AI 中文总结
研究奇数股土耳其头结的等变有理切片性和克莱因两性性，通过构造环境对合证明q为奇数时结的性质，利用等变福克斯 - 米尔诺平方条件和相关计算证明q为偶数时结不具有相应性质。
AI 中文摘要
我们为奇数股土耳其头结的等变有理切片性和克莱因两性性建立了一个清晰的奇偶二分法。当扭转参数q为奇数时，我们通过带符号、带箭头的高斯图的显式对称经由相关抽象链图和支撑球面提升，构造了一对可交换的环境对合。这证明了这些结是克莱因两性的，因此是等变有理切片的。当q为偶数时，我们证明这些结对于任何强反转都不是等变有理切片的。障碍是迪·普里萨 - 萨夫克的等变福克斯 - 米尔诺平方条件：我们表明亚历山大多项式不是洛朗平方。偶数q的论证结合了来自加赛德共轭的布劳平方分解和模2塞弗特矩阵计算。
英文摘要
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-Şavk: 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.
Comments21 pages, 2 figures
