Graduate and Postdoctoral Studies
Derivatives of genus one and three knots
Tuesday, April 11, 2017
to 4:00 PM
301 Sewall Hall
A derivative L of an algebraically slice knot K is an oriented link disjointly embedded in a Seifert surface of K such that its homology class forms a basis for a metabolizer H of K. For genus one knots, we produce a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. In order to do so, we define an operation on a homology B^4 that we call an n-twist annulus modification. Further, we give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. For genus three knots, we show that the set of possible Milnor's triple linking number associated to a metabolizer H contains nZ, where n is an integer determined by a Seifert form of K and a metabolizer H. As a corollary, we show that it is possible to realize any integer as the Milnor's triple linking number of a derivative of the unknot on a fixed Seifert surface and with a fixed metabolizer.