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Series: Geometry Topology Seminar

This is joint work with Hyam Rubinstein. Matveev and Piergallini independently show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 Pachner moves, excepting triangulations with only one tetrahedron. We give a more direct proof of their result which (in work in progress) allows us to extend the result to triangulations of four-manifolds.

Series: Geometry Topology Seminar

I will present the recent result with P.Albers and D.Hein that every graphical hypersurface in a prequantization bundle over a symplectic manifold M pinched between two circle bundles whose ratio of radii is less than \sqrt{2} carries either one short simple periodic orbit or carries at least cuplength(M)+1 simple periodic Reeb orbits.

Series: Geometry Topology Seminar

A foundational result in the study of contact geometry and Legendrian knots is Eliashberg and Fraser's classification of Legendrian unknots They showed that two homotopy-theoretic invariants - the Thurston-Bennequin number and rotation number - completely determine a Legendrian unknot up to isotopy. Legendrian spatial graphs are a natural generalization of Legendrian knots. We prove an analogous result for planar Legendrian graphs. Using convex surface theory, we prove that the rotation invariant and Legendrian ribbon are a complete set of invariants for planar Legendrian graphs. We apply this result to completely classify planar Legendrian embeddings of the Theta graph. Surprisingly, this classification shows that Legendrian graphs violate some proven and conjectured properties of Legendrian knots. This is joint work with Danielle O'Donnol.

Series: Geometry Topology Seminar

Exploring when a closed oriented 3-manifold has vanishing reduced Heegaard Floer homology---hence is a so-called L-space---lends insight into the deeper question of how Heegaard Floer homology can be used to enumerate and classify interesting geometric structures. Two years ago, J. Rasmussen and I developed a tool to classify the L-space Dehn surgery slopes for knots in 3-manifolds, and I later built on these methods to classify all graph manifold L-spaces. After briefly discussing these tools, I will describe my more recent computation of the region of rational L-space surgeries on any torus-link satellite of an L-space knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables. More generally, I will discuss the region of L-space surgeries on iterated torus-link satellites and algebraic link satellites, along with implications for conjectures involving co-oriented taut foliations and left-orderable fundamental groups.

Series: Geometry Topology Seminar

I will discuss joint work with Hutchings which gives a rigorousconstruction of cylindrical contact homology via geometric methods. Thistalk will highlight our use of non-equivariant constructions, automatictransversality, and obstruction bundle gluing. Together these yield anonequivariant homological contact invariant which is expected to beisomorphic to SH^+ under suitable assumptions. By making use of familyFloer theory we obtain an S^1-equivariant theory defined with coefficientsin Z, which when tensored with Q recovers the classical cylindrical contacthomology, now with the guarantee of well-definedness and invariance. Thisintegral lift of contact homology also contains interesting torsioninformation.

Series: Geometry Topology Seminar

I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit non-trivial L-space surgeries. A corollary of the classification is that 1-bridge braids in these manifolds admit non-trivial L-space surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.

Series: Geometry Topology Seminar

In this talk we associate a combinatorial dg-algebra to a cubic planar graph. This algebra is defined by counting binary sequences, which we introduce, and we shall provide explicit computations. From there, we study the Legendrian surfaces behind these combinatorial constructions, including Legendrian surgeries and the count of Morse flow trees, and discuss the proof of the correspondence between augmentations and constructible sheaves for this class of Legendrians.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

This is joint work with Jeff Meier. The Gluck twist operation removes an S^2XB^2 neighborhood of a knotted S^2 in S^4 and glues it back with a twist, producing a homotopy S^4 (i.e. potential counterexamples to the smooth Poincare conjecture, although for many classes of 2-knots theresults are in fact known to be smooth S^4's). By representing knotted S^2's in S^4 as doubly pointed Heegaard triples and understanding relative trisection diagrams of S^2XB^2 carefully, I'll show how to produce trisection diagrams (a.k.a. Heegaard triples) for these homotopy S^4's.(And for those not up on trisections I'll review the foundations.) The resulting recipe is surprisingly simple, but the fun, as always, is in the process.

Series: Geometry Topology Seminar

We use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We also give a formula for how this theory behaves under connected sum, and use it to give examples not homology cobordant to anything computable via our surgery formula. This is joint work with C. Manolescu; the last part of is also joint with I. Zemke.