Abstract. We use bordered Floer homology to give a formula for HFK (Kp,pn+1) of any (p, pn + 1)-cable of a thin knot K in terms of ∆K(t), τ (K), p, and n. We also give a formula for the Ozsváth-Szabó concordance invariant τ (Kp,q) in terms of τ (K), p, and q, for all relatively prime p and q.Mathematics Subject Classification (2010). 57M27, 57R58.

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.Comment: 106 pages, 44 figure

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism φ : F → ∂Y , a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invariant lying in this group for every bordered 3-manifold (Y, ∂Y, φ). We prove that this invariant recovers the kernel of the inclusion i * :is finite, and is 0 otherwise. We also study the properties of this invariant corresponding to gluing. As one application, we show that the pairing theorem for bordered Floer homology categorifies the classical Alexander polynomial formula for satellites.

We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold (M, ξ, F) whose convex boundary is equipped with a signed singular foliation F closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux Correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes cD and cA in the corresponding bordered sutured modules.Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation F in favor of the dividing set, and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda-Kazez-Matić. CONTENTS1. Introduction 1 2. Preliminaries 4 3. Construction of the contact invariant 18 4. Invariance 23 5. Gluing 29 6. Relationship to the HKM contact element 33 References 36 2010 Mathematics Subject Classification. 57M27.

We identify the Grothendieck group of the tangle Floer dg algebra with a tensor product of certain Uq(gl 1|1 ) representations. Under this identification, up to a scalar factor, the map on the Grothendieck group induced by the tangle Floer dg bimodule associated to a tangle agrees with the Reshetikhin-Turaev homomorphism for that tangle. We also introduce dg bimodules which act on the Grothendieck group as the generators E and F of Uq(gl 1|1 ).

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In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle T a differential graded bimodule CT(T ). If L is obtained by gluing together T 1 , . . . , T m , then the knot Floer homology HFK(L) of L can be recovered from CT(T 1 ), . . . , CT(T m ). In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floer homology.2010 Mathematics Subject Classification. 57M58 (Primary); 57M25, 57M27 (Secondary).e 1 − 1 2 as δ-graded type DD structures. Analogous statements hold for type DA, AD, and AA structures.Remark. Following [33,34], our δ-gradings differ from those in [25,38] by a factor of −1.By taking the box tensor product, we immediately obtain a combinatorially computable unoriented skein exact triangle for knot Floer homology, recovering a version of the results in [24,38]. Suppose L ∞ , L 0 , and L 1 are three oriented links that are identical (after forgetting the orientations) except near a point, so that they form an unoriented skein triple. Let ℓ ∞ , ℓ 0 , and ℓ 1 be the number of components of L ∞ , L 0 , and L 1 respectively, and define neg(L k ), e 0 , and e 1 in a fashion analogous to neg(T k ), e 0 , and e 1 above.Corollary 4. For sufficiently large m, there exists a δ-graded exact trianglewhere V is a vector space of dimension 2 with grading 0, and W is a vector space of dimension 2 with grading −1.Remark. Due to a difference in the orientation convention, the arrows in the exact triangle point in the opposite direction from those in [24,25]. We follow the convention in [26][27][28]38], where the Heegaard surface is the oriented boundary of the α-handlebody.In another direction, Theorem 3 may also provide a way to further the development of knot Floer homology in the framework of categorification. Precisely, tangle Floer homology has been shown by Ellis, Vértesi, and the first author [5] to categorify the Reshetikhin-Turaev invariant for the quantum group U q (gl 1|1 ). This puts tangle Floer homology on a similar footing as the tangle formulation of Khovanov homology [3,13,39], which categorifies the Organization. We review the necessary algebraic background and the definition of tangle Floer homology in Section 2. We prove the ungraded unoriented skein relation, Theorem 2, in Section 3. We then determine the δ-gradings in Section 4 to prove the graded skein relation, Theorem 3. Theorems 5 and 6 is proven in Section 5.

We observe that Khovanov homology detects causality in (2 + 1)-dimensional globally hyperbolic spacetimes whose Cauchy surface is homeomorphic to R 2 .2010 Mathematics Subject Classification. 53C50 (Primary); 57M27, 83C75, 83C80 (Secondary).