Dartmouth Topology Seminar

Date	Speaker	Title
Apr 21	Adria Marin Salvador Universitat Autonoma de Barcelona, Spain and University of Oxford, UK	On the contact manifold of null geodesics of a spacetime Zoom ID 870 912 2782 if you are not from this Department please ask Vladimir Chernov for the password
Apr 7	Lois Alberto Ibort Latre Universidad Carlos III de Madrid, Spain	A new class of conformal invariants: the sky-invariant of a spacetime Zoom ID 870 912 2782 if you are not from this Department please ask Vladimir Chernov for the password
Mar 31	Lizzie Buchanan Dartmouth College	Towards an Infinite Family of Knots with Minimal Almost-Alternating Diagrams Zoom ID 870 912 2782 if you are not from this Department please ask Vladimir Chernov for the password
Mar 3	Stephan Wehrli Syracuse University	Cooper-Krushkal projectors and annular Khovanov homology
Feb 24	Viktória Földvári Alfréd Rényi Institute of Mathematics	Legendrian double twist knots
Feb 17	Ryan Maguire Dartmouth College	Conjectures on Legendrian and Transversally Simple Knots and Links and an Iterative Algorithm for the Jones Polynomial Zoom ID 870 912 2782, if you are not from this department please ask Vladimir Chernov for the password
Feb 3	Roman Golovko Charles University, Prague, Czech Republic	On infinite number of diffeomorphic and Hamiltonian non-isotopic exact Lagrangian fillings in high dimensions Zoom ID 918 8779 3663 if you are not a Department member, please ask Vladimir Chernov for a password
Nov 4	Sergey Maksimenko Corresponding member of the National Academy of Science of Ukraine, President of the Kiev Mathematical Society, Institute of mathematics of NAS of Ukraine, Kyiv, Ukraine	Combinatorial symplectic symmetries of smooth functions on surfaces
Oct 28	Samuel Tripp, Dartmouth College	Integral Grid Homology for Links in Lens Spaces
Oct 21	Zachary Winkeler, Dartmouth College	Multipunctured Khovanov homology
Oct 14	Daniel Alvarez-Gavela, MIT	Legendrian invariants from algebraic K-theory
Oct 7	Misha Tyomkin, Dartmouth College	Various facets of torsion theory: Whitehead, Reidemeister and Milnor
Sept 30	Misha Tyomkin, Dartmouth College	Numbers on barcode of a strong Morse function
Sept 23	Emanuele Zappala, University of Tartu, Estonia	Cocycle invariants of knots and knotted surfaces

April 21, 2022: Adria Marin Salvador "On the contact manifold of null geodesics of a spacetime"

Abstract:The space of null geodesics of a spacetime (a Lorentzian manifold with a choice of future) sometimes has the structure of a smooth manifold. When this is the case, it comes equipped with a canonical contact structure. In this talk I will show that this is the case for the family of spacetimes {(S²×S¹, g_o-dt²/c²)}_c∈N⁺, with g_o the round metric on S² and t the S¹-coordinate. Their spaces of null geodesics are actually the lens spaces L(2c,1) together with the pushforward of the canonical contact structure on STS²≅L(2,1) under the natural projection L(2,1)→L(2c,1). These results can also be extended to trivial bundles over Zoll manifolds. On the other hand, motivated by these examples, I will comment on how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation, and how this gives an idea on how to retrieve the spacetime from its contact manifold of null geodesics. This is joint work with F. Presas and R. Rubio.

April 7, 2022: Lois Alberto Ibort Latre "A new class of conformal invariants: the sky-invariant of a spacetime"

Abstract:A categorical notion of conformal invariants for a suitable class of spacetimes will be introduced. Using it, a new conformal invariant, the sky-invariant, using the space of light rays of the given spacetime as well as its contact structure, will be constructed. Some relevant properties and its explicit computation on a few non-trivial examples will be discussed.

March 31, 2022: Lizzie Buchanan "Towards an Infinite Family of Knots with Minimal Almost-Alternating Diagrams "

Abstract:We would ultimately like to produce an infinite family of knots that have a minimal (with respect to crossing number) almost-alternating diagram. Almost-positive-alternating knot diagrams appear to be a good candidate of sub-families to work with. While working on this problem, we believe we have found a new upper bound on the maximum degree term of the Jones polynomial of a certain class of positive knots. To the best of our knowledge, this work completes the classification of all knots with crossing number less or equal to 12 as positive or not positive.

February 24, 2022: Viktória Földvári "Legendrian double twist knots "

Abstract: Legendrian knot theory is a rich field of contact topology which, although widely investigated, has still many fundamental open questions. For example, we hardly know any results on the natural problem of classifying Legendrian knots up to Legendrian isotopy. In this talk, I am going to present a lower and an upper bound on the number of different Legendrian realizations of a family of double twist knots with prescribed tb invariant. The lower bound provides an infinite family of Legendrian non-simple knot types, while the upper bound will hopefully yield future classification results.

March 3, 2022: Stephan Wehrli "Cooper-Krushkal projectors and annular Khovanov homology "

Abstract: Jones-Wenzl idempotents play an important role in the definition of the quantum sl(2) invariants for knots and 3-manifolds. Around 2010, Cooper and Krushkal gave an axiomatic definition of categorified Jones-Wenzl projectors. To prove that such categorified projectors exist, they used a recursion formula of Frenkel and Khovanov. In my talk, I will discuss an alternative construction, which instead uses the classical Jones-Wenzl recursion. Focusing on the case of the annular unknot, I will then show that the colored sl(2) knot homology theories of Cooper-Krushkal and Khovanov become equivalent when formulated in the quantum annular setting. This is joint work with Anna Beliakova, Matt Hogancamp, and Kris Putyra.

February 24, 2022: Viktória Földvári "Legendrian double twist knots "

February 17, 2022: Ryan Maguire "Conjectures on Legendrian and Transversally Simple Knots and Links and an Iterative Algorithm for the Jones polynomial "

Abstract: A theorem of Kronheimer and Mrowka states that Khovanov homology is able to detect the unknot. That is, if a knot has the Khovanov homology of the unknot, then it is isomorphic to it. A similar result holds for Knot Floer homology. The unknot is the simplest of the torus knots, which is a class of knots known to be Legendrian simple. It is conjectured that Khovanov and Knot Floer homology are able to distinguish Legendrian and Transversally simple knot. An iterative algorithm for computing the Jones' polynomial of a knot is described and an analysis of the computational complexity is given. This algorithm is then used to efficiently compare the Khovanov homologies of all knots of up to 15 crossings with all torus knots of up to 50 crossings, providing numerical evidence for this conjecture on the first 313,230 knots.

February 3, 2022: Roman Golovko "On infinite number of diffeomorphic and Hamiltonian non-isotopic exact Lagrangian fillings in high dimensions "

Abstract: We will discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of diffeomorphic and Hamiltonian non-isotopic exact Lagrangian fillings. Zoom ID 918 8779 3663 if you are not a Department member, please ask Vladimir Chernov for a password

November 4, 2021: Sergy Maksimenko "Combinatorial symplectic symmetries of smooth functions on surfaces"

Abstract:Let M be a compact orientable surface equipped with a symplectic form w, P be either the real line of the circle, and f:M→P a smooth function. Let also S(f,w) = {h ∈ Diff(M) | foh=f and h^*w=w} be the group of diffeomorphisms of M which mutually preserve f and w, and p:S(f,w)→π₀S(f,w) be the natural projection into the group π₀S(f,w) of path components of S(f,w). It is shown that for a large class of smooth maps f:M→P, which includes all Morse maps, the homomorphism p admits a section homomorphism s: π₀S(f,w)→S(f,w). In other words, in each isotopy class γ of diffeomorphisms mutually preserving f and w one can choose a representative h_γ ∈ S(f,w) so that the collection {h_γ}_γ constitute a subgroup of S(f,w), i.e. h_γh_γ' = h_{γ γ'}. A variant of such result for non-orientable surfaces is also given. As an application it is also shown that every finite group is a subgroup of S(f,w) for some M, f and w.

October 28, 2021: Samuel Tripp "Integral Grid Homology for Links in Lens Spaces"

Abstract: Baker, Grigsby and Hedden presented a combinatorial description for the knot Flor homology of links in lens spaces. We provide a combinatorial proof of invariance of this grid homology, and adapt this to prove the invariance of the generalization of this grid homology to integral coefficients.

October 21, 2021: Zachary Winkeler "Multipunctured Khovanov homology"

Abstract: Khovanov homology is a knot invariant that categorifies the Jones polynomial. For knots in a thickened annulus, we can define a filtration on the Khovanov chain complex that gives us an invariant called annular Khovanov homology. In this talk, we will discuss a generalized notion of filtration that allows us to construct an analogous invariant for knots in thickened disks with any number of punctures.

October 14, 2021: Daniel Alvarez-Gavela "Legendrian invariants from algebraic K-theory"

Abstract: The theory of generating functions allows one to build Legendrian invariants out of Morse theoretic invariants. While such homological invariants have been extensively studied and compared to pseudo-holomorphic curve invariants, K-theoretic invariants are somewhat more mysterious. In this talk I will discuss K-theoretic torsion invariants for Legendrians in 1-jet spaces, focusing on an example which appeared in previous work of Igusa and Klein, where it was used to provide a picture for a generator of K_3(Z)=Z/48. Joint work with K. Igusa.

October 7, 2021: Misha Tyomkin "Various facets of torsion theory: Whitehead, Reidemeister and Milnor"

Abstract: Torsion is a topological invariant defined algebraically by an appropriate generalization of a notion of determinant. Some of its incarnations are:
* Whitehead torsion is an obstruction for a famous h-cobordism theorem to hold true in the non-simply connected case.
* Reidemeister torsion is an invariant which is able to detect homotopy equivalent, but non-homeomorphic manifolds, such as lens spaces.
* Milnor torsion is a low-dimensional variation of a Reidemeister one, which in some cases equals to the Alexander polynomial of knot.
In this overview talk I will try to touch on these incarnations without delving into proofs or technical details.

September 30, 2021: Misha Tyomkin "Numbers on barcode of a strong Morse function"

Abstract: Morse function f on a manifold M is called strong if all its critical values are pairwise distinct. For a given field F, Barannikov decomposition (a.k.a. barcode) is a canonical pairing of some critical points of such f with neighboring indices. We present a construction which naturally associates a number (i.e. an element of F) to each Barannikov pair (a.k.a. bar in the barcode), defined up to a sign. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f (up to sign). We then proceed to consider homology with twisted coefficients (still in some field F). It is in this setting where the Reidemeister torsion is defined. We construct the twisted barcode and prove that the mentioned product equals to the Reidemeister torsion (provided that the latter is defined); in particular, it's again independent of f. Based on a joint work with Petya Pushkar

September 23, 2021: Emanuele Zappala "Cocycle invariants of knots and knotted surfaces "

Abstract: In this talk I will introduce the notions of racks and quandles, as well as their cohomology. I will discuss the diagrammatic interpretation of these algebraic structures, and show how to use this to construct a partition function that is an invariant of Reidemeister moves. Using the same principles, I will also show that it is possible to obtain an invariant of knotted surfaces in 4-space, defining certain partition functions based on quandle 3-cocycles. These results were introduced in Trans. Amer. Math. Soc. 355 (2003) 3947-3989 by Carter, Jelsovsky, Kamada, Langford and Saito. Then, I will discuss how to generalize the previous results by means of ternary quandle cohomology to get invariants of framed links and compact oriented surfaces with boundary (embedded in 3-space). I will present applications of all these invariants throughout the talk, including a generalization of the cocycle invariants to quantum invariants.