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We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon-invariant to provide boundson cobordisms between knots that `contain full-twists'. In particular, werecover and generalize a classical consequence of the Morton-Franks-Williamsinequality for knots: positive braids that contain a positive full-twistrealize the braid index of their closure. We also establish that quasi-positivebraids that are sufficiently twisted realize the minimal braid index among allknots that are concordant to their closure. Finally, we provide inductiveformulas for the Upsilon invariant of torus knots and compare it to theLevine-Tristram signature profile.

On cobordisms between knots, braid index, and the Upsilon-invariant