Overview
This study investigates the equivariant $\mathbb Q$-sliceness and Klein amphichirality of odd-stranded Turk's head knots, establishing a dichotomy based on the parity of the twisting parameter $q$. The research differentiates the behavior of these knots depending on whether $q$ is odd or even, utilizing distinct mathematical approaches for each case.
Approach
For the case where the twisting parameter $q$ is odd, the researchers constructed a commuting pair of ambient involutions. This construction involved lifting explicit symmetries observed in the signed, arrowed Gauss diagram through the associated abstract link diagram and supporting sphere. This specific methodology allowed for the demonstration of both Klein amphichirality and equivariant $\mathbb Q$-sliceness.
When the twisting parameter $q$ is even, the methodology shifted to proving that the knots are not equivariantly $\mathbb Q$-slice with respect to any strong inversion. The primary obstruction identified for this case was the equivariant Fox-Milnor square condition, attributed to Di Prisa-Savk. To establish this obstruction, the researchers showed that the Alexander polynomial of these knots is not a Laurent square. The argument for even $q$ combined two computational elements: a Burau square factorization, derived from Garside conjugacy, and a mod-$2$ Seifert-matrix computation.
Findings
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Odd Twisting Parameter ($q$ odd): Odd-stranded Turk's head knots are demonstrated to be Klein amphichiral when the twisting parameter $q$ is odd. This condition further implies that these knots are equivariantly $\mathbb Q$-slice. The proof relies on the construction of a commuting pair of ambient involutions, derived from the symmetries of their signed, arrowed Gauss diagrams and their associated abstract link diagrams and supporting spheres.
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Even Twisting Parameter ($q$ even): Odd-stranded Turk's head knots are proven not to be equivariantly $\mathbb Q$-slice with respect to any strong inversion when the twisting parameter $q$ is even. This conclusion is based on the equivariant Fox-Milnor square condition, which states that the Alexander polynomial must be a Laurent square for equivariant $\mathbb Q$-sliceness. The research specifically found that the Alexander polynomial for these knots is not a Laurent square when $q$ is even. The supporting arguments for this finding involved a Burau square factorization, derived from Garside conjugacy, and a mod-$2$ Seifert-matrix computation.
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Sharp Parity Dichotomy: The research establishes a sharp parity dichotomy regarding the equivariant $\mathbb Q$-sliceness and Klein amphichirality of odd-stranded Turk's head knots, with the behavior precisely determined by the odd or even nature of the twisting parameter $q$.