में बचाया:
| मुख्य लेखकों: | , |
|---|---|
| स्वरूप: | Preprint |
| प्रकाशित: |
2018
|
| विषय: | |
| ऑनलाइन पहुंच: | https://arxiv.org/abs/1812.06930 |
| टैग: |
टैग जोड़ें
कोई टैग नहीं, इस रिकॉर्ड को टैग करने वाले पहले व्यक्ति बनें!
|
| _version_ | 1866910623210668032 |
|---|---|
| author | Lamb, Kevin Weed, Patrick |
| author_facet | Lamb, Kevin Weed, Patrick |
| contents | For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslashη(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E(K)$ into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of $E(K)$. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) $E(K)$ can't contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_06930 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A Distance for Circular Heegaard Splittings Lamb, Kevin Weed, Patrick Geometric Topology For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslashη(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E(K)$ into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of $E(K)$. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) $E(K)$ can't contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy. |
| title | A Distance for Circular Heegaard Splittings |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/1812.06930 |