Research on knot theory using contact structures and holomorphic curves

基于接触结构和全纯曲线的纽结理论研究

• 批准号: 21740041
• 负责人: KALMAN Tamas
• 金额: $ 2.91万
• 依托单位: Tokyo Institute of Technology (2010-2012)The University of Tokyo (2009)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21740041/
• 关键词: 微分トポロジー; トポロジー; 接触構造; 複素曲線; ホモロジー; 結び目; ハイパーグラフ; 結び目理論

In this research, I developed a theory that describes a surprising new relation between Heegaard Floer homology and the Homfly polynomial. The connection is well understood in the case of special alternating links, with partial results toward a generalization to all oriented links. It uses a theory in algebraic combinatorics (on hypertrees, polymatroids, and their interior polynomials) that I developed and which is very interesting in its own right. In joint work with Juhasz and Rasmussen, we related these notions to Heegaard Floer homology, and in joint work with Murakami, we described their connection to quantum invariants. All these works were presented at conferences both inside and outside of Japan, and with the exception of the joint work with Murakami, they were published in refereed journals.

在这项研究中，我开发了一个理论，描述了一个令人惊讶的新关系Heegaard Floer同源性和Homfly多项式。在特殊的交替链接的情况下，连接是很好理解的，部分结果走向推广到所有定向链接。它使用了我开发的代数组合学理论（关于超树，多拟阵及其内部多项式），这本身就非常有趣。在与Juhasz和Rasmussen的合作中，我们将这些概念与Heegaard Floer同调联系起来，在与Murakami的合作中，我们描述了它们与量子不变量的联系。所有这些作品都在日本国内外的会议上发表，除了与村上春树的合作作品外，它们都发表在参考期刊上。

项目成果

Relating Jones-type and Ozsvath-Szabo-type knot invariants

关联 Jones 型和 Ozsvath-Szabo 型结不变量

• 发表时间: 2013
• 作者: Kalman;Tamas.
• 通讯作者: Tamas.

Relating Jones-type and Ozsvath-Szabo-type knot invariants, Algebraic Geometry and Differential Topology Seminar

关联Jones型和Ozsvath-Szabo型结不变量，代数几何和微分拓扑研讨会

• 发表时间: 2012
• 作者: Kalman;Tamas.
• 通讯作者: Tamas.

Tutte's polynomial for hypergraphs and polymatroids

超图和多拟阵的塔特多项式

• 发表时间: 2011
• 作者: Kalman;Tamas.
• 通讯作者: Tamas.

Rulings and Lagrangian caps of Legendrian knots

传奇结的规则和拉格朗日帽

• 发表时间: 2009
• 作者: Kalman;Tamas
• 通讯作者: Tamas

A new type of combinatorics in knot theory

纽结理论中的一种新型组合学

• 发表时间: 2011
• 作者: Kalman;Tamas.;KALMAN Tamas
• 通讯作者: KALMAN Tamas