Comparing invariants of Legendrian knots
To a Legendrian knot \(L\) in a closed contact \(3\)-manifold \(Y\) one associates a contact \(3\)-manifold with convex boundary \(Y_L\) defined as the complement of a standard neighborhood of \(L\) in \(Y\). The contact invariant of \(Y_L\), defined by Honda, Kazez and Matić [\textit{K. Honda} et al., Invent. Math. 176, No. 3, 637--676 (2009; Zbl 1171.57031)] and denoted by \(EH(L)\), can be seen as an invariant of the Legendrian isotopy class of \(L\). In addition, Lisca, Ozsváth, Stipsicz and Szabó in [\textit{P. Lisca} et al., J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307--1363 (2009; Zbl 1232.57017)] using open book decompositions constructed an invariant of an oriented nullhomologous Legendrian knot \(L\) in a closed contact 3-manifold \((Y,\xi)\) which is called \({\mathfrak L}^{-}(L)\). The author proves the equivalence of the invariants \(EH(L)\) and \({\mathfrak L}^{-}(\pm L)\) for oriented Legendrian knots \(L\) in \((S^3, \xi_{st})\), where \(\xi_{st}\) is the standard contact structure on \(S^3\). Here \(-L\) denotes \(L\) with reversed orientation. More precisely, given two oriented, topologically isotopic Legendrian knots \(L_0\), \(L_1\) in \((S^3,\xi_{st})\), the author proves that the following two statements are equivalent: \(EH(L_0)=EH(L_1)\) and \({\mathfrak L^{-}}(\pm L_0) = {\mathfrak L}^{-}(\pm L_1)\).
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