Algebraic independence of topological Pontryagin classes
We show that the topological Pontryagin classes are algebraically independent in the rationalised cohomology of BTop(d) for all $d \geq 4$.
The descendant colored Jones polynomials
We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the f...
Zero-surgery characterizes infinitely many knots
We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all (β3, 3, 2n + 1) pretzel knots. This was previously only known for 52 and its mirror, as a corollary of that classification, an...
Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory
We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at $\theta=\pi$ (SU(2)$_{\theta=\pi}$ YM), by turning on background fields for...
Airfoil Optimization using Design-by-Morphing
We present Design-by-Morphing (DbM), a novel design methodology applicable to creating a search space for topology optimization of 2D airfoils. Most design techniques impose geometric constraints and sometimes designers' bias on the design space itself, thus restricting the novelty of the designs created, and only allowing for small local changes. ...
Heegaard Floer homology for manifolds with torus boundary: properties and examples
This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion....
On the order of Dehn twists
This note records the order of a higher dimensional Dehn twist in a range of topologically significant groups.
Instanton L-spaces and splicing
We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L-spaces, by a map which identifies meridians with Seifert longitudes, cannot be an instanton L-space. This recovers the recent theorem of Lidman, PinzΓ³n-Caicedo, and Zentner that the fundamental group of every closed, oriented, to...
Framed instanton homology and concordance
We define two concordance invariants of knots using framed instanton homology. These invariants πβ― and πβ― provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the π invariant in Heegaard Floer homology. We use πβ― and πβ― to compute the framed instanton homolo...
THE POSITIVE SCALAR CURVATURE COBORDISM CATEGORY
We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connec...
