Title: Concordances to prime hyperbolic virtual knots
Speaker: Micah Chrisman, OSU
Abstract: In 1979, Kirby and Lickorish proved that every knot in S^3 is concordant to a prime knot. Later, in 1981, Livingston proved that every knot is concordant to a prime satellite knot. Myers proved (1983) that every knot is concordant to a hyperbolic knot. Furthermore, these concordances can be realized so that the Alexander polynomial is preserved (Bleiler 1982, Nakanishi 1983). There has been much recent work on hyperbolicity of knots in thickened surfaces (see e.g. Adams et al. 2018, Champanekar-Kofman-Purcell 2018). We will sketch a proof extending the above mentioned results to this context. It will be shown that every knot in a thickened surface is (virtually) concordant to a prime satellite virtual knot and a prime hyperbolic virtual knot. Moreover, if a knot is homologically trivial in some thickened surface, then it is concordant to a prime satellite virtual knot and a prime hyperbolic virtual knot having the same Alexander polynomial. The main difficulty in each case lies in proving primeness. Indeed, there are hyperbolic knots in thickened surfaces that are not prime as virtual knots.
This talk is based on the paper at https://arxiv.org/pdf/1904.05288 [pdf]
Seminar URL: https://www.coreyjonesmath.com/qaqt-seminar-osu
