On loose Legendrian knots in rational homology spheres (Q684003)

If \((M,\xi)\) is a contact manifold, then a closed, connected, embedded submanifold \(L\to M\) so that \(TL\subset\xi\) is a Legendrian knot. Two such knots are said to be equivalent if they are Legendrian isotopic. A contact structure \(\xi\) on \(M\) is called overtwisted if there is a disk \(D\) embedded in \(M\) such that the tangent plane \(T_xD\) to \(D\) is the same as \(\xi_x\) for every \(x\in\partial D\), otherwise it is called tight. A Legendrian knot \(K\) in an overtwisted contact 3-manifold \((M,\xi)\) is said to be loose if also its complement is overtwisted, while it is non-loose if the complement is tight. There are three classical invariants of Legendrian knots. The first one is the smooth isotopy class of an oriented Legendrian knot \(K\) and is called the knot type. The other two classical invariants are the Thurston-Bennequin number, which is defined as the linking number of the contact framing of \(K\) with respect to a Seifert framing of \(K\), and the rotation number which is the numerical obstruction to extending a non-zero vector field, everywhere tangent to the knot, to a Seifert surface of \(K\). In this paper, the author considers loose Legendrian knots in rational homology spheres. It is shown that if \((M,\xi)\) is a rational homology contact 3-sphere, \(K_1\) and \(K_2\) are two loose Legendrian knots in \(M\) such that there exists a pair of disjoint overtwisted disks \((E_1,E_2)\), where \(E_i\) is contained in the complement of \(K_i\), then \(K_1\) and \(K_2\) are Legendrian isotopic if and only if they have the same classical invariants. Even though the assumption that disks are overtwisted is needed, this result can be applied in many interesting cases like disjoint unions of Legendrian knots. A Legendrian 2-component link \(L\) is called split if \((M,\xi)\) can be decomposed into \((M_1\#M_2,\xi_1\#\xi_2)\) and \(K_i\hookrightarrow(M_i,\xi_i)\), that is, \(L\) is the disjoint union of \(K_1\) and \(K_2\). The author proves that if \(K_1\) and \(K_2\) are two loose Legendrian knots in the rational homology contact sphere \((M,\xi)\) such that \(K_1\cup K_2\) is a split Legendrian link, then they are Legendrian isotopic if and only if they have the same classical invariants.