Slice implies mutant ribbon for odd 5-stranded pretzel knots
Abstract
The Slice-Ribbon Conjecture, posed by Fox in 1966, is a long-standing open conjecture that posits that every slice knot is a ribbon knot. It is known and easily seen that every ribbon knot is a slice knot, implying that the conjecture is really a statement about the equivalence of the two notions of `slice' and `ribbon'. In 2011, Greene and Jabuka showed that the Slice-Ribbon Conjecture holds for the infinite family of odd 3-stranded pretzel knots. In their work, they give a complete characterization of the slice/ribbon knots in that infinite family. This dissertation is motivated by their work and proves that the family of odd 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice odd 5-stranded pretzel knots are mutant ribbon.
The two extra strands in this case add a level of complexity not seen in the 3-stranded case, precisely with respect to mutation. The main result is obtained through use of the knot signature, Donaldson's Diagonalization Theorem from gauge theory, and d-invariants from Heegaard-Floer theory. From each of these tools, a necessary condition for sliceness can be extracted and this thesis shows that for odd 5-stranded pretzel knots that are not mutant ribbon, these conditions are not met and hence do not apply to the Slice-Ribbon Conjecture.