Lens space surgeries and a conjecture of Goda and Teragaito

The Goda-Teragaito conjecture: an overview

In math.GT/0106017 it was shown that thin position on Heegaard spines can be a useful tool for analyzing the topology of knots in 3-space. The proof there (specifically, of the Goda-Teragaito conjecture) requires masses of technical detail; it is easy to lose track of the underlying ideas. The present paper gives an overview of the argument, stripped of technical detail and emphasizing the mathematical background that seems naturally to motivate the proof.

寻找交替手术

The search for alternating surgeries

本文研究了交替手术的算法可计算性及结构特性，通过计算多种结的交替手术斜率，揭示了强可逆结和非对称结的交替手术现象，并给出了超曲面结的交替手术斜率上界。

On the Alexander polynomial of lens space knot

Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This cu…