Thesis advisor: John A. Baldwin / Contact geometry has played a central role in many recent advances in low-dimensional topology; e.g. in showing that knot Floer homology detects the genus of a knot and whether a knot is fibered. It has also been used to show that the unknot, trefoil, and figure eight knot are determined by their Dehn surgeries. An important problem in 3-dimensional contact geometry is the classification of Legendrian and transverse knots. Such knots come equipped with some classical invariants. New invariants from knot Floer homology have been effective in distinguishing Legendrian and transverse knots with identical classical invariants, a notoriously difficult task. The Giroux correspondence allows contact structures to be studied via purely topological constructs called open book decompositions. Transverse links are then braids about these open books, which in turn may be thought of as mapping tori of diffeomorphisms of compact surfaces with boundary having marked points, which we refer to as pointed monodromies. In the first part of this thesis, we investigate properties of the transverse invariant in knot Floer homology, in particular its behavior for transverse closures of pointed monodromies possessing certain dynamical properties. The binding of an open book sits naturally as a transverse link in the supported contact manifold. We prove that the transverse link invariant in knot Floer homology of the binding union any braid about the open book is non-zero. As an application, we show that any pointed monodromy with fractional Dehn twist coefficient greater than one has non-zero transverse invariant, generalizing a result of Plamenevskaya for braids about the unknot. In the second part of this thesis, we define invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing those defined by Ozsvath, Szabo and Thurston. We show that our invariants are equivalent to those defined by Lisca, Ozsvath, Szabo and Stipsicz for Legendrian and transverse links in arbitrary contact 3-manifolds. Our argument involves considering braids about rational open book decompositions and filtrations on knot Floer complexes. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

We investigate various forms of link positivity: braid positivity, strong quasipositivity, and quasi- positivity. On the one hand, this investigation is undertaken in the context of braid simplification: we give sufficient conditions under which a given braid word is conjugate to a braid word with strictly fewer negative bands. On the other hand, we use the famous Bennequin inequality to define a new link invariant: the defect of the Bennequin inequality, or 3-defect, and give criteria in terms of the 3-defect under which a given link is (strongly) quasipositive. Moreover, we use the 4-dimensional analogue of the Bennequin inequality, the slice Bennequin inequality in order to define the analogous defect of the slice Bennequin inequality, or 4-defect. We then investigate the relationship between the 4-defect and the most complicated class of 3- braids, Xu’s NP-form 3-braids, and establish several bounds. We also conjecture a formula for the signature of NP-form 3-braids which uses a new and easily computable NP-form 3-braid invariant, the offset. Finally, the appendices provide lists of all quasipositive and strongly quasipositive knots with at most 12 crossings (with two exceptions, 12n239 and 12n512), along with accompanying quasipositive or strongly quasipositive braid words. Many of these knots did not have previously established positivities or braid words reflecting these positivities—these facts were discovered using various criteria (conjectural or proven) expressed throughout this thesis.

Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.

In this thesis the author develops techniques to study contact structures via Riemannian geometry. The main observation is a relation between characteristic surfaces of contact structures and zero sets of solutions to certain subelliptic PDEs. This relation makes it possible to derive, under a symmetry assumption, necessary and sufficient conditions for tightness of contact structures arising from a certain class of invariant curl eigenfields. Further, it has implications in the energy relaxation of this special class of fluid flows. Specifically, the author shows existence of an energy minimizing curl eigenfield which is orthogonal to an overtwisted contact structure. It provides a counterexample to the conjecture of Etnyre and Ghrist posed in their work on hydrodynamics of contact structures.

In the first paper, we prove that for a closed Legendrian submanifold L of dimension n>2 with a loose chart of size η, any Legendrian isotopy starting at L can be C0-approximated by a Legendrian isotopy with energy arbitrarily close to η/2. This in particular implies that the displacement energy of loose displaceable Legendrians is bounded by half the size of its smallest loose chart, which proves a conjecture of Dimitroglou Rizell and Sullivan. In the second paper, we show that the Legendrian lift of an exact, displaceable Lagrangian has vanishing Shelukhin-Chekanov-Hofer pseudo-metric by lifting an argument due to Sikorav to the contactization. In particular, this proves the existence of such Legendrians, providing counterexamples to a conjecture of Rosen and Zhang. After completion of the manuscript, we noticed that Cant (arXiv:2301.06205) independently proved a more general version of our main result.

A contact structure is a maximally non-integrable hyperplane field $\xi$ on an odd-dimensional manifold $M$. In $3$-dimensional contact geometry, there is a fundamental dichotomy, where a contact structure is either tight or overtwisted. Making use of Giroux's correspondence between contact structures and open books for $3$-dimensional manifolds, Honda, Kazez, and Mat\'{i}c proved that verifying whether a mapping class is right-veering or not gives a way of detecting tightness of the compatible contact structure. As a counter-part to right-veering mapping classes, right-veering closed braids have been studied by Baldwin and others. Ito and Kawamuro have shown how various results on open books can be translated to results on closed braids; introducing the notion of quasi right-veering closed braids to provide a sufficient condition which guarantees tightness. We use the related concept of $P$-bigon right-veeringness for closed braids to show that given a $3$-dimensional contact manifold $(M, \xi)$ supported by an open book $(S, \phi)$, if $L \subset (M, \xi)$ is a non-$P$-bigon right-veering transverse link in pure braid position with respect to $(S, \phi)$, performing $0$-surgery along $L$ produces an overtwisted contact manifold $(M', \xi')$. Furthermore, if we suppose $L \subset (M, \xi)$ is a pure and non-quasi right-veering braid with respect to $(S, \phi)$, performing $p$-surgery along $L$, for $p \geq 0$, gives rise to an open book $(S', \phi')$ which supports an overtwisted contact manifold $(M', \xi')$.

We will discuss what is known about the construction of contact structures via branched covers, emphasizing the search for universal transverse knots. Recall that a topological knot is called universal if all 3-manifold can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transverse knot in the standard contact structure on S³ from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21537号 / 理博第4444号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 小野 薫, 教授 向井 茂, 教授 望月 拓郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM