Abstract. We use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that 'contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-FranksWilliams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon-invariant of torus knots and compare it to the LevineTristram signature profile.

We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot $K$. As an application to connected sums, we prove that if a knot in the three-sphere admits an $L$-space surgery, it must be a prime knot. As an application of the computation of $d$-invariants, we show that the Alexander polynomial is a concordance invariant within the class of $L$-space knots, and show the four-genus bound given by the $d$-invariant of +1-surgery is independent of the genus bounds given by the Ozsv\'ath-Szab\'o $\tau$ invariant, the knot signature and the Rasmussen $s$ invariant.Comment: 41 pages, 14 figures; changed formatting, updated references, added some clarifying remarks, results unchange

Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by L-space knots.

We give infinitely many 2-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any 2-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

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