Equivariant rational sliceness and Klein amphichirality of odd-stranded Turk's head knots

Abstract

We establish a sharp parity dichotomy for the equivariant $\mathbb Q$-sliceness and Klein amphichirality of odd-stranded Turk's head knots. When the twisting parameter $q$ is odd, we construct a commuting pair of ambient involutions by lifting explicit symmetries of the signed, arrowed Gauss diagram through the associated abstract link diagram and supporting sphere. This proves that the knots are Klein amphichiral and hence equivariantly $\mathbb Q$-slice. When $q$ is even, we prove that the knots are not equivariantly $\mathbb Q$-slice with respect to any strong inversion. The obstruction is the equivariant Fox-Milnor square condition of Di Prisa-\c{S}avk: we show that the Alexander polynomial is not a Laurent square. The even-$q$ argument combines a Burau square factorization arising from Garside conjugacy with a mod-$2$ Seifert-matrix computation.