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Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the u^-(K). Moreover, the difference l(K)- u^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.
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We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_isubset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $widehat{mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $widehat{mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$big|(h_infty^1-h_1^1)(h_infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)big|.$$ We also show that if splicing the complement of a knot $Ksubset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.
Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called bordered knot Floer homology, which has an explicit algebraic and combinatorial construction. In the present paper, we extend the holomorphic theory to bordered Heegaard diagrams for partial knot projections, and establish a pairing result for gluing such diagrams, in the spirit of the pairing theorem of bordered Floer homology. After making some model calculations, we obtain an identification of a variant of knot Floer homology with its algebraically defined relative. These results give a fast algorithm for computing knot Floer homology.
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of knot Floer homology.
In this paper we introduce a chain complex $C_{1 pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K, and that the total homology is a link invariant which we conjecture is isomorphic to delta-graded knot Floer homology. The complex can be refined to a tangle invariant for braids on 2n strands, where the associated invariant is a bimodule over an algebra A_n. We show that A_n is isomorphic to B(2n+1, n), the algebra used for the DA-bimodule constructed by Ozsvath and Szabo in their algebraic construction of knot Floer homology.
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