{"id":"hexagon/2610.00475","pdf":"https://hexagonmath.org/pdf/2610.00475v1","url":"https://hexagonmath.org/abs/2610.00475v1","tags":["math.GR","msc:57K10","msc:57K40","msc:20C33","msc:11E12","Donaldson diagonalisation","Goeritz lattices","KnotInfo","bridge index","meridional rank"],"score":0,"title":"The bridge index and the nonorientable four-genus of knots with at most thirteen crossings","domain":"math.GT","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.GT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}