March 2017 |

2:30 pm Tuesday, April 11, 2017 Thesis Defense: Derivatives of genus one and three knotsby JungHwan Park, Ph.D. Candidate (Rice University) in Sewall Hall, Rm. 301- 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 B4 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 SK,H = {µ¯L(123) − µ¯LI (123) | L, LI ∈ dK/dH} contains n · Z, where dK/dH is the set of all the derivatives associated with a metabolizer H and 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.
Submitted by lpl1@rice.edu |