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Geometry and Topology Seminar

Friday, May 20, 2022
1:00pm to 2:00pm
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Linde Hall 187
The alternation number and the Upsilon-invariant at 1 of positive 3-braid knots
Paula Trüol, Department of Mathematics, ETH Zurich,

The alternation number of a knot is the minimal number of crossing changes needed to deform the knot into an alternating knot, i.e. a knot with a diagram where the crossings alternate between over- and under-crossings as one travels around the knot. The tau- and the Upsilon-invariant from knot Floer homology give a lower bound on the alternation number of any knot. We use this lower bound and an upper bound by Abe and Kishimoto to determine the alternation number of all positive 3-braid knots.

The key tool and a result of independent interest is an explicit calculation of the Upsilon invariant at 1 of all 3-braid knots. We determine this integer-valued (concordance) invariant - which was defined by Ozsváth, Stipsicz and Szabó - by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. In particular, we will only work with properties of Upsilon and not its definition, so no background in knot Floer homology will be assumed.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected] or visit https://sites.google.com/site/caltechgtseminar/home.