math.GT — Geometric Topology

The bridge index and the nonorientable four-genus of knots with at most thirteen crossings

Contributed by Deep Bhattacharjee

We determine two invariants for knots that the KnotInfo database leaves open. KnotInfo lists the bridge index of \btotal{} prime knots with thirteen crossings as either $3$ or $4$; we prove that \bclosed{} of them have bridge index and meridional rank $4$, so the meridional rank conjecture holds for them. The lower bound comes from representations of the knot group in $\GL_4(\F_q)$ that send a meridian to a transvection or a reflection and fix no nonzero vector. KnotInfo leaves the smooth nonorientable four-genus open for \ntotal{} prime knots with eleven to thirteen crossings; we determine it for \nsettled{} of them and narrow the interval for \nnarrowed{} more. Upper bounds come from pinch moves, lower bounds from the Goeritz-lattice obstructions of Jabuka and Kelly evaluated by an exhaustive search for lattice embeddings. Every bound is a certificate checked by a program independent of the search that found it.