Untwisting 3-strand torus knots

We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus of torus knots from 2/3 to 14/27.


Introduction
The braid group on three strands B 3 is generated by two elements a, b satisfying the braid relation aba = bab. In this note, we are interested in the natural closure of the positive braid (ab) n in S 3 , known as torus link of type T (3, n). Whenever n ∈ N is a multiple of 3, the link T (3, n) has three components; otherwise it is a knot. The topological 4-genus g t (K) of a knot K ⊂ S 3 is defined to be the minimal genus among all surfaces Σ ⊂ D 4 , embedded in a locally flat way, with boundary ∂Σ = K. As with the smooth version of the 4-genus invariant, the topological 4-genus of knots K is bounded below by the signature invariant [12]: g t (K) ≥ |σ(K)|/2. The same lower bound holds with the signature invariant replaced by the maximum value of the Levine-Tristram signature function outside of the set of roots of the Alexander polynomial ∆ K (t) of K σ(K) := max |σ ω (K)|.
We believe that the equality g t = σ/2 holds for a much larger class of torus knots, possibly for all. This can be seen as a topological counterpart of the local Thom conjecture, which states that the smooth 4-genus g s of torus knots coincides with their Seifert genus [9,15,13]. Unlike in the smooth case, where the hard part is finding suitable lower bounds, the difficulty in the topological case is figuring out genus-minimising surfaces (see [14] and [2] for first attempts in this direction). We will not see any of these surfaces. Rather, we will find a precise upper bound for the topological 4-genus via an operation called null-homologous twisting, which has recently received some attention [6,8,10,11]. A null-homologous twist is an operation on oriented links that inserts a full twist into an even number 2m of parallel strands, m of which point upwards, and m of which point downwards (see e.g. Figure 3). Throughout this paper, we will use the term twist for a null-homologous twist. The case of two strands corresponds to a simple crossing change. For a knot K, we define the untwisting number t(K) to be the minimal number of twists needed to transform K into the trivial knot, as in [6]. Relying on Freedman's disc theorem [3], McCoy proved that the untwisting number is an upper bound for the topological 4-genus of knots [10]. This is the tool we use to construct the genus-minimising surfaces in Theorem 1.
Let us take another look at the resemblance of the smooth and topological setting. Writing s and u for the Rasmussen invariant and unknotting number, respectively, it follows from the (smooth) local Thom conjecture that the inequalities which hold for all knots K, become equalities for all torus knots: s(T (p, q))/2 = g s (T (p, q)) = u(T (p, q)).
Thus in the topological setting, the untwisting number apparently takes the place that the unknotting number has in the smooth setting. Untwisting might very well lead to the equality g t = σ/2 = t for all torus knots. For the time being, we show that the equality is off by at most 1 for torus knots with four and six strands.
Moreover, for all natural numbers n coprime to 6, McCoy also developed an induction scheme that allows to estimate the asymptotic ratio between the topological 4-genus and the Seifert genus of torus knots [10]: Theorem 3.
The proof of Theorem 1 makes use of a calculus for positive 3-braids introduced in [1], which we present in the next section. Sections 3 and 4 contain the proofs of Theorem 1 and Proposition 2. The latter follows from the former by untwisting torus knots on four and six strands via torus knots on three strands. Theorem 3 follows from McCoy's induction scheme, which we briefly review in the last section.
For later use, we record a family of positive braid presentations for iterated full twists on three strands.
The key observation is that the two braids abbaabba and bb are related by a sequence of two twists, as shown in Figure 1. Here the first arrow stands for a twist on four strands, while the second arrow is a simple crossing change.
= → → Figure 1. Sequence of two twists As a consequence, the double full twist on three strands, (ab) 6 = abbaabbabbbb, is related to the braid b 6 (and also to a 6 ) by a sequence of two twists. For the first knot, T (3, 7), we turn the braid (ab) 7 into a 7 b by two twists, and into ab by another three crossing changes, thus showing t(T (3, 7)) ≤ 5. In order to deal with the other two knots, we use the notation A = a −1 , B = b −1 . We write (ab) 10 = (ab) 12 BABA = (ab) 12 ABAA = AB(ab) 12 AA, which transforms into ABb 6 a 6 AA = Ab 5 a 4 by a sequence of four twists, and into Aba 2 by another three twists. The closure of the last braid is the trivial knot; this shows t(T (3, 10)) ≤ 7. For the knot T (3, 13), we observe that the braid (AB) 5 = (AB) 6 ba = A 3 BA 3 BA 3 Bba = A 3 BA 3 BA 2 represents the torus knot T (3, −5). Therefore, we can write (ab) 13 = (ab) 18 A 3 BA 3 BA 2 = A 3 (ab) 12 BA 3 (ab) 6 BA 2 , which transforms into A 3 a 6 b 6 BA 3 a 6 BA 2 = a 3 b 5 a 3 BA 2 by a sequence of six twists, and into a 3 baBA 2 = a 2 bA by another three twists. This shows t(T (3, 13)) ≤ 9.
By Lemma 4, we have which contains precisely 2k + 4 pure powers of A. We slide one double full twist (ab) 6 to the right of each power of A and transform it into a 6 by a sequence of 2(2k + 4) twists, in total. This leaves us with the braid Sliding the half-twist aba from the left to the middle yields Then we transform the middle part b 2 abaBa 3 B = b 3 a 4 B into the empty braid by three crossing changes. What remains is the braid b 2 Aa 2 B, whose closure is the trivial knot. Therefore t(T (3, 6k +16)) ≤ 2(2k +4)+3 = 4k +11, in accordance with statement (1).
To be more precise, one has to take into account the parity of k in the above two formulas: for even k, the signature function attains its maximum in the classical signature, σ = σ; for odd k, we have σ = σ + 2. This is an immediate consequence of the Dedekind sum formula for the Levine-Tristram signature function of torus links (see Proposition 1 in [7] or Proposition 5.1 in [4]).

Untwisting torus knots with four and six strands
To prove Proposition 2, we essentially untwist torus knots with four and six strands to torus knots on three strands, and conclude using Theorem 1. As is the case for torus knots on three strands, the formulas for σ follow quickly from [7,4]. We denote the standard Artin generators of the braid groups B 4 and B 6 by a, b, c and a, b, c, d, e, respectively.
For torus knots with four strands the crucial move is to transform three full twists on four strands, (abc) 12 , into four full twists on three strands, (bc) 12 (or (ab) 12 ), by four twist operations. This can be seen by composing the braids in Figure 2 with (bc) 9 . = → → Hence, for n = 12k + ε, with ε ∈ {±1, ±3, ±5}, one may change T (4, n) = L[(abc) n ] by 4k twists into L[(ab) 12k (abc) ε ] =: K ε . We now consider the possible values of ε one by one, showing T (4, n) ≤ n + 1 in each case, thus completing the proof of the first part of Proposition 2.

Asymptotic genus ratio
The key point in McCoy's induction scheme is that a positive full twist in a braid with 2n strands can be transformed into two parallel copies of positive double full twists in n strands, with a single twist operation (see Lemma 13 in [10]). This is shown in Figure 2, for n = 2, and was used in the previous section for n = 2 and n = 3. When iterating this operation on successive powers of two, one gets 2/3 as an upper bound for the asymptotic ratio g t /g for torus knots with increasing parameters. We will apply the same procedure, starting from braids with 3 strands, successively multiplying the strand number by two: (1) T (6, 6) transforms into the disjoint union of two copies of T (3, 6) by one twist, (2) T (12, 12) transforms into the disjoint union of two copies of T (6,12) by one twist, then into the disjoint union of four copies of T (3, 12) by 4 more twists, (3) T (24, 24) transforms into the disjoint union of two copies of T (12,24) by one twist, then into the disjoint union of eight copies of T (3, 24) by 4(1 + 4) = 20 more twists, (4) T (3 · 2 k , 3 · 2 k ) transforms into the disjoint union of 2 k copies of T (3, 3 · 2 k ) by a total number of 1 + 4 + 16 + . . . + 4 k−1 = 1/3 · (4 k − 1) twists. By Theorem 1, the untwisting number of T (3, 3 · 2 k ) is of the order 2/3 · (3 · 2 k ) = 2 k+1 .
We are left with the strong belief that the ratio tends to 1/2, in accordance with the asymptotic behaviour of the signature invariant: lim n→∞ σ(T (n, n + 1)) 2g(T (n, n + 1)) = 1 2 .