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