Legendrian singular links and singular connected sums

A Chekanov–Eliashberg algebra for Legendrian graphs

“…It is easy to see from Theorem 2.3 that one can equip a cyclic order between half-edges at each vertex v by using the co-orientation with respect to the Reeb direction ∂ z for the contact form dz + xdy − ydx. This is observed in [21,Remark 3.3] and also in [1,Section 3.2].…”

“…On the other hand, for two adjacent edges e, e sharing common vertex v, the end points De (1) and De (0) are exactly the same as end points of a path…”

“…) determines an half edge of the vertex γ ± gr (1). We simply denote the induced half edge by h(γ ± gr ), then (4.5) implies…”

“…1 is inward. Indeed, there are no orientation-preserving isotopies between 4 × 1 and −4 × 1 as seen in [1], and therefore 4 × 1 is irreversible. In terms of DGAs and peripheral structures, one can prove that generalized stable-tame isomorphism classes of pairs (…”

“…We first regard both M(g) (1) and M(g ) (1) as subsets of M by choosing the separator as follows: For f ∈ M t (g) and f ∈ M t (g ), we choose separators as 0 and (t + 1), respectively. That is, Conversely, it is obvious that we can regard all hybrid disks in M v,most L (g) and M v,most R (g) as admissible disks in M(g) (1) and M(g ) (1) , respectively. with g = ι(g), we have bijections…”