With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

We define polynomial tangle invariants ∇ s T via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for ∇ s T of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants ∇ s T can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of ∇ s T : a Heegaard Floer homology HFT for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on HFT and prove symmetry relations for HFT of 4-ended tangles that echo those for ∇ s T .Kauffman states and Heegaard diagrams for tangles 3 of L and we call R the mutating tangle in this mutation. If L is oriented, we choose an orientation of L that agrees with the one for L outside of R. If this means that we need to reverse the orientation of the two open components of R, then we also reverse the orientation of all other components of R during the mutation; otherwise we do not change any orientation. For an alternative, but equivalent definition, see definition 3.3 and remark 3.4.Theorem 0.2 along with the glueing formula for ∇ s T gives rise to the following result. Corollary 0.4 (3.5) The multivariate Alexander polynomial is invariant under Conway mutation after identifying the variables corresponding to the two open strands of the mutating tangle.This result has long been known for the univariate Alexander polynomial, see for example [LM87, proposition 11], but I have been unable to find a corresponding result for the multivariate polynomial in the literature. The fact that mutation invariance follows so easily from the symmetry relations of theorem 0.2 suggests that ∇ T is well-suited for studying the "local behaviour" of the Alexander polynomial. The homological tangle invariant HFTHeegaard Floer homology theories were first defined by Ozsváth and Szabó in 2001 [OS01]. With an oriented, closed 3-dimensional manifold M , they associated a family of homological invariants, the simplest of which is denoted by HF(M). Given an oriented (null-homologous) knot or link L in M , Ozsváth and Szabó, and independently J. Rasmussen, then defined filtrations on the chain complexes which give rise to the respective flavours of knot and link Floer homology [OS03a, Ras03, OS05], the simplest of which is denoted by HFL(L). The Alexander polynomial can be recovered from these groups as the graded Euler characteristic.Given corollary 0.4, it is only natural to ask for a Heegaard-Floer theoretic categorification of ∇ s T . To this end, we define a homology theory HFT as follows: given a Heegaard diagram for a tangle T (see definition 4.1) along with a site s of T , we define a finitely generated Abelian group which comes with two gradings: a relative homological Z-grading and an Alexander grading, which is an additional relative Z-grading for each component of the tangle:CFT(T, s) = h∈Z ←homological grading a∈Z |T| ←Alexander grading CFT h (T, s, a).

We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles T multicurves BN(T ), that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use BN(T ) to define two immersed curve invariants Kh(T ) and Kh(T ), which satisfy similar gluing theorems that recover reduced and unreduced Khovanov homology of links, respectively. As a first application, we prove that Conway mutation preserves reduced Bar-Natan homology over the field with two elements and Rasmussen's s-invariant over any field. As a second application, we give a geometric interpretation of Rozansky's categorification of the two-stranded Jones-Wenzl projector. This allows us to define a module structure on reduced Bar-Natan and Khovanov homologies of infinitely twisted knots, generalizing a result by Benheddi. Contents 1. Introduction 1 2. Algebraic preliminaries 11 3. Khovanov-theoretic invariants of links 17 4. Bar-Natan's universal cobordism category and tangle invariants 25 5. Classification results 36 6. Immersed curve invariants 56 7. Pairing theorem 61 8. The mapping class group action 71 9. Signs and mutation 79 10. Khovanov homology of infinitely twisted knots 84 References 93 AK is supported by an AMS-Simons travel grant. LW is supported by an NSERC discovery/accelerator grant and was partially supported by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program.

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [J. Topol. 13 (2020)]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mutation preserves the hat flavour of the relatively \delta -graded Heegaard Floer theory of links.

There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author (Adv. Math. 313 (2017), 915-946). In particular, we provide a counterexample to Conjecture 29 therein, as well as a refinement of and additional evidence for Conjecture 28.The Brieskorn spheres (2, q, 2nq ∓ 1) may be obtained by Dehn surgery on a torus knot in the three-sphere, namely, these are the integer homology spheres S 3 ±1/n (T 2,q ), where T 2,q is the positive (2, q) torus knot. These homology spheres admit Seifert fibrations, with base orbifold S 2 (2, q, 2nq ∓ 1). Denoting by (A, b) the two-fold branched cover of A with branch set b, each of these manifolds admits two descriptions as a two-fold branched cover:. This construction might be best termed as classical; for our purposes it is helpful to review the notation introduced in [9]. The first of these two-fold branched covers results from an involution on the Seifert fibred space that preserves an orientation on the fibres -we refer to this as the Seifert involution. The second of these arises from the Montesinos involution, which reverses an orientation on the fibres; the branch set in question arises from the Montesinos trick, that is, by first constructing a tangle (B 3 , τ ) over which the exterior of T 2,q is realized as a two-fold branched cover. We review the construction below as it is central to our enumeration of Kotelskiy is supported by an AMS-Simons travel grant. Watson is supported by an NSERC discovery/accelerator grant. Zibrowius is supported by the Emmy Noether Programme of the DFG, Project number 412851057. MSC2020: 57K10, 57K18.