New lower bounds for the non-orientable four-genus: one hundred and eighteen knots of at most thirteen crossings

Abstract

The non-orientable four-genus γ₄(K) of a knot K ⊂ S³ is the least first Betti number of a smooth, compact, properly embedded surface in B⁴ with boundary K; γ₄(K)=1 says that K bounds a Möbius band. Lee and Sabloff have recently surveyed γ₄ over all knots of at most 14 crossings, implementing the Murakami–Yasuhara linking-form obstruction only under the hypothesis that det(K) is square-free; their data records 14,319 knots on which that obstruction was in consequence never evaluated at all. We evaluate it on the part of that set for which KnotInfo publishes a diagram and the first homology of the double branched cover is cyclic, and obtain γ₄(K) ≥ 2 for 118 knots whose γ₄ their pipeline left undetermined — five with 11 crossings, six with 12, and one hundred and seven with 13. For 91 of them the upper bound recorded in KnotInfo is 2, so γ₄(K)=2 exactly; the remaining 27 have γ₄(K) ∈ {2,3}. These 118 values raise the count in Lee–Sabloff's Theorem 1.1 from "at least 22,412" to "at least 22,530". The obstruction argument is not new: for 87 of the 118 the applicable published statement is a corollary of Gilmer and Livingston, for 23 it is a lemma of Fairchild, and for 8 it is Lee–Sabloff's own corollary. What is new is the values. We also record the uniform decidable form of those statements for cyclic first homology, exhibit 322 knots in Lee–Sabloff's own data showing that the arithmetic of their corollary is unsound if applied outside its square-free hypothesis, and re-verify the arithmetic behind their corrections to six KnotInfo entries. Every finite computation reported here is machine-checked in Lean 4.

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