A note on the Upsilon invariant, its homogenization, and the braid index of knots (Feller/Krcatovich)

We use Ozsvath, Stipsicz, and Szabo's Upsilon invariant to provide bounds on cobordisms between knots that `contain full-twists'. This generalizes previous results and allows us to recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a full twist realize the braid index of their closure. We also provide inductive formulas for the Upsilon invariants of torus knots and compare the Upsilon function to the Levine-Tristram signature profile.Non UBCUnreviewedAuthor affiliation: Boston CollegePostdoctora