Legendrian knots and monopoles

Differential algebras of Legendrian links

The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them is: which combinations of the three classical invariants can be realized by a Legendrian knot? (It is well-known that each invariant by itself can be realized by a Legendrian knot). The first step in this direction was done by Bennequin who proved that the Bennequin number of a Legendrian knot is less than twice its genus. The second subproblem is whether there exist a pair of Legendrian knots whic…

关于纽结环路的勒让德实现

On the Legendrian realisation of loops of knots

本研究针对T. Kálmán提出的勒让德嵌入到光滑嵌入基本群同态是否为满射的公开问题，对三大主族无穷多纽结类型及所有稳定化勒让德代表元给出了肯定解答。

不能参数化地成为Legendrian的纽结族

Families of knots that cannot be made Legendrian parametrically

本文证明对于每个n≥3，任何纽结类型的Legendrian代表及其形式代表在π_n层面上的群同态均非满射，并指出π_2层面的满射性依赖纽结类型，从而揭示了参数族在更高同伦层面的刚性。

Poincar{é} series and linking of Legendrian knots

On a negatively curved surface, we show that the Poincar{é} series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincar{é} series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesi…

Legendrian 纽结在海森堡群中的 CR 不变能量

CR-invariant energy of Legendrian knots in the Heisenberg group

本文研究了海森堡群中 Legendrian 纽结的 CR 不变能量，通过正则化 Korányi 距离下的 -2 阶势能积分来定义能量函数，并展示了实数圆作为能量最小化者的特性，同时给出了海森堡群中的 Doyle--Schramm 余弦公式类比。