Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

中文翻译：

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结的手性平滑

可以平滑结图中的单个交叉点将其转换为结的镜像图吗？Zeković 为圆环结找到了这样的平滑方法吨(2, 5) 和 Moore-Vazquez 证明了其他圆环结不存在这种平滑吨(2,米） 和米奇数和平方免费。这种平滑的存在意味着ķ#ķ将莫比乌斯带限制在乙4. 我们使用 Casson-Gordon 理论来为这种手性平滑的存在提供新的障碍。特别是，我们删除了约束米在 Moore-Vazquez 定理中是自由的，除了米= 9，这仍然是一个开放的案例。Heegaard Floer 理论提供了进一步的障碍；对于以下形式的圆环结，这些不会提供新信息吨(2,米)，但它们确实为其他圆环结族提供了强大的约束。一个更一般的问题，对于每对结ķ和Ĵ, 转换图表所需的最小平滑次数是多少ķ合二为一Ĵ. 此处介绍的方法可用于提供此数字的下限。