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Based on work of Rasmussen [Ras03], we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsv ́ath-Szab ́o τ invariant.

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We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S^3∖K. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of SL(2,C) Chern-Simons theory, one can define perturbative invariants of knot K in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of K and K. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.

We consider the flow of closed convex hypersurfaces in Euclidean space R^{n+1} with speed given by a power of the k-th mean curvature E_k plus a global term chosen to impose a constraint involving the enclosed volume V_{n+1} and the mixed volume V_{n+1−k} of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.