Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots
Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots
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Chekanov-Eliashberg 不变量和 Legendrian 结的横向近似
DOI：
10.2140/pjm.2001.201.89
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发表时间：
2001
影响因子：
0.6
通讯作者：
Maike Meyer
中科院分区：
文献类型：
--
作者：
J. Epstein;D. Fuchs;Maike Meyer
在本文中，我们考虑标准接触空间中的勒让结和横结，即在 R3 中，接触结构全局由 1-形式 α = ydx− dz 给出。众所周知，在接触结构内沿其正法线方向稍微推动定向勒让德结 Γ 会将其转变为横向结 Γ+，其自然方向由 α 确定，与选择的 Γ 方向相匹配；沿负法线方向推动 γ 会产生相反方向的横向结 γ−。任何横结都是从某个定向的勒让结 Γ 导出的 Γ+ 的横同位素，并且如果勒让结 Г,' 是勒让德同位素，则 Г+,'+ 是横同位素。除了拓扑不变量之外，勒让结和横结还具有接触起源的经典不变量：勒让结 Γ 的瑟斯顿-贝内金数 τβ(Г) 和马斯洛夫数 μ(Г) 以及横结 Г 的瑟斯顿贝内金数 τβ(Г)。另外，τβ(Γ)＝τβ(τ)+μ(τ)，τβ(τ-)＝τβ(τ)-μ(τ)。
In this article, we consider Legendrian and transverse knots in the standard contact space, that is in R3 with the contact structure globally given by the 1-form α = ydx− dz. It is well-known that a little push of an oriented Legendrian knot Γ in the direction of its positive normal within the contact structure changes it into a transverse knot Γ+, whose natural orientation, as determined by α, matches the chosen orientation of Γ; pushing Γ in the direction of its negative normal produces the transverse knot Γ− of the opposite orientation. Any transverse knot is transverse isotopic to Γ+ derived from some oriented Legendrian knot Γ, and if Legendrian knots Γ,Γ′ are Legendrian isotopic, then Γ+,Γ′+ are transverse isotopic. Besides their topological invariants, Legendrian and transverse knots have classical invariants of contact origin: The Thurston-Bennequin number τβ(Γ) and the Maslov number μ(Γ) for a Legendrian knot Γ and the ThurstonBennequin number τβ(Γ) for a transverse knot Γ. Furthermore, τβ(Γ) = τβ(Γ) + μ(Γ), τβ(Γ−) = τβ(Γ)− μ(Γ).