Involutions of knots that fix unknotting tunnels

Pretzel 结的 Delta-解结数

Delta-Unknotting Number for Pretzel Knots

本文计算了正 pretzel 结的 Δ-解结数，并给出了奇型正 pretzel 结和环面结的最大值，以及特定负型 pretzel 结的 Δ-解结数。

The untwisting number of a knot

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The untwisting number is the minimum number (over all diagrams of a knot) of right- or left-handed twists on even numbers of strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot…

解开亏格为4和5的不可定向曲面

Unknotting Nonorientable Surfaces of Genus 4 and 5

本文利用Sage计算修正手术障碍，证明D^4中纽结群为Z_2、亏格4或5且边界行列式绝对值为1的不可定向局部平坦曲面相对边界环境同痕，从而对应S^4闭曲面拓扑不打结，并指出方法在亏格6、7时失效。

Complexity of Unknotting of Trivial 2-knots

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $K_i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation. These results contrast with the situatio…

直角 Coxeter 群边界上的无不动点对合

Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups

本文证明 Nica 关于边界存在无不动点对合的猜想在直角 Coxeter 群情形下的变体，并为原词双曲群问题提供方法启发。

Equivariant Rho-Invariants and Instanton Homology of Torus Knots

The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p,q)-torus knots with odd p and q.