Contact surgeries and the transverse invariant in knot Floer homology

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and n>3.Comment: Corrected naturality discussion

Condensation of quasiparticles and density modulation beyond the superfluid critical velocity

We investigate the effect of a constant external velocity field on the ground state of a bosonic quasiparticle Hamiltonian. Below a critical velocity the ground state is a quasiparticle vacuum, corresponding to a pure superfluid phase at zero temperature. Beyond the critical velocity energy minimization leads to a macroscopic condensation of quasiparticles at a nonzero wave vector k_v parallel to the velocity v. Simultaneously, physical particles also undergo a condensation at k_v and, to a smaller extent, at -k_v. Together with the BEC at k=0, the three entangled condensates give rise to density modulations of wave vectors k_v and 2k_v. For larger |v| our model predicts a bifurcation of k_v with corresponding two pure condensates and no density modulation.Comment: Proof of quasiparticle condensation complete

Concordance homomorphisms from knot Floer homology

We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.Comment: Minor revision, corrected typos and added explanation to Chapter