Obstructions to Lagrangian concordance

Abstract

Abstract. We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in R 3 . In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

“…. We do not have a closed-form description of f , but one can work out that f takes the values 0, 4,8,9,13, and 16 for t = 0, 1, 2, 3, 4, and 5 respectively. It is not hard to see that if we allow real values for d i , then the minimum of d 2 i occurs when m = 1 and d 1 = d, and satisfies d 2 − d = 2t.…”

“…Proof of Proposition 1.8. In [8], it was shown that if there is a Lagrangian concordance from L to L then there is also a Lagrangian concordance from a given Legendrian satellite of L to the same satellite of L (in that paper it is assumed the concordance is a product at the end, but this hypothesis is not necessary, in light of Lemma 3.2, see [22]). It is easy to check that the (n, 1)cable of the maximal Thurston-Bennequin invariant unknot U described in Theorem 1.8 results in U .…”

“…As indicated above, there are many knots that satisfy the condition of Theorem 1.1, although there are not too many with small crossing number. In [8] it was shown that for knots with 12 or fewer crossings, the only examples are 9 46 , 10 140 , 11n 139 , 12n 582 , 12n 768 , and 12n 838 , see Figure 1. Here, the names are the ones given in KnotInfo [3], and a bar over the name indicates the mirror of that knot.…”

Non-loose Legendrian spheres with trivial contact homology DGA

“…Consider the Legendrian knot K in Figure 6. This knot belongs to the isotopy class 9 46 and its Lagrangian disk fillings are described in [2,3].…”

“…We prove Theorem 1.1 as follows. First, elaborating on results from [2,3], we find a Legendrian (n − 1)-sphere Σ ⊂ R We also consider the Legendrian spheres Λ j , j = 0, 1, constructed exactly as Λ but using instead the same filling W Mo j on both sides.…”