Determine whether the threshold parameter is rational

Determine whether the threshold parameter \(\alpha^*\in(1/3,1/2)\) governing the Morse-index formula for the six-ended Karcher saddle-tower quotients \(\Xi_{3,m}\) is rational, and consequently whether the associated index-increment pattern is eventually periodic.

Background

For the quotients Ξ3,m\Xi_{3,m}, the paper proves that the Morse index is 6m−4⌊mα∗⌋−36m-4\lfloor m\alpha^*\rfloor-3, where α∗∈(1/3,1/2)\alpha^*\in(1/3,1/2) is a threshold determined by an eigenvalue crossing. As mm increases, the index increment is either six or two, according to whether ⌊mα∗⌋\lfloor m\alpha^*\rfloor remains constant or increases.

The increment pattern would eventually repeat if α∗\alpha^* were rational, but the paper explicitly states that the rationality of α∗\alpha^* is unknown. Thus determining this arithmetic property would settle whether the observed Morse-index increment pattern is eventually periodic.

References

By \ref{k=3} the index of \ks_{3,m} increases by either six or two as m advances by one (and the genus correspondingly by two), depending on whether \lfloor m\alpha* \rfloor stays constant or jumps; only if \alpha* is rational, which we do not know, is this increment pattern eventually periodic.

— Morse index of Karcher saddle towers in $\mathbb{R}^2 \times \mathbb{S}^1(m)$  (2608.13451 - Wiygul, 13 Aug 2026) in Remark following Corollary, Section 1 (Introduction)