Floer homology of surgeries on two-bridge knots

By Jacob Rasmussen. 2002

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We compute the Ozsv ath-Szab o Floer homologies HF and d HF for three-manifolds obtained by integer surgery on a two-bridge knot. AMS Classication 57R58; 57M27

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/9b3d7500559573b8712b7ac54bd3f1c12923aaa0

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Citations in the corpus

Floer homology and knot complements (2003, methodology): “ration on CFd r(K) is the same as the ﬁltration induced by the homological grading. In this case, we have the following theorem, which generalizes the calculation of HF+ for two-bridge knots given in [25]: FLOER HOMOLOGY AND KNOT COMPLEMENTS 3 Theorem 2. If K is perfect, the groups HF+(K n,s k) are ”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, background): “ a four-pointed Heegaard diagram with genus zero, and hence the holomorphic curve counts deﬁning the diﬀerential take place in the Riemann sphere. Thus, these counts are purely combinatorial, compare [23] and also [20, Section 6].) Consider the Hopf link H~. There are two orientations for H~, distin”
Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “The reformulation of knot Floer homology (introduced originally in [35, 39]) through grid diagrams [26, 27] provided not only a combinatorial way of computing the knot Floer homology groups of knots in S3, but also showed a natural way of defining invariants of Legendrian and transverse knots in the”
Knot polynomials and knot homologies (2005, methodology): “Given this data, the generators of ĈFK(K) can be computed by a process which is more or less the same as computing the Alexander polynomial via Fox calculus [28].”
Heegaard Floer homologies and contact structures (2002, methodology): “ucture. (see also Subsection 2.2 below for a more detailed discussion of these moves) To describe the Floer homology class associated to ξ, we use the knot invariants introduced in [14] (compare also [15]). When a three-manifold is equipped with an oriented knot Kand a Seifert surface F for K, there”
Holomorphic disks and knot invariants (2002, methodology): “are used to deﬁne invariants of contact structures for three-manifolds. 1.3. Further remarks. The fact that a knot in Y induces a ﬁltration on CF∞(Y) has been discovered independently by Rasmussen in [25], where he uses the induced ﬁltration to compute the Floer homologies of three-manifolds obtaine”
On the Khovanov and knot Floer homologies of quasi-alternating links (2007, background): “Rasmussen [26] showed that 2-bridge knots are Floer homologically σ-thin; and this result was generalized in [16] to all alternating knots.”
Grid Homology for Knots and Links (2015, methodology): “Knot Floer homology for alternating knots was computed in [169], generalizing a result of Rasmussen [190].”
Knot Floer homology and the four-ball genus (2003, methodology): “Our aim here is to define an integer invariant of a knot K which gives a lower bound on g∗(K), using the knot filtration from [17], see also [22] and [21].”
Heegaard Floer homology and alternating knots. (2002, methodology): “PSO was supported by NSF grant number DMS 9971950 and a Sloan Research Fellowship; ZSz was supported by NSF grant number DMS 0107792 and a Packard Fellowship.”
Transverse knots distinguished by knot Floer homology (2007, methodology): “Considering the Alexander polynomial, and using the structure of knot Floer homology of two-bridge knots [30], see also [26], one sees that the transverse invariant θ̂ for the knots in”
Knot Floer homology, genus bounds, and mutation (2003, methodology): “We also calculate ĤFK for certain pretzel knots, and knots with small crossing number (n ≤ 9).”
Knot Floer Homology of (1, 1)-Knots (2003, methodology): “Furthermore Heegaard Floer homologies for 3-manifolds obtained by Dehn surgeries on 2-bridge knots in S(3) are calculated by Rasmussen [26], using the method presented in [18, Proposition 3.”
Cables of thin knots and bordered Heegaard Floer homology (2009, background): “If the homology is supported on the diagonal δ = −σ/2, where σ denotes the knot signature, then we say the knot is σ-thin, or perfect [12].”
Knot Floer homology of Whitehead doubles (2006, methodology): “Proof The refinement is proved by the somewhat standard technique of “moving the basepoint,” see Ozsváth–Szabó [28] and Rasmussen [32].”