Department of Mathematics Calendar
Allison Miller, UT Austin, Winding number of satellite operators and concordance
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Historically, the study of the collection of concordance classes of knots has focused on understanding its group structure while devoting relatively little attention to the natural metric induced by the 4-genus. Cochran and Harvey investigated the metric properties of the maps on concordance induced by satellite operators, asking when two patterns P and Q are of bounded distance in their action on concordance. In this talk, I will describe the proof that two patterns are of bounded distance if and only if they have the same algebraic winding number [Cochran-Harvey, M.]. I’ll focus on the subtlest case, when P has winding number m and Q has winding number -m, and use Casson-Gordon signatures to show that for m>0 the 4-genus of P(K)#-Q(K) can be arbitrarily large.