Lipshitz-Sarkar对镜像的精细处理优于Khovanov
Lipshitz-Sarkar refines mirrors more than Khovanov
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中文总结 AI 辅助
针对Lipshitz和Sarkar提出的同伦论Khovanov同调精细处理能否检测普通整系数Khovanov同调无法检测的镜像反射的问题，研究借助chat-GTP 5.6 sol Pro和Regina Census，证明该精细处理可区分特定素纽结及其镜像，还给出复合例子。
中文摘要 AI 辅助
Lipshitz和Sarkar在2018年国际数学家大会（ICM）上提出问题：他们对Khovanov同调的同伦论精细处理，能否检测出普通整系数Khovanov同调无法检测的镜像反射。我们给出肯定回答，提供一个素纽结，其整系数Khovanov同调在所有双次数（包括挠率）下与镜像的一致，但在指定双次数下，对该纽结，$\boldsymbol{\rm \bf F}_2$上的第二Steenrod平方秩为1，对其镜像则为0。我们还给出一个复合例子，构造、搜索和计算借助chat-GTP 5.6 sol Pro，梳理Regina Census中的纽结。
英文摘要
Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of {\it chat-GTP 5.6 sol Pro} combing over the {\it Regina Census} of knots.
发表机构
- London Institute for Mathematical Sciences（伦敦数学科学研究所）
- Royal Institution（皇家研究院）
- Merton College, Oxford（牛津大学默顿学院）
机构由 AI 辅助整理，请以论文原文为准。
