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Part 1 Transverse Invariants in Khovanov-Type MY soul is an entangled knot, Upon a liquid vortex wrought By Intellect, in the Unseen residing, And thine cloth like a convict sit, With marlinspike untwisting it, Only to find its knottiness abiding; Since all the tools for its untying In four-dimensioned space are lying Extract from “A Paradoxical Ode“ by James Clerk Maxwell Contents Introduction iii 1. Links in contact manifolds and effective invariants iii 2. Transverse invariants in Khovanov-Rozansky homologies iv 3. Outline of contents and main results v Part 1. Transverse invariants in Khovanov-type sl2-theories xi Chapter 1. Frobenius algebras and Khovanov-type homologies 1 1. Frobenius algebras 1 2. Link homology theories 7 Chapter 2. Bar-Natan Theory 17 1. The structure of Bar-Natan homology I: the free part 17 2. The structure of Bar-Natan homology II: torsion 31 Chapter 3. Transverse invariants in the sl2-theory 39 1. Transverse knots and braiding 39 2. Transverse invariants from Bar-Natan Homology I: the b-invariant 46 3. Transverse invariants from Bar-Natan Homology II: the invariant c 68 4. Some philosophical remarks 77 Chapter 4. A Bennequin s-inequality from Bar-Natan homology 89 1. A Bennequin s-inequality from the c-invariants 89 2. Comparisons and examples 91 3. Final remarks 100 Part 2. Transverse invariants in sl3-homologies 103 Chapter 5. The universal sl3-link homology via foams 105 1. Webs and Foams 105 2. Foams and Local relations 110 3. The sl3-link homologies via Foams 112 4. Grading and filtration 118 Chapter 6. Transverse invariants in the universal sl3-theory 123 1. The b-chains 123 2. The transverse invariance of the b-chains 126 3. The c3-invariants 133 i ii CONTENTS Bibliography 137 Index 141 Appendices 143 Appendix A. Algebra 145 1. Duality and modules 145 2. Grading and Filtration 147 3. Filtered degree and bases 150 4. Homogeneous lifts 152 5. Spectral sequences 155 Appendix B. Computations 159 1. The knot 13n1336 159 2. Bar-Natan homology of T(2, n) torus links 161 3. Bar-Natan homology of the links T(3, k) 162 4. The rational Bar-Natan homology of the prime knot with less than 12 crossing 165 5. The computation of the s-invariant for some families of links 255 Appendix C. Self-duality of the Bar-Nathan Frobenius Algebra 259 Introduction The aim of this thesis is the study of transverse link invariants coming from Khovanov sl2- and sl3-homologies and from their deformations. As a by-product of our work we get computable estimates on some concordance invariants com- ing from Khovanov sl2-homologies. In order to explain the meaning of the the previous sentences it is necessary to give some definitions. 1. Links in contact manifolds and effective invariants A contact manifold (Definition 3.1) is an odd-dimensional manifold en- M dowed with a totally non-integrable hyperplane field x – intuitively speaking, a plane field which twists too much to allow the existence of a co-dimension one local sub-manifold of admitting x as tangent space. The simplest con- M3 tact manifold is probably R endowed with the symmetric contact structure xsym (see Chapter 3, Section 1, for the definition). Despite its simplicity, the study of 3 the contact manifold (R , xsym) is fundamental to understand the “zoology” of 3 the contact structures over any 3-manifold. In fact, on the one hand (R , xsym) provides a local model for all contact 3-manifolds (cf. Darboux’s theorem [16, Theorem 2.5.1] and [14, Example 2.1]). On the other hand, any contact closed 3 3-manifold can be obtained from the one point compactification of (R , xsym), 3 namely (S , xsym), via surgical constructions along links which are compatible, in some sense, with the contact structure (cf. [16, Theorems 6.4.4 & 7.3.5]). There are two natural ways for a link l in a 3-manifold to be compatible M with a contact structure x on : M . l is everywhere tangent to the contact structure; . l is be everywhere transverse to the contact structure; the first type of links are called Legendrian, while the second type of links are called transverse. Two members of each of these families of links are considered equivalent if they are ambient isotopic through links in the same family. Legendrian and transverse links have been extensively studied, and they have some classical invariants: the link type, the Thurston-Bennequin number and the rotation number for Legendrian links, and the link type and the self-linking number for transverse links. These invariants are complete invariants for some families of links (e.g. the unknot, the torus links, the figure eight knot etc.) but in general they are not good at distinguishing transverse or Legendrian links. There are infinite families of links (or even knots) whose members share the same classical invariants even though they are distinct as transverse or Legendrian links iii iv INTRODUCTION (e.g. some cablings of torus knots). An invariant of Legendrian, or transverse, links which can tell apart two different Legendrian, respectively transverse, links with the same classical invariants is called effective. The study of Legendrian and transverse links gives also information on the topology of the contact three manifold. For instance, Rudolph in [49] used an inequality to relate the classical invariants of Legendrian and transverse links with the slice genus of the underlying link. Other similar inequalities, called Bennequin-type inequalities, have been produced through the years. For Legendrian links there are lots of well known effective invariants, for example the Chekanov-Eliashberg DGA [12], the LOSS invariants coming from knot Floer Homology [36], the invariants l coming from the grid version of ± knot Floer Homology [43] etc. and some of them give rise to effective transverse invariants, e.g. the q invariant coming from grid Floer homology [43]. The work of various authors (for example [36, 43, 45, 34, 60, 59]) has estab- lished that link homologies, such as HFK and Khovanov-Rozansky homologies, contain not only topological but also Legendrian and transverse information. The Legendrian and transverse information contained in Heeegaard Floer type the- ories is, in some sense, more explicit. Perhaps this is due to the more direct connection of the Heegaard Floer theories with the topology of three manifolds. On the other hand, the Legendrian and transverse information in the Khovanov- Rozansky homologies is less understood. So, in the hope of shedding new light on this subject we investigated the presence of transverse invariants in (the de- formations of) Khovanov sl2- and sl3-homologies. 2. Transverse invariants in Khovanov-Rozansky homologies Let R be the ring F[U1, ..., Us], where F is a field and deg(Ui)=2ki.A potential w for the (equivariant) Khovanov-Rozansky sl(N)-homology is an ho- mogeneous polynomial in the ring R[X], where deg(X)=2, of the form N+1 w(X, U1, ..., Us)=X + XF(X, U1, ..., Us), F(X, 0, ..., 0)=0. Given a oriented link diagram L, one can associate to L a bi-graded chain complex , (Cw• •(L), dc) over the ring R. Where the first grading is the homological grading and the second grading is induced by the polynomial grading on R[X]. The differential dc is homogeneous with respect to both gradings. The homology of this complex is called the equivariant Khovanov-Rozansky sl(N)-homology (over R with respect to w). , The chain module Cw• •(L) is itself obtained as the homology of another com- plex, here denoted (Mw(L), ∂). This complex is obtained from the diagram D by 1 associating to each MOY-resolution of L, say G, a matrix factorization (M(G), ∂G), and to the diagram L the direct sum of these complexes. The complex Mw(L) is 1A web resolution with markers. For the definition of web resolution the reader may consult Chapter 5 Section 1. 3. OUTLINE OF CONTENTS AND MAIN RESULTS v triply graded: it has a Z/2Z-degree coming from the degree of the matrix factor- izations, an homological degree coming from the type of resolution performed, and a polynomial grading. The whole construction is technically quite complex. The complexity of the theory makes it difficult to work with the chain complex. This is the main reason why in this thesis we work with a potential w of degree either 3 or 4. In these cases it is possible to bypass some of the technical difficulties using different approaches. The general case will be the subject of a future work by the author. When degX(w)=3 we will use an approach similar to the original definition of Khovanov homology (cf. [25, 27]). When degX(w)=4 we use the approach introduced by Khovanov ([26]) for w = X4, and generalized by Mackaay and Vaz ([40]) to arbitrary potentials of degree 4, which uses foams. The construction via matrix factorizations described above will be hidden in both these approaches. Even the choice of the potential will be partially hidden; both the constructions we are going to describe in this thesis will depend on the choice of a polynomial, which corresponds to the derivative (with respect to X) of the potential. The idea of defining transverse invariants using Khovanov-Rozansky homo- logies is not new. In 2006 Olga Plamenevskaya defined a transverse invariant y which is an homology class in the Khovanov homology of a transverse link (presented as a closed braid). The Plamenevskaya invariant y was generalized in 2008, by Hao Wu (see [59]). Wu defined an invariant yn which is an homology class in the non-deformed Khovanov-Rozansky slN-homology (that is the theory with potential w = XN).
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