Title: ON THE HEEGAARD FLOER HOMOLOGY OF S 3 p/q (K)
Abstract: Assume that the oriented 3-manifold M = S 3 p/q(K) is obtained by a rational surgery (with coefficient p/q < 0) along an algebraic knot KS 3 . We compute the Heegaard Floer homology of M in terms of p/q and the Alexander polynomial of K. In this article we compute the Heegaard Floer homology HF + (−M) (introduced by Ozsvath and Szabo (13)) for the oriented 3-manifold M = S 3 −p/q (K) obtained by a negative rational surgery (with coefficient−p/q) along an algebraic knot K ⊂ S 3 . In this case, since H1(M, Z) = Zp, the spin c -structures {σa}a of M can be parametrized by integers a = 0,1, . . . p − 1. The main result of the article establishes HF + (−M, σa) in terms of the integers p, q, a, and the Alexander polynomial � of K ⊂ S 3 . Notice that the Alexander polynomial of an algebraic (any) knot is well-understood, it can be easily computed from most of the other invariants of the knot (e.g., in the present algebraic case, from Puiseux or Newton pairs, or from the semigroup associated with the corresponding local analytic germ). In particular, the input of the theorem is the simplest what one can hope. Since (in some sense) all the coefficients ofare effectively involved i n the description of the Heegaard Floer homology, in fact, the result is optimal. In the very recent manuscript (16), Ozsvath and Szabo computed HF + (S 3(K)) - for any knot K and any integer surgery coefficientp - in terms of the filtered chain homotopy type of the Heegaard Floer complex associated with the pair (S 3 , K). Compared with this, our starting data, and also the description of HF + (−M), are simpler, and totally elementary; as a price for this we have to impose the 'negativity restrictions' for the surgery coefficient and for K. The proof (and the structure of the article) is based on the results and constructions of (10) valid for plumbed 3-manifolds associated with some negative definite plumbing graphs - in fact, this also explains the source of our restrictions. Although (10) presents a precise algorithm how one should compute HF + , its implementations in different situations sometimes is not straightforward. In the present case too, the proof and additional constructions run over many sections. In fact, with the present article, we also wish to advertise the efficiency, novelty and the power of (10). This method, in fact, determines the 'graded roots' (some graded trees) associated with the plumbing graph of M, from which one can read easily the Heegaard Floer homology. The advantage of these graded roots is that (in all the cases known by the author) their structure reflects perfectly the corresponding geometrical construction which provides M. In particular, in many cases, from the topology of M one can identify the roots rather conceptually. Section 2 recalls the classical invariants of algebraic knots and connects them with the plumbing of M. The next section recalls the definition and first properties of graded roots - necessary to formulate the main theorem, which appears in section 4. Section 5 presents two relevant results of (10), as general principles to compute HF + . (In fact, section 5 can serve as a general recipe for the 1991 Mathematics Subject Classification. Primary. 57M27, 57R57, 58Kxx, 32Sxx; Secondary. 14E15, 14Bxx.
Publication Year: 2004
Publication Date: 2004-10-27
Language: en
Type: preprint
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Cited By Count: 4
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