring moves (A0 , f0 ) → (A1 , f1 ) → . . . → (Ac(D)−k , fc(D)−k ). We associate to this succession of moves the coloring 4 WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS c(D) sequence {αj }j=1 given by αj = sij for 1 ≤ j ≤ k and αj ∈ Aj−k \ Aj−(k+1) for k + 1 ≤ j ≤ c(D) − k. Furthermore, given a coloring sequence {αj } we define its height function h : s(D) → Z by h(αj ) := −j. Introducing the negative sign here serves merely to indulge the authors’ mild preference for focusing on local maxima, rather than local minima, in our construction. We also remark that any diagram D of a non-trivial link can give rise to a multitude of distinct coloring sequences. When a collection of seeds suffices to extend a partial coloring across all of D, the order in which moves are performed involves making arbitrary choices; the color a strand attains can also vary depending on the chosen order. However, once a succession of coloring moves is chosen, the associated coloring sequence is unique. We review a couple of terms used in the proof of the next proposition. Let A be some subset of s(D). We say the strands of A are connected if there exists a reordering of the strands si1 , si2 , . . . , sia in A such that sij is adjacent to sij+1 for all j, 1 ≤ j ≤ a − 1. Note the set of all strands in D is connected if K is a knot. In this case, if A = s(D), then sia is adjacent to si1 . Secondly, let {si }ni=1 be a sequence of adjacent strands ordered by adjacency and let g : {s1 , s2 , . . . , sn } → R be a one-to-one map. We say g has a local maximum at sj if the function g ′ : {1, 2, . . . , n} → R defined by g ′ (i) := g(si ) has a local maximum at j. We now summarize the relevant properties of the functions fi and h. Because links require some additional considerations, we begin by studying the case of knots. Proposition 2.2. Let D be a diagram of a non-trivial knot. Assume D can be k-meridionally colored via c(D) (A0 , f0 ) → . . . → (An , fn ), where n = c(D) − k, and let {αm }m=1 and h be as above. The following hold: (1) For every j ∈ {1, 2, . . . , k}, δ ∈ {0, 1, . . . , n}, fδ−1 (j) is connected. (2) For any j ∈ {1, 2, . . . , k}, h has a unique local maximum on {si | si ∈ fn−1 (j)} when this set is ordered sequentially by adjacency. (3) Let sp , sq ∈ s(D) be adjacent understrands at a crossing in D, and denote the overstrand at this crossing by sr . If fn (sp ) = fn (sq ), then h(sr ) > min{h(sp ), h(sq )}. Proof. Since D is the diagram of a nontrivial knot, whenever sp , sq ∈ s(D) are adjacent understrands at a crossing in D, sp 6= sq . However, it is possible for the overstrand and an understrand at a crossing of D to be the same strand (i.e. take D to be the result of a type one Reidemeister move that increases crossing number). (1) Colloquially, the assertion here is that at every stage δ of the coloring process, each color in the diagram corresponds to a connected arc of K. We verify this claim by induction on u, where u denotes the stage of the coloring process. For u = 0, f0−1 (j) = {sij } is connected. Now assume fu−1 (j) is connected for all u < t, and let u = t with t > 0. By definition of the coloring move, ∃si ∈ s(D) such that si is the unique strand to which a color is assigned at stage t. That is, {si } = At \ At−1 and si is adjacent to some sl ∈ At−1 . −1 Moreover, by definition of the coloring move, ft (si ) = ft−1 (sl ) = c. Therefore, ft−1 (c) = ft−1 (c) ∪ {si } is −1 −1 connected since ft−1 (j) is connected by assumption, and si is adjacent to a strand in ft−1 (c). Similarly, by −1 definition of the coloring move, ∀r 6= c, ft−1 (r) = ft−1 (r), which is connected by the inductive hypothesis. (2) The statement is that h attains a unique local maximum along each color; in fact, the local maximum in every color is the seed strand. Intuitively, this follows from the fact that, by the definition of h, at every stage u > 0 of the coloring process, the single strand {sa } = Au \ Au−1 has the property that h(sa ) = min{h(sc )| sc ∈ Au }, so sa can not possibly introduce a new local maximum in its color. We formalize this argument by induction on u. At stage u = 0, each color j ∈ {1, 2, . . . , k} corresponds only to its seed strand. That is, f0−1 (j) = {sij }, and h trivially attains a single local maximum on this set. Now assume that for u < t, h has a unique local maximum on fu−1 (j). Set u = t. There exists a strand {si } = At \ At−1 with the property that si is adjacent to some sl ∈ At−1 and ft (si ) = ft (sl ) = c. But h(si ) = −t and h(sl ) > −t, since sl was colored before stage t of the coloring process. Because si and sl are adjacent, si is not a local maximum in ft−1 (c) and the number of local maxima in each color remains unchanged. (3) This claim can be rephrased by saying that if sp and sq , two strands adjacent at a crossing, have been assigned the same color, then the over-strand sr at this crossing cannot have been the last one of the three to attain a color. The intuitive reason is that the definition of the coloring move dictates that sr must have been assigned a color in order for the coloring to be extended from sp to sq or vice-versa. To prove this assertion, WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS 5 let h(sc ) = −τc for c ∈ {p, q, r} and assume for contradiction that h(sr ) ≤ min{h(sp ), h(sq )}. Recall that sq 6= sp . Hence, without loss of generality, we can assume τp > τq ≥ τr . Denote fn (sp ) = fn (sq ) = j and consider iq such that {sq } = Aiq \ Aiq −1 . By definition of the coloring move, at stage δiq , the color j was extended to the strand sq from an adjacent strand sl . That is, ∃sl ∈ Aiq −1 such that sl is adjacent to sq and fiq −1 (sl ) = j. By assumption, τr ≤ τq , so sr ∈ / Aiq −1 . In particular, since D is the diagram of a non-trivial knot, sl 6= sp . Moreover, sq is adjacent to both sl and sp . Additionally, since τp > τq , we have sl , sp ∈ fi−1 (j) whereas sq ∈ / fi−1 (j). But we know from (1) that fi−1 (j) is connected. Thus, since D is q −1 q −1 q −1 −1 the diagram of a knot, fiq −1 (j) must contain all arcs in D except sq . If sr and sq are distinct strands, this contradicts the assumption that sr has not been colored by stage iq − 1. If sr = sq , it follows that, at stage iq , the entire diagram is colored and f (si ) = j for all i ∈ {0, 1, . . . , n}. This implies that ω(D) = 1, so the meridional rank of K is 1, contradicting our assumption that D is a diagram of a non-trivial knot.  Remark. The connectedness of K plays an essential role in the proof of (3), and this argument does not generalize without modification to the case of links. In fact, the Hopf link violates (3). In order to extend Proposition 2.2 to links, we need to consider links which exhibit the above exception. Definition 2.3. A link L is S 3 is cut-split if there exists an unknotted component U of L such that U bounds an embedded disk B 2 in S 3 with int(B 2 ) ∩ L = ∅ or L meets int(B 2 ) transversely in a single point. We call U the splitting component of L. A link diagram D is cut-split if there exists sp , sq ∈ s(D) that are adjacent at some crossing of D such that sp = sq or if there exists an element of s(D) that is a simple closed curve. The standard diagram of the Hopf link is cut-split, with either of the link components as a splitting component. More generally, if D is a cut-split diagram of a link L, then L is cut-split. Indeed, a self-adjacent strand of D corresponds to a component of L that bounds a disk B 2 ⊂ S 3 whose interior meets L transversely in one or zero (see Figure 1) points. We leave it to the reader to verify the following easy facts about cut-split links and diagrams. Remark 2.4. Let L ⊂ S 3 be a link. (1) If L is cut-split with splitting component U , then β(L) = β(L \ U ) + 1. (2) If D is a cut-split link diagram of L, U is the splitting component of L that projects to the self adjacent strand sp or to a simple closed curve, and D′ is the the natural diagram of L \ U corresponding to the removal of sp from D, then ω(D) = ω(D′ ) + 1. Proposition 2.5. Let D be a diagram of a link L such that D is not cut-split. Assume D can be kc(D) meridionally colored via (A0 , f0 ) → . . . → (An , fn ), where n = c(D) − k, and let {αm }m=1 and h be as above. The following hold: (1) For every j ∈ {1, 2, . . . , k}, δ ∈ {0, 1, . . . , n}, fδ−1 (j) is connected. (2) For any j ∈ {1, 2, . . . , k}, h has a unique local maximum on {si | si ∈ fn−1 (j)} when this set is ordered sequentially by adjacency. (In the special case when {si | si ∈ fn−1 (j)} is the set of all strands corresponding to the projection of a single component of L, this set is ordered cyclically by adjacency) (3) Let sp , sq ∈ s(D) be adjacent understrands at a crossing c in D, and denote the overstrand at this crossing by sr . If fn (sp ) = fn (sq ), then one of the following holds: (a) h(sr ) > min{h(sp ), h(sq )}, (b) the set {fn−1 (fn (sp ))} corresponds to the projection of one component U of L, and c is the unique crossing incident to p(U ) with the property that h(sr ) ≤ min{h(sp ), h(sq )}. Proof. (1) and (2) follow without modification from the proof of Proposition 2.2 parts (1) and (2). (3) First, we reestablish the setup for the proof of Proposition 2.2 part (3). Let h(sc ) = −τc for c ∈ {p, q, r} and assume that h(sr ) < min{h(sp ), h(sq )}. Recall that sq 6= sp since D is not cut-split. Hence, without loss of generality, we can assume τp > τq ≥ τr . Denote fn (sp ) = fn (sq ) =: j and let iq be such that {sq } = Aiq \ Aiq −1 . By definition of the coloring move, at stage δiq , the color j was extended to the strand sq from an adjacent strand sl . That is, ∃sl ∈ Aiq −1 such that sl is adjacent to sq and fiq −1 (sl ) = j. By assumption, τr ≤ τq , so sr ∈ / Aiq −1 . 6 WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS If sl = sp , then sp is the only strand adjacent to sl , and {sp , sq } = {fn−1 (j)} is the set of all strands corresponding to the projection of a single component U of L. The projection p(U ) is then incident to exactly two crossings, c and c′ . Moreover, by the definition of coloring move, the overstrand at c′ is contained in Aiq −1 . This establishes that situation (b) described in the proposition holds. Now assume that sl 6= sp and note that sq is adjacent to both sl and sp . Additionally, since τp > τq , as in the proof of Proposition 2.2 (3), we have sl , sp ∈ fi−1 (j) whereas sq ∈ / fi−1 (j). But we know from (1) that q −1 q −1 −1 −1 fiq −1 (j) is connected. Thus, fiq −1 (j) must contain all strands corresponding to the projection of a single component U of L except sq . Hence, iq is the first stage of the coloring process at which every strand of fn−1 (j) is colored. Assume that there exists a second crossing c′ such that h(s′r ) ≤ min{h(s′p ), h(s′q )} where s′r is the overstrand at c′ , s′p and s′q are adjacent understands at c′ and s′p , s′q ∈ fn−1 (j) (i.e. s′p and s′q are contained in the projection of U ). By repeating the above argument for the strands incident to the crossing c′ , at stage δi′q , the color j was extended to the strand s′q from an adjacent strand s′l and i′q is the first stage of the coloring process at which every strand of fn−1 (j) is colored. Thus, i′q = iq and s′q = sq . By definition, the strand sq is an understrand at the crossings c and c′ . Moreover, sq is an understrand at exactly two crossings, one of which has an uncolored overstrand at stage iq and one of which has a colored overstrand at stage iq . Since both c and c′ have uncolored overstands at stage iq , then c = c′ .  Proof of the Main Theorem. We prove the theorem by induction on N , the number of components of L. Step I. Let N = 1, that is, L =: K is a knot. If K is trivial, then ω(K) = β(K) = 1, so we can assume K is non-trivial. It suffices to show that, if K admits a diagram D which is k-meridionally colorable, then β(K) ≤ k. We use Proposition 2.2 to construct from D a smooth embedding of K in R3 with exactly k local maxima. We begin by embeding D in the plane z = −c(D) − 2 in R3 . By assumption, D can be k-meridionally colored c(D) via some succession of coloring moves (A0 , f0 ) → . . . → (An , fn ), where n = c(D) − k, Let {αm }m=1 be the associated coloring sequence and let and h be its height function, as in Definition 2.1. Note that the range of h is the set {−1, −2, . . . , −c(D)}. Next, embed a copy, denoted ŝi , of each strand si of D in the plane z = h(si ) in such a way that the orthogonal projection of R3 to the plane z = −c(D) − 2 maps ŝi to si . We call ŝi the lift of si . In what follows, we show that the strands ŝi can be connected in such a way that the resulting knot has D as the diagram of its projection to the plane z = −c(D) − 2. That is, we construct arcs sij in R3 connecting the S c(D) c(D) lifts ŝi and ŝj of adjacent strands si , sj of D, in such a way that (∪r=1 ŝr ) (∪r=1 srr+1 mod c(D) ) ∼ = S1 and the sij correspond to the arcs of K which are not visible in D, i.e. to the deleted underpasses at each crossing. Set-up: Let c be an arbitrary crossing in D. Label the overstrand at c by sk and the understrands by si and sj , in some order. Pick a small ε > 0 in such a way that the ball Bε2 (c) in the plane z = −c(D) − 2 has non-trivial connected intersection with each strand si , sj , sk and is disjoint from all other strands of D. Consider the infinite cylinder Bε2 (c) × R, where R denotes the z direction. By construction, this cylinder intersects ŝi , ŝj and ŝk and is disjoint from the lifts of the remaining strands. We will embed an arc sij into Bε2 (c)×R in such a way that ŝi ∪sij ∪ ŝj is a continuous arc and the orthogonal projection of ŝk ∪(ŝi ∪sij ∪ ŝj ) to the plane z = −c(D) − 2 coincides with the corresponding section of D. (Informally, sij will connect ŝi to ŝj , and it will pass “under” ŝk .) Case 1: Assume fn (si ) = fn (sj ), that is, at the end of the coloring process, the strands si and sj are assigned the same color. Let sij be a smooth, monotonically decreasing curve which connects the endpoints of ŝi and ŝj that are contained in the cylinder Bε2 (c) × R and which has the property that sij itself is contained entirely within the cylinder. Recall that, by Proposition 2.2, h(sk ) > min{h(si ), h(sj )}. This implies that sij can be chosen so that the orthogonal projection of ŝk ∪ (ŝi ∪ sij ∪ ŝj ) to the plane z = −c(D) − 2 is the subset of D, as desired. (Precisely, for any ε1 ∈ (0, ε), one can guarantee that the intersection of sij and half-space z ≥ h(sk ) is contained entirely outside the cylinder Bε21 (c) × R.) See Figure 3. Case 2: Assume fn (si ) 6= fn (sj ), that is, at the end of the coloring process, the strands si and sj are assigned distinct colors. Let xij denote point in (Bε2 (c) × R) ∩ {z = −c(D) − 1} with the property that the orthogonal WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS 7 Figure 3. The construction of sij in Case 1. Figure 4. The construction of sij in Case 2. projection of xij to the plane z = −c(D) − 2 coincides with the crossing c in the diagram. Construct sij as the union of two smooth, monotonic arcs, contained entirely within Bε2 (c) × R, connecting xij to those endpoints of ŝi and ŝj which are themselves contained in the cylinder. Because h(sk ) ≥ −c(D) > −c(D) − 1, these two monotonic arcs can be chosen so that the orthogonal projection of ŝk ∪ (ŝi ∪ sij ∪ ŝj ) to the plane z = −c(D) − 2 is once again a subset of D. (Precisely, for any ε1 ∈ (0, ε), one can guarantee that the intersection of sij and the cylinder Bε21 (c) × R is contained in Bε21 (c) × [c(D) − 1, c(D) − 21 ].) See Figure 4. In both Case 1 and Case 2, the above construction amounts to a careful way of joining a pair of adjacent strands in the diagram so that the overstrand at the crossing where they meet is preserved. Performing this construction at every crossing of D therefore reconstructs an embedding of K. In order to produce from here an embedding with the desired number of local extrema, we perturb each lift ŝi to obtain a new lift ŝ′i in the following way. Let cii+1 denote the point in sii+1 that projects to a vertex of p(K). If h(si ) is not the unique local maximum of h on the set fn−1 (fn (si )), we let the subarc of si−1i ∪ ŝ′i ∪ sii+1 from ci−1i to cii+1 be a smooth monotonic arc, strictly increasing or strictly decreasing as dictated by the values of h(si−1 ) and h(si+1 ). On the other hand, if h(si ) is the unique local maximum of h on the set fn−1 (fn (si )), we let the subarc of si−1i ∪ ŝ′i ∪ sii+1 from ci−1i to cii+1 be a smooth arc increasing monotonically to the midpoint of ŝ′i and decreasing monotonically thereafter. 8 WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS This construction produces a smooth embedding of K in R3 with exactly k local maxima, corresponding to the seed strands in each color. The local minima correspond to the points xij , and thus project to those crossings in D at which the diagram changes color. Step II. Now let L be a link of N > 1 components, and assume that ω(L′ ) = β(L′ ) for all links L′ of fewer than N components. First consider the case that L has a diagram with ω(D) = ω(L) and such that D is not cut-split. In this situation, Proposition 2.5 applies. A minor adaptation of the proof given in Part I will establish that ω(L) = β(L). We adopt the identical setup and begin by re-examining the cases. Case 1: Assume fn (si ) = fn (sj ) and h(sk ) > min{h(si ), h(sj )}. Construct sij exactly as in Case 1 of Step I. Case 2: Assume fn (si ) 6= fn (sj ). Construct sij exactly as in Case 2 of Step I. Case 3: Assume fn (si ) = fn (sj ) and h(sk ) ≤ min{h(si ), h(sj )}. By Proposition 2.5, the set {fn−1 (fn (si ))} corresponds to the projection of a single component U of L and c is the unique crossing incident to p(U ) with the property that h(sk ) ≤ min{h(si ), h(sj )}. Then, exactly as in Case 2 of Proposition 1.3, we construct sij as the union of two smooth, monotonic arcs, connecting xij to endpoints of ŝi and ŝj . Moreover, these two monotonic arcs can be chosen so that the orthogonal projection of ŝk ∪ (ŝi ∪ sij ∪ ŝj ) to the plane z = −c(D) − 2 is once again a subset of D. Note that U is monochromatic, and the arc sij contains the unique local minimum in this color of the constructed embedding. Performing the above construction at every crossing of D reconstructs an embedding of L. As in Step I, we can perturb this embedding slightly to produce a smooth embedding of L in R3 with exactly ω(D) local maxima, corresponding to the seed strands in each color. This completes the proof of the proposition in the case when L is a link with a diagram D that is not cut-split and with ω(D) = ω(L). Now allow L to be an arbitrary link of N components and let D be a diagram of L such that ω(L) = ω(D). If D is not cut-split, then ω(L) = β(L) by our previous argument. Hence, we can assume both D and L are cut-split with splitting component U . Recall that, by Remark 2.4, β(L) = β(L \ U ) + 1. Moreover, U is the splitting component of L that projects to a self-adjacent strand or a simple closed curve in s(D). By Remark 2.4, if D′ is the the natural diagram of L \ U corresponding to the removal of the self-adjacent strand or simple closed curve from D, then ω(D) = ω(D′ ) + 1. Hence, ω(L \ U ) ≤ ω(D) − 1 = ω(L) − 1. By the induction hypothesis, we have ω(L \ U ) = β(L \ U ) = β(L) − 1. Thus, β(L) − 1 = ω(L \ U ) ≤ ω(L) − 1. Since ω(L) ≤ β(L) for all L, it follows that β(L) = ω(L), completing the proof.  3. Applications and further questions We begin with the proofs of Corollary 1.4 and Theorem 1.5, which relate the bridge number to the twist number of links and the hyperbolic volume of prime alternating links. Subsequently, we discuss applications of Theorem 1.3 to the tabulation of bridge number. Proof of Corollary 1.4. Given a link L, it follows from the definition of t(L) that L admits a diagram D with exactly t(L) twist regions. These twist-regions are connected via 2t(L) strands to form D. In other words, there are at most 2t(L) strands in D which are not properly contained in a twist region. Declare each of these 2t strands to be a seed strand. This defines a 2t(L)-partial coloring of D with the property that all four strands of D incident to the boundary of any given twist region of D have received a color. Recall that a twist region constitutes either a single crossing or a collection of bigons. Therefore, the coloring move by definition allows us to extend the coloring of strands incident to the boundary of a twist region across the entire region. It follows that β(L) = ω(L) ≤ ω(D) ≤ 2t(L), as claimed.  Proof of Theorem 1.5. Let L be a prime alternating link and let D be a reduced alternating diagram for L. By [13], 12 v3 (t(D) − 2) ≤ vol(L), where v3 is the volume of a regular hyperbolic ideal 3-simplex. By Corollary 1.4, β(L) ≤ 2t(L). Hence, 21 v3 ( 12 β(L) − 2) ≤ vol(L). In order to eliminate the constant term in the previous inequality, we note that Cao and Meyerhoff have shown that the minimum volume of any hyperbolic knot is 2v3 [6]. Hence, we can set C = 61 v3 to insure that Cβ(L) ≤ max( 12 v3 ( 12 β(L) − 2), 2v3 ) ≤ vol(L) for all values of β(L).  The Wirtinger number can also be used to compute bridge numbers of links. Since for any diagram D of a link L we have ω(D) ≥ ω(L) = β(L), the Wirtinger number provides an approach to calculating upper bounds on the bridge number of L. Moreover, as previously noted, the ω(D) for a given diagram D is readily WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS 9 computed; for knot diagrams, this can be done via the computer algorithm outlined in the Appendix. The upper bounds obtained in this manner have turned out to be astonishingly strong. At the start of this project, according to KnotInfo [7], bridge numbers were tabulated for prime knots up to and including 11 crossings. By comparing these bridge numbers to the Wirtinger numbers of minimal diagrams, we verified that for all prime knots with up to 11 crossings the upper bounds on β obtained by computing ω(D) for representative minimal diagrams are sharp. We computed Wirtinger numbers for minimal diagrams of all 12-, 13- and 14- crossing knots as well. The number of knots among them whose minimal diagrams have Wirtinger number 2 coincided exactly with the number of two-bridge knots of 12, 13 and 14 crossings [10]. Therefore, our calculations identify all two-bridge knots in this range. It also follows that all diagrams D with ω(D) = 3 represent three-bridge knots. To complete the tabulation of bridge number for prime 12-crossing knots, we calculated that all such knots have Wirtinger number at most 4, and we checked that the knots whose minimal diagrams have Wirtinger number 4 are not three-bridge. This was done by hand using methods of Jang [11]. (We believe that the same method would allow us to complete the tabulation of bridge number for knots of 13 crossings as well.) Altogether, our computations so far have newly determined the bridge number of approximately 50, 000 prime knots of less than 15 crossings. In addition, as a corollary of these computations, we have verified that for all prime knots of less than 13 crossings, the Wirtinger number of some minimal diagram realizes the bridge number. We propose the following: Question 3.1. (Property M) For which links L is ω(L) realized in a minimum-crossing diagram of L? Our calculations show that all prime knots of up to and including 12 crossings have Property M. We conjecture that all prime 13-crossing knots do as well, and that the Wirtinger numbers found equal the bridge numbers. Furthermore, by taking connected sums of two-bridge knots, one can construct families of knots which have Property M and whose crossing number and bridge number are unbounded. The question of completely characterizing knots with Property M remains open. Let us now turn an eye back to the Meridional Rank Conjecture. The main theorem of this paper reduces the Conjecture to the following: Question 3.2. Does every link admit a minimal meridional presentation in which all relations arise as iterated Wirtinger relations in a diagram? A positive answer to this question for a class of links would mean that µ(L) = ω(L), which, together with our result β(L) = ω(L), would imply the conjecture for these links. In particular, our point of view casts the Meridional Rank Conjecture as a question about the type of relations in a meridional presentation. 4. Appendix: Computing ω(D) We sketch the algorithm by which we obtained the computational results discussed previously. From now on we work only with knots. Furthermore, we make the following simplifying assumption. Note that coloring a knot diagram D in several colors allowed us to study the combinatorics of the coloring process, which in turn enabled us to count the number of local maxima in the knot embedding we reconstructed from D. This analysis is a bit more subtle than what we need if we are merely asking whether a set A of meridional elements generates the knot group via iterated application of the Wirtinger relations in D. Therefore, for the purpose of calculating ω(D), we do not keep track of the different colors. Instead, we simply ask if a given partial coloring of D can be extended to all of D. (Formally, we compose the function f : A → {1, 2, . . . , k} with the constant function c : {1, 2, . . . , k} → {1}, then we define the coloring move as before.) The algorithm can be broken down into three steps. (1) From the Gauss code of a non-trivial knot diagram D, extract information about which strands are over- and under-strands at every crossing of D. (2) Given a subset A of set of strands s(D), determine if choosing the strands in A as seeds would allow the entire diagram to be colored by iterating the coloring move. (3) Running across all subsets of size k ≥ 2 of s(D), determine if D admits a Wirtinger generating system of size k. The algorithm terminates as soon as the first valid coloring occurs. Now we describe in some detail how these steps are performed. 10 WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS (1) Creating a knot dictionary. Let K be a knot with diagram D. By convention, we label the strands of D by letters. Represent each crossing of D by the (unordered) tuple (a, b), where a and b are the understrands at that crossing. The knot dictionary DK is a map which assigns to each element c of s(D) a subset of the crossings of D. The map is given by DK (c) = {(a, b) | c is the overstrand of (a, b)}.In terms of data structures, the knot dictionary is a map whose keys are the strands of the knot diagram and whose values are subsets of v(D). Example 4.1. The trefoil has knot dictionary D31 = {a → {(b, c)}, b → {(a, c)}, c → {(a, b)}}. We can derive the knot dictionary DK of a knot K by examining its Gauss code GK , using a function we call knot-dictionary(GK ). To illustrate how this function works, we return to the diagram of the trefoil, which has Gauss code G31 = [−1, 3, −2, 1, −3, 2]. Since the negative numbers in the Gauss code correspond to a strand going under a crossing, we see that each strand is described by a subsequence of G31 beginning and ending with a negative number (“wrapping around” the sequence if needed). Since there are three strands in this diagram of the trefoil, there are three corresponding subsequences of G31 , which we have labeled to be consistent with the knot dictionary representation in the example above. These three subsequences are a = [−1, 3, −2], b = [−2, 1, −3], and c = [−3, 2, −1]. Once we have determined which subsequences correspond to which strands, we next determine the crossings at which they are overstrands. We do this by examining the positive integers in each subsequence. For example, since a = [−1, 3, −2], the strand labeled a is the overstrand at the crossing labeled 3. Then, since b and c contain −3, this indicates that they are under this same crossing and are thus the two strands under strand a at crossing 3. We then assign the tuple (b, c) to a. Since a contains no more positive integers, we have found all the crossings a is over and have completed the knot dictionary entry for a, which is D31 (a) → {(b, c)}, as in our above example. Repeating this process for the remaining subsequences results in the same knot dictionary as in our above example: D31 = {a → {(b, c)}, b → {(a, c)}, c → {(a, b)}}. (2) Extending a partial coloring. Once we have a knot dictionary DK , we can determine whether a given set of seed strands A leads to a coloring of every strand in the diagram. We do this using a function called color. Consider a crossing (p, q) of the diagram and assume that p is not colored. The partial coloring can be extended at this crossing if and only if both q and the overstrand are colored. Running through the list of crossings of D in any order allows us to determine if a coloring move can be performed. The function color works as follows. Make a copy, C, of the seed strands A, and iterate through the keys of DK that are in C. For each of these keys in C, say a, examine each crossing in DK (a). For each (b, c) ∈ DK (a), if C contains either of b or c, add the other one to C. Repeat this step until either all the strands of D are added to C (in which case, A has been shown to be a Wirtinger generating system) or one entire iteration through all a ∈ C and all (b, c) ∈ DK (a) is completed without adding new strands to C (in which case, A has been shown to not be a Wirtinger generating system). (3) Finding a minimal coloring. We define a function calculate-w which determines the Wirtinger number for a knot diagram D. Given Gauss code GK , we first call knot-dictionary(GK ) to create the corresponding knot dictionary DK . Then, for n ranging from 1 to DK .size (the number of keys in DK , i.e., the number of strands in the knot diagram), we repeat the following: we call combinations(DK .keys, n), which returns X, the set of all combinations of n strands; then, for each set of seed strands A ∈ X, we call color(DK , A). If color(DK , A) results in coloring the entire diagram, we return A.size, the number of strands in A. Otherwise, we pick a new set of seed strands A′ from X and repeat this process. If none of the combinations in X lead to a complete coloring, we increment n and repeat the process until such a combination is found. Note that every non-trivial knot diagram is colorable by c(D) − 1 strands, so this algorithm is guaranteed to return DK .size − 1 in the worst case. Since the function color is applied to every subset of s(D) of a given size k ≤ ω(D), the algorithm runs in factorial time. However, ω(D) << c(D) in general, and the algorithm terminates when the first valid coloring occurs. As a result, the running time is short in practice. Computing the Wirtinger numbers of all diagrams in the Knot Table of up to 14 crossings took approximately 10 minutes on a weak fashionable laptop. That said, it is evident that the algorithm performs many redundant checks, and its efficiency can definitely be improved, should the running time increase unreasonably with c(D). We also remark that the above procedure for calculating ω(D) can be extended to link diagrams, by implementing a few modifications to handle Gauss code for multiple-component links. WIRTINGER SYSTEMS OF GENERATORS OF KNOT GROUPS 11 Acknowledgement The authors would like to thank Michel Boileau for suggesting the name Wirtinger number. References [1] M. Boileau, Ederson Dutra, Y. Jang, and R. Weidmann. Meridional rank of knots whose exterior is a graph manifold. arXiv preprint arXiv:1608.01570. [2] M. Boileau, Y. Jang, and R. Weidmann. Meridional rank and bridge number for a class of links. preprint. [3] Michel Boileau and Heiner Zieschang. Nombre de ponts et générateurs méridiens des entrelacs de Montesinos. Comment. Math. Helv., 60(2):270–279, 1985. [4] Michel Boileau and Bruno Zimmermann. The π-orbifold group of a link. Math. Z., 200(2):187–208, 1989. [5] Richard Sean Bowman, Scott Taylor, and Alexander Zupan. Bridge spectra of twisted torus knots. Int. Math. Res. Not. IMRN, (16):7336–7356, 2015. [6] Chun Cao and G. Robert Meyerhoff. The orientable cusped hyperbolic 3-manifolds of minimum volume. Invent. Math., 146(3):451–478, 2001. [7] J.C. Cha and C. Livingston. Knotinfo: Table of knot invariants. http://www.indiana.edu/ knotinfo, December 5, 2016. [8] Abhijit Champanerkar, David Futer, Ilya Kofman, Walter Neumann, and Jessica S. Purcell. Volume bounds for generalized twisted torus links. Math. Res. Lett., 18(6):1097–1120, 2011. [9] C. R. Cornwell and D. R. Hemminger. Augmentation rank of satellites with braid pattern. arXiv preprint arXiv:1408.4110, 2014. [10] David De Wit. The 2-bridge knots of up to 16 crossings. Journal of Knot Theory and Its Ramifications, 16(08):997–1019, 2007. [11] Yeonhee Jang et al. Classification of 3-bridge arborescent links. Hiroshima Math. J, 41(1):89–136, 2011. [12] Rob Kirby and Edited Rob Kirby. Problems in low-dimensional topology. In Proceedings of Georgia Topology Conference, Part 2. Citeseer, 1995. [13] Marc Lackenby. The volume of hyperbolic alternating link complements. Proc. London Math. Soc. (3), 88(1):204–224, 2004. With an appendix by Ian Agol and Dylan Thurston. [14] Martin Lustig and Yoav Moriah. Generalized Montesinos knots, tunnels and n-torsion. Math. Ann., 295(1):167–189, 1993. [15] Markus Rost and Heiner Zieschang. Meridional generators and plat presentations of torus links. J. London Math. Soc. (2), 35(3):551–562, 1987.