Legendrian singular links and singular connected sums (Q1720128)

A Legendrian singular link is an immersion of a closed \(1\)-manifold into the \(3\)-sphere \(S^3\) that is everywhere tangent to the standard contact structure \(\xi_{st}\) on \(S^3\). That means, in contrast to the theory of Legendrian links, transverse double points are allowed. Legendrian singular links occur naturally in some areas of contact topology, for example in the context of \(4\)-valent Legendrian graphs [\textit{S. Baader} and \textit{M. Ishikawa}, Ann. Fac. Sci. Toulouse, Math. (6) 18, No. 2, 285--305 (2009; Zbl 1206.57005)] or by considering Vassilev type invariants of Legendrian knots [\textit{D. Fuchs} and \textit{S. Tabachnikov}, Topology 36, No. 5, 1025--1053 (1997; Zbl 0904.57006)]. In this article, the authors study Legendrian singular links on their own. Note, that Legendrian singular links up to regular Legendrian homotopy fulfill an \(h\)-principle and therefore depend only on the underlying algebraic topology [\textit{Y. Eliashberg} and \textit{N. Mishachëv}, Introduction to the \(h\)-principle. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1008.58001)]. The main result in this article says that this is not true if we study Legendrian singular links up to ambient contact isotopy. First, several results from the theory of ordinary Legendrian links and topological singular links are carried over to this setting. For example a Reidemeister-type theorem is proven, stabilization, resolutions and classical invariants for Legendrian singular links are defined and discussed. At other places the theory of Legendrian singular links differs from the situation for ordinary Legendrian links. Several results in this direction are discussed. For example it is in general not true that two Legendrian singular links of the same topological singular knot type do admit a common stabilization. However, it is proven that this is true if we allow another move -- the so called flip -- at a singular point. Finally, the authors prove their main result, stating that there exist non-isotopic Legendrian singular knots \(L_1\) and \(L_2\) with the same classical invariants and the same Legendrian link types of all resolutions. To distinguish these Legendrian singular knots, the authors define the so-called singular connected sum of two Legendrian singular links by removing a standard neighborhood of a singular point in each link and identifying the resulting boundaries. The result follows then by finding a Legendrian singular knot \(L\) such that the singular connected sum of \(L_i\) with \(L\) yields non-isotopic Chekanov Legendrian twist knots [\textit{Y. Chekanov}, Invent. Math. 150, No. 3, 441--483 (2002; Zbl 1029.57011)].