aa r X i v : . [ m a t h . G T ] N ov THE VOLUME OF POSITIVE BRAID LINKS
SEBASTIAN BAADER
Abstract.
Based on recent work by Futer, Kalfagianni and Pur-cell, we prove that the volume of sufficiently complicated positivebraid links is proportional to the signature defect ∆ σ = 2 g − σ . Introduction
A positive braid on n strings is a product of positive standard gener-ators of the braid group B n . The canonical closure of a positive braid β ∈ B n is a link ˆ β with at most n components. Let b be the first Bettinumber of the canonical fibre surface of the link ˆ β . Furthermore, let σ be the signature invariant of the link ˆ β . The quantity ∆ σ ( ˆ β ) = b − σ is called the signature defect of the link ˆ β . In the case of knots, this issimply 2 g − σ , where g is the minimal genus of ˆ β . The main result ofthis note is a volume estimate in terms of the signature defect. Theorem 1.
Let ˆ β ⊂ S be a hyperbolic link associated with a suffi-ciently complicated positive braid β . Then v ∆ σ ( ˆ β ) ≤ vol ( S \ ˆ β ) < v ∆ σ ( ˆ β ) , where v = 1 . ... and v = 3 . ... are the volumes of a regularideal tetrahedron and octahedron, respectively. Here a positive braid is sufficiently complicated, if it is a product ofpowers σ ki , with k ≥
3. The twist number t ( β ) of a sufficiently com-plicated braid β is the minimal number of factors in such a product.As discussed in [2], the link ˆ β is hyperbolic, if and only if β is prime.Equivalently, β contains at least two non-consecutive factors of theform σ ki , for all i ≤ n −
1. In the recent monograph [2], Futer, Kalfa-gianni and Purcell determined tight bounds for the volume of variousfamilies of hyperbolic links, in terms of the twist number. In particular,they proved the following volume estimates for sufficiently complicatedpositive braid links.
Theorem 2 ([2], Theorem 9.7) . Let ˆ β ⊂ S be a hyperbolic link asso-ciated with a sufficiently complicated positive braid β . Then v t ( β ) ≤ vol ( S \ ˆ β ) < v ( t ( β ) − . Theorem 1 is an immediate consequence of this and the following esti-mates for the twist number, which we will prove in the next section.
Theorem 3.
Let ˆ β ⊂ S be a hyperbolic link associated with a suffi-ciently complicated positive braid β . Then
12 ∆ σ ( ˆ β ) ≤ t ( β ) ≤
212 ∆ σ ( ˆ β ) . Acknowledgements
I would like to thank Julien March´e for drawing my attention to thework of Futer, Kalfagianni and Purcell.2.
Twist number and signature defect
The signature σ ( L ) of a link L is defined as the signature of anysymmetrised Seifert matrix V + V T associated with L . Let b be thefirst Betti number of a minimal genus Seifert surface for L . Then − b ≤ σ ( L ) ≤ b . In particular, the signature defect ∆ σ = b − σ is positive. The proofof Theorem 3 relies on three elementary facts about the signature in-variant.(1) The signature defect of the closure of σ n is zero.(2) The signature defect of the closure of σ k σ k σ k σ k is two, pro-vided all k i ≥ σ = 0, see [1]).(3) Let Σ ⊂ e Σ be an inclusion of Seifert surfaces which induces aninjection on the level of first homology groups. Then∆ σ ( ∂ e Σ) ≤ ∆ σ ( ∂ Σ) + 2( b ( e Σ) − b (Σ)) . Proof of Theorem 3.
Let β be a sufficiently complicated positive braidwith twist number t ( β ); let e Σ ⊂ S be the canonical fibre surface ofthe link ˆ β (see Stallings [3]). By cutting the surface e Σ along an intervalon the left of every twist region σ ki , as sketched in Figure 1, we obtaina subsurface Σ ⊂ e Σ. HE VOLUME OF POSITIVE BRAID LINKS 3 −→ Figure 1.
The boundary of Σ is a disjoint union of connected sums of torus linkson two strings. According to fact (1), the signature defect ∆ σ ( ∂ Σ) iszero. Using fact (3), we conclude∆ σ ( ˆ β ) = ∆ σ ( ∂ e Σ) ≤ b ( e Σ) − b (Σ)) ≤ t ( β ) . This is the first inequality of Theorem 3.For the second inequality, we first consider the case of 3-string braids.Let β ∈ B be a sufficiently complicated positive braid with hyperbolicclosure. Then t ( β ) ≥
4. Moreover, β contains at least t ( β )7 consecutivesubwords of the form σ k σ k σ k σ k (a better estimate would be ( t ( β ) − β contains at least t ( β )7 disjointsubsurfaces whose homology groups are orthogonal with respect to thesymmetrised Seifert form. All these contribute two to the signaturedefect. In particular, 27 t ( β ) ≤ ∆ σ ( ˆ β ) . Now let us turn to the case of higher braid indices. Our goal is to finda large number of non-interfering subwords of the form σ k i σ k i +1 σ k i σ k i +1 in β . For this purpose, let us partition the strings of the braid β ∈ B n into three subsets S , S , S , according to their index modulo 3. Asimple counting argument shows that one of the subsets S j carries atleast t ( β )21 disjoint subwords of the desired form. Here ‘carrying’ meansthat the central string i + 1 of the word σ k i σ k i +1 σ k i σ k i +1 belongs to S j .Indeed, let us put dots on the strings of the closed braid ˆ β , one betweeneach pair of adjacent twist regions, as in σ ki σ li +1 or σ ki σ li − . Altogether,there are at least t ( β ) dots, otherwise the number of factors σ ki would SEBASTIAN BAADER not be minimal. One of the subsets S j carries at least t ( β )3 dots, thusat least t ( β )3 · disjoint subwords of the form σ k i σ k i +1 σ k i σ k i +1 . Here it isimportant that neighbouring strings of S j are at distance three. Asbefore, we conclude 221 t ( β ) ≤ ∆ σ ( ˆ β ) . (cid:3) References [1] S. Baader:
Positive braids of maximal signature , arXiv:1211.4824.[2] D. Futer, E. Kalfagianni, J. Purcell:
Guts of surfaces and the colored Jonespolynomial , Lecture Notes in Mathematics, 2069. Springer, Heidelberg, 2013.[3] J. Stallings: