Loose Legendrian and pseudo-Legendrian knots in \(3\)-manifolds. (English) Zbl 1482.57018
This paper is concerned with the classification of loose Legendrian knots in contact \(3\)-manifolds, i.e.Legendrian knots which have an overtwisted disk in their complement. The existence of such an overtwisted disk in the complement ensures that loose Legendrian knots are classified by their formal data coming from algebraic topology.
In the present paper, such an \(h\)-principle classification result is explicitly worked out in a very general situation as follows:
Let \(L_1\) and \(L_2\) be two loose Legendrian knots in a cooriented, overtwisted (not necessarily closed) contact \(3\)-manifold \((M,\xi)\), which represent the same smooth knot type \(K\) in the complement of an overtwisted disk \(D\). Then \(L_1\) and \(L_2\) are isotopic as Legendrian knots if the following three conditions are satisfied:
Furthermore, explicit examples are worked out to demonstrate that each of these three conditions is indeed needed.
For nullhomologous Legendrian knots, Conditions (1) and (2) are equivalent to requiring that \(L_1\) and \(L_2\) have the same Thurston–Bennequin invariants and rotation numbers. Condition (3) is always fulfilled for nullhomologous knots. Therefore, the above result directly generalizes earlier classification results for nullhomologous knots by K. Dymara [Ann. Global Anal. Geom. 19, No. 3, 293–305 (2001; Zbl 0985.57009)], J. B. Etnyre [Quantum Topol. 4, No. 3, 229–264 (2013; Zbl 1281.57016)], and the more general result by F. Ding and H. Geiges [J. Topol. 2, No. 1, 105–122 (2009; Zbl 1169.57025)].
In their proof, the authors show that the classification of loose Legendrian knots can be reduced to the study of knots transverse to a given nowhere vanishing vector field. And a large part of the paper deals with the classification of the latter (by algebraic topology), which might be of independent interest.
In the present paper, such an \(h\)-principle classification result is explicitly worked out in a very general situation as follows:
Let \(L_1\) and \(L_2\) be two loose Legendrian knots in a cooriented, overtwisted (not necessarily closed) contact \(3\)-manifold \((M,\xi)\), which represent the same smooth knot type \(K\) in the complement of an overtwisted disk \(D\). Then \(L_1\) and \(L_2\) are isotopic as Legendrian knots if the following three conditions are satisfied:
- (1)
- \(L_1\) and \(L_2\) are isotopic as framed knots (where the framing is induced by a vector field transverse to the contact structure).
- (2)
- \(L_1\) and \(L_2\) are homotopic as Legendrian immersions.
- (3)
- The possible values of the Euler class of \(\xi\) evaluated on smooth isotopies of \(K\) agree with the evaluations on homotopies of \(K\).
Furthermore, explicit examples are worked out to demonstrate that each of these three conditions is indeed needed.
For nullhomologous Legendrian knots, Conditions (1) and (2) are equivalent to requiring that \(L_1\) and \(L_2\) have the same Thurston–Bennequin invariants and rotation numbers. Condition (3) is always fulfilled for nullhomologous knots. Therefore, the above result directly generalizes earlier classification results for nullhomologous knots by K. Dymara [Ann. Global Anal. Geom. 19, No. 3, 293–305 (2001; Zbl 0985.57009)], J. B. Etnyre [Quantum Topol. 4, No. 3, 229–264 (2013; Zbl 1281.57016)], and the more general result by F. Ding and H. Geiges [J. Topol. 2, No. 1, 105–122 (2009; Zbl 1169.57025)].
In their proof, the authors show that the classification of loose Legendrian knots can be reduced to the study of knots transverse to a given nowhere vanishing vector field. And a large part of the paper deals with the classification of the latter (by algebraic topology), which might be of independent interest.
Reviewer: Marc Kegel (Berlin)
