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Corollary 2. Let P ֒→ S1 × D2 be a knot in the solid torus. Suppose that

• For any K, P (K) has tangle one. • P (K) ≃ U if and only if K ≃ U, where U is the unknot.

Then rk Kh(f P (K)) = 1 if and only if K ≃ U. In particular, the reduced Khovanov homology of the satellite operation defined by P detects the unknot.

For instance, the above corollary shows that the Khovanov homology of the untwisted Whitehead double, the (2, 1)-cable or, more generally, the infinite family of satellites defined by the figure in Section 4, all detect the unknot. Theorem 1 relies on the fact that there is a spectral sequence from the reduced Khovanov homology of K to the Ozsv´ath-Szab´oFloer homology of the branched double cover of K. Tangle unknotting number one knots are those knots whose branched double covers can be obtained by surgery on (other) knots in the three- sphere (see Lemma 4). The theorem then follows from formulas which compute the of manifolds obtained by Dehn surgery on a knot in terms of its knot Floer homology invariants, together with known properties of these latter invariants (e.g. they detect whether the knot is fibered). The corollary should be compared to [4] which uses the fact that Floer homology detects the Thurston norm to show that the Khovanov homology of the 2-cable detects the unknot. In particular, it would be interesting to understand the full extent to which a combination of surgery formula techniques and analysis of the Thurston norm of branched covers can be applied to understand whether Khovanov homology detects the trivial link.

Acknowlegement. This work benefitted from useful discussions with Steve Boyer.

2. Tangles and two-fold branched covers

Let τ ֒→ B3 be a pair of embedded arcs in the 3-ball, intersecting the boundary ∂B3 = S2 transversally in 4 points. The pair T = (B3, τ) is referred to as a tangle. We consider tangles up to homeomorphism in the sense of Lickorish [10]; such a homeomorphism need not fix the boundary in general. Tangles arise naturally as component pieces of knots. Given a knot K ֒→ S3 and an embedding S2 ֒→ S3 such that S2 intersects K transversely in 4 points, the resulting decomposition of S3 into 3-balls restricts to a decomposition of K into tangles, denoted K = T0 ∪ T1. DOES KHOVANOV HOMOLOGY DETECT THE UNKNOT? 3

2 3 If ΣK denotes the two-fold branched cover of S branched over the knot K, then the 2 decomposition K = T0 ∪ T1 lifts to a decomposition of ΣK into manifolds with torus boundary. The two-fold branched cover of a tangle is always a manifold with torus boundary, and a tangle is rational if and only if this cover is a solid torus [10]. Definition 3. A knot K has tangle unknotting number one if there is a decomposition K = T0 ∪ T1 with the property that T0 ∪ T2 is the unknot, where T1 and T2 are rational tangles.

2 ∼ 3 ′ Notice that if K has tangle unknotting number one then ΣK = Sp/q(K ) where 3 ′ p ′ 3 ′ Sp/q(K ) denotes q -framed surgery on a knot K ֒→ S . Moreover, K must be strongly invertible i.e. there is an orientation-preserving involution of S3 which preserves K′ as a set and reverses the orientation of K′. Conversely, to any strongly K′ we may associate a tangle T by taking the quotient of the 3 ′ S r ν(K ) by the Z2-action from the strong inversion. Recall that a strong inversion on K′ gives an involution on S3 rν(K′) fixing a pair of arcs intersecting the boundary transversally in 4 points [22]. The strands τ ֒→ B3 of the tangle then correspond to the image of the fixed point set in the quotient. In summary we have the following. Lemma 4. The two-fold branched cover of a knot K ֒→ S3 is obtained by surgery on a knot K′ ֒→ S3 if and only if K has tangle unknotting number one. Moreover, K′ is strongly invertible.

3. The proof of Theorem 1

The proof relies heavily on the machinery of Heegaard-Floer homology introduced by Ozsv´ath and Szab´o[13]. We will make use of the “hat” version of the theory HF,c and take coefficients in Z2. In [14, Corollary 1.2], Ozsv´ath and Szab´oshow the following: 2 2 H1(ΣL; Z) ≤ rk HF(Σc L) ≤ rk Kh(f L)

This inequality follows from the fact that that there is a spectral sequence with E2 term given by the reduced Khovanov homology of the mirror of L converging to 2 HF(Σc L) [14, Theorem 1.1]. 2 ∼ 3 ′ Now suppose that K is a tangle unknotting number one knot. Then ΣK = Sp/q(K ), p and by passing to the mirror image if necessary we may assume that q > 0 (notice p that since we are considering knots the case q = 0 is omitted). In this setting we obtain 3 ′ p ≤ rk HF(c Sp/q(K )) ≤ rk Kh(f K) 4 MATTHEW HEDDEN AND LIAM WATSON

and Theorem 1 follows immediately if p> 1. Therefore we may reduce to the case of 1 3 ′ Z q -framed surgeries so that S1/q(K ) is a -homology sphere. Specifically, our task is 3 ′ to consider the case rk HF(c S1/q(K )) = 1. That is, the case when surgery on a knot in S3 yields a Z-homology sphere L-space. Recall that a Q-homology sphere Y is an L-space whenever |H1(Y ; Z)| = rk HF(c Y ). In the spirit of [16], we have the following. ′ 3 ′ Proposition 5. Suppose K is a non-trivial knot and S1/q(K ) is an L-space. Then q =1 and K′ is the .

3 ′ 3 ′ Proof. If S1/q(K ) is an L-space, [15, Proposition 9.5] implies that S1 (K ) is an L- space as well. Applying [9, Corollary 8.5] (see also [5, 17]) gives the bound g ≤ 1, where g denotes the genus of the knot K′. Since K′ is non-trivial by assumption, g = 1 and it follows from [3] (see also [12]) that K′ is the trefoil.  Remark 6. An alternate approach comes from the mapping cone formula for rational 3 ′ surgeries [15]. Indeed, a direct calculation shows that rk HF(c S1/q(K )) ≥ 2q − 1, with equality if and only if K′ is the trefoil. Moreover, when the genus of K′ is greater 3 ′ than 1, rk HF(c S1/q(K )) ≥ 4q +1.

Thus we are left to deal with the case when K′ is the trefoil. This is a strongly invertible knot, and the tangle associated to the quotient of S3 rν(K′) has the form shown on the right. This can be worked out by hand as in [2], for example. Alternatively, we remark that this tangle is a sum of two rational tangles: This reflects the Seifert fiber structure in the cover. Indeed, the two tangles lift to a pair of solid tori identified along an essential annulus, the cores of which are singular fibers of order 2 and 3. That this is the unique such tangle follows from the fact that torus knots admit a unique strong inversion [20]. From here it is straightforward to identify the (−2, 3, 5)-pretzel knot (the knot 10124 2 ∼ 3 ′ in Rolfsen’s table [19]) as the appropriate K for which ΣK = S1 (K ). The conclusion now follows by direct calculation: KhoHo [21] confirms that rk Kh(10f 124) = 7, proving Theorem 1.

4. Satellite knots whose Khovanov homology detect the unknot

An interesting consequence of Theorem 1 is that it allows us to show that the Kho- vanov homology of many satellite constructions detects the unknot. To make this pre- ,cise, recall that associated to any non-trivial knot in a solid torus P ֒→ V ∼= S1 × D2 we obtain an operation P (−): {Knots} −→{Knots}, DOES KHOVANOV HOMOLOGY DETECT THE UNKNOT? 5

where P (K) is defined as the image of P under an identification of V with ν(K).1 P (K) is called a satellite of K with pattern P .

Observe that if K1 ≃ K2 then P (K1) ≃ P (K2), so that the operation defined by P does indeed descend to isotopy classes of knots. Given an invariant of a knot, K, this observation allows us to define infinite families of invariants: Simply apply the invariant to all the various satellites of K.

In the case at hand, the invariant we are considering is the reduced Khovanov homol- ogy. Suppose that we choose a pattern P so that P (K) has tangle unknotting number one for every knot K, and so that P (K) ≃ U, if and only if U is the unknot. In this situation, Theorem 1 applies to show that rk Kh(f P (K)) = 1 if and only if P (K) ≃ U which, in turn, happens if and only if K ≃ U. Thus we arrive at Corollary 2.

A simple infinite family of satellite constructions whose Khovanov homologies detect the unknot are provided by the patterns shown in the solid torus on the right, ··· where n denotes the number of half twists. It is | {z } straightforward to verify that each of these patterns n satisfies the hypotheses of Corollary 2. Note that the (2, ±1)-cable of K is obtained for n = ±1. Similarly, the positive (respectively negative) clasp, untwisted Whitehead double of K is ob- tained for n = 2 (respectively n = −2). The case n = 0 is always the unknot, while 1 the convention n = 0 gives rise to the 2-cable of the knot K (this latter satellite is a link, and was handled by a different technique in [4]). We remark that for the satellites specified by the figure, it is straightforward to determine the knot in S3 on which one performs surgery to obtain the double branched covers. If we let Kn denote the satellite of K with pattern given by the figure with n half-twists, then 2 ∼ 3 r ΣKn = S1/n(K#K ) where Kr denotes K with the orientation reversed. Indeed, there is an obvious strong inversion on K#Kr exchanging the two summands. Extending this strong inversion 3 r over the surgery torus yields a strong inversion on S1/n(K#K ). From this, one can 3 see that the quotient is S and the image of the fixed-point set is Kn. See Akbulut and Kirby [1] or Montesinos and Whitten [11] for details. Coupled with Remark 6, this yields the explicit bound

rk Kh(f Kn) ≥ 4n +1,

1 This identification requires a choice of framing which we obtain from a for K. 6 MATTHEW HEDDEN AND LIAM WATSON whenever K is non-trivial. This bound, together with the skein exact sequence for Khovanov homology [7, 18], recovers the fact that the reduced Khovanov homology of the 2-cable of a knot detects the unknot [4].

References

[1] Selman Akbulut and Robion Kirby. Branched covers of surfaces in 4-manifolds. Math. Ann., 252(2):111–131, 1980. [2] Steven A. Bleiler. Prime tangles and composite knots. In and manifolds (Vancouver, B.C., 1983), volume 1144 of Lecture Notes in Math., pages 1–13. Springer, Berlin, 1985. [3] Paolo Ghiggini. Knot Floer homology detects genus-one fibred knots. To appear in Amer. J. Math. [4] Matthew Hedden. Khovanov homology of the 2-cable detects the unknot. arxiv.org/abs/0805.4418v1. [5] Matthew Hedden. On Floer homology and the Berge conjecture on knots admitting lens space surgeries. math.GT/0710.0357. [6] Vaughan F. R. Jones. A polynomial invariant of knots via von Neumann algebras. Bul. Amer. Math. Soc., 12:103–111, 1985. [7] . A categorification of the . Duke Math. J., 101(3):359–426, 2000. [8] Mikhail Khovanov. Patterns in knot . I. Experiment. Math., 12(3):365–374, 2003. [9] P. B. Kronheimer, T. S. Mrowka, P. S. Ozsv´ath, and Z. Szab´o. Monopoles and lens space surgeries. Ann. of Math., 165(2):457–546, 2007. [10] W. B. Raymond Lickorish. Prime knots and tangles. Trans. Amer. Math. Soc., 267(1):321–332, 1981. [11] Jos´eM. Montesinos and Wilbur Whitten. Constructions of two-fold branched covering spaces. Pacific J. Math., 125(2):415–446, 1986. [12] Yi Ni. Knot Floer homology detects fibred knots. Invent. Math., 170(3):577–608, 2007. [13] Peter Ozsv´ath and Zolt´an Szab´o. Holomorphic disks and topological invariants for closed three- manifolds. Ann. of Math. (2), 159(3):1027–1158, 2004. [14] Peter Ozsv´ath and Zolt´an Szab´o. On the Heegaard Floer homology of branched double-covers. Adv. Math., 194(1):1–33, 2005. [15] Peter S. Ozsv´ath and Zolt´an Szab´o. Knot Floer homology and rational surgeries. math.GT/0504404. [16] Peter S. Ozsv´ath and Zolt´an Szab´o. The Dehn surgery characterization of the trefoil and the figure eight knot. math.GT/0604079. [17] Jacob Rasmussen. Lens space surgeries and L-space homology spheres. math.GT/0710.2531. [18] Jacob Rasmussen. Knot polynomials and knot homologies. In Geometry and topology of man- ifolds, volume 47 of Fields Inst. Commun., pages 261–280. Amer. Math. Soc., Providence, RI, 2005. [19] Dale Rolfsen. Knots and links. Publish or Perish Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. [20] Otto Schreier. Uber¨ die Gruppen AaBb = 1. Abh. Math. Sem. Univ. Hamburg, 3:167–169, 1924. [21] Alexander Shumakovitch. KhoHo – a program for computing and studying Khovanov homology, http://www.geometrie.ch/KhoHo. [22] Friedhelm Waldhausen. Uber¨ Involutionen der 3-Sph¨are. Topology, 8:81–91, 1969. DOES KHOVANOV HOMOLOGY DETECT THE UNKNOT? 7

Department of Mathematics, Massachusetts Institute of Technology, MA.

E-mail address: [email protected]

URL: http://www-math.mit.edu/~mhedden

Departement´ de Mathematiques,´ Universite´ du Quebec´ a` Montreal,´ Montreal´ Canada.

E-mail address: [email protected]

URL: http://www.cirget.uqam.ca/~liam