aa r X i v : . [ m a t h . G T ] M a y ON KNOTS WITH INFINITE SMOOTHCONCORDANCE ORDER
ADAM SIMON LEVINE
Abstract.
We use the Heegaard Floer obstructions defined byGrigsby, Ruberman, and Strle to show that forty-six of the sixty-seven knots through eleven crossings whose concordance orderswere previously unknown have infinite concordance order.
Let K be an oriented knot in S . If K bounds a smoothly embeddeddisk in D , we say that K is (smoothly) slice . Two knots K, K ′ aresaid to be (smoothly) concordant if K K ′ is slice, where K ′ denotesthe mirror of K ′ . The set of concordance classes of knots forms agroup C under the connect sum operation with identity the unknot.The concordance order of a knot K is the order of K in C . Thestructure of the torsion in C is of considerable interest; see, for instance,Livingston-Naik [6, 7] and Jabuka-Naik [4].Let Y K = Σ ( K ) be the double branched cover of K , and let e K be the inverse image of K in Y . Grigsby, Ruberman, and Strle [3]defined numerical invariants D n ( K ) and T n ( K ) ( n ∈ N ) coming fromthe Heegaard Floer homology of Y K and the knot Floer homology of e K . They proved: Theorem 1.
Let K be a knot in S . Let p be prime, and supposethat p m is the largest power of p that divides det( K ) . If K has finiteconcordance order, then for each integer ≤ e ≤ (cid:4) m +12 (cid:5) , we have D p e ( K ) = T p e ( K ) = 0 . In practice, we are usually interested in D p ( K ) and T p ( K ), where p is 1 or a prime that divides det( K ), so we restrict our discussion to thiscase.According to Livingston’s database KnotInfo [2], the smooth concor-dance orders of sixty-seven knots with up to eleven crossings, listed inTable 1, were previously unknown. We show here that forty-six of theseknots, listed in Tables 2 and 3, have at least one nonzero D p invariantand hence have infinite concordance order. For the remaining knots, allof the relevant D p invariants vanish, so the concordance orders of theseknots remains unknown. The T p invariants for several of these knots a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a n n n n n n n n n n n n n n Table 1.
Knots through eleven crossings with unknownconcordance order.can be obtained using the author’s computations of [ HFK( Y K , e K ) [5],but we do not obtain any new concordance information in this manner.For the remainder of this paper, we describe the techniques used tocompute the D p and T p invariants for the knots considered here.Let us briefly recall the definition of these invariants in the casewhere H ( Y K ; Z ) is cyclic. (For the general case, see [3, Definition4.1].) Let s ∈ Spin c ( Y K ) be the so-called canonical spin c structure on Y K , uniquely characterized by the property that c ( s ) = 0. Recallthat Spin c ( Y K ) is an affine space for H ( Y K ; Z ), so we may identifySpin c ( Y K ) with H ( Y K ; Z ) via the identification s c ( s ). Let G m bethe unique order- p subgroup of H ( Y K ; Z ). The invariants D p ( K ) and T p ( K ) are then defined as D p ( K ) = X s ∈ s + G p d ( Y K , s ) T p ( K ) = X s ∈ s + G p τ ( Y K , e K, s ) . Here d ( Y K , s ) is the correction term for HF + ( Y K , s ), and τ ( Y K , e K, s ) isthe τ -invariant for [ HFK( Y K , e K, s ). (See Ozsv´ath-Szab´o [8, 11] for thedefinitions of d and τ .)In many cases, the results of Ozsv´ath and Szab´o [9, 12, 13] may beused to compute the correction terms d ( Y, s ) combinatorially. Given aprojection of K , let G be its Goeritz matrix (defined in [12, section 3]).Let | G | denote the rank of G . The double cover Y K bounds a 4-manifold X G whose intersection form on H , Q = Q X G , is given by G (withrespect to a basis of spheres). Let Char( G ) ⊂ H ( X G ; Z ) denote theset of characteristic vectors for Q , i.e., vectors α ∈ H ( X G ; Z ) such that N KNOTS WITH INFINITE SMOOTH CONCORDANCE ORDER 3
Knot K det( K ) Nonzero GRS invariants9 D = 49 D = 410 D = 410 D = 410 D = − D = − a D = − a D = − a D = 1211 a D = 1211 a D = 1211 a D = 1211 a D = 1611 a D = − a D = − a D = − a D = − a D = − a D = 4 , D = 411 a D = − a D = 1211 a D = − a D = − a D = 1611 a D = − a D = − a D = 2411 a D = 1211 a D = 1211 a D = − a D = 4 , D = 411 a D = 411 a D = − a D = 3611 a D = 1211 a D = 4 , D = 4 Table 2.
Alternating knots with non-vanishingGrigsby-Ruberman-Strle D p invariants. ADAM SIMON LEVINE
Knot K det( K ) Nonzero GRS invariants9 D = 410 D = 411 n D = − n D = − n D = − n D = 1211 n D = − n D = − n D = 1211 n D = − Table 3.
Non-alternating knots with non-vanishingGrigsby-Ruberman-Strle D p invariants. h α, v i ≡ Q ( v, v ) (mod 2) for every v ∈ H ( X G ; Z ). The restriction map i ∗ : H ( X G ) → H ( Y K ) partitions Char( G ) into equivalence classesChar( G, s ) corresponding to the spin c structures on Y K . Given certainhypotheses on G , including that G is negative-definite, Ozsv´ath andSzab´o [9, Corollary 1.5] proved that the correction terms for HFK + ( Y K )are given by the formula(1) d ( Y K , s ) = max α ∈ Char( G, s ) α + | G | . Ozsv´ath and Szab´o provide an algorithm for finding the vectors ineach equivalence class that realize this maximum. Moreover, since H ( Y K ; Z ) ∼ = coker( G ), we may easily identify the group structure onSpin c ( Y K ) (specifically, which spin c structures are in the special sub-group G p ) using the Smith normal form for G .As shown in [12], Equation 1 holds whenever G is computed from analternating projection. More generally, if K admits a projection thatis alternating except in a region that consists of left-handed twists,Ozsv´ath and Szab´o [13] show how to use Kirby calculus on X G toobtain a matrix e G for Q that satisfies the correct hypotheses. (See alsoJabuka-Naik [4] for a concise explanation.) All of the non-alternatingknots in Table 1 satisfy this hypothesis, so we may compute the D p invariants as described above.Finally, to compute the T p invariants of a knot, one must com-pute the integers τ ( Y K , e K, s ) associated to the spectral sequence from That K = 11 n has infinite concordance order also follows from the simplerfact that τ ( S , K ) = 1, as was computed by Baldwin and Gillam [1]. N KNOTS WITH INFINITE SMOOTH CONCORDANCE ORDER 5 [ HFK( Y K , e K, s ) to c HF( Y K , s ). When [ HFK( Y K , e K, s ) and c HF( Y K , s ) aresufficiently simple, one can sometimes determine τ without knowing allthe differentials in the spectral sequence. For instance, if c HF( Y K , s ) hasrank 1 and [ HFK( Y K , e K, s ) is supported on a single diagonal, τ ( Y K , e K, s )is equal to the Alexander grading of the nonzero group in Maslov grad-ing d ( Y K , s ). The author [5] has shown how to compute HFK( Y K , e K )(with coefficients in Z /
2) for any knot K using grid diagrams and hascomputed the values of τ for several of the non-alternating knots con-sidered here (9 , 10 , 10 , 10 , 11 n , and 11 n ). However, the T p invariants all vanish in these cases, so we do not obtain any newconcordance information. References [1] J. A. Baldwin and W. D. Gillam,
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