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Slice genera of knots in thickened surfaces

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Micah Chrisman (Monmouth)

Abstract: Let S be a compact oriented surface and K a knot in $S \times [0,1]$. Turaev defined the slice genus of K to be the smallest genus among all smooth surfaces with boundary K that can be embedded in a thickened 3-manilfold $M \times [0,1]$, taken over all compact oriented 3-manifolds M with boundary S. In this talk, we will discuss a number of techniques for determining the slice genus of knots in thickened surfaces. This problem is best viewed from the perspective of virtual knots. There are 92800 virtual knots having classical crossing number at most six. Applying these methods determines the slice genus in a large number of cases. In conclusion, we discuss some conjectures whose resolution would further improve the tabulation. This project is joint work with H. U. Boden and R. Gaudreau.