Journal of Symplectic Geometry

Volume 20 (2022)

Number 1

Ruling invariants for Legendrian graphs

Pages: 49 – 98



Byung Hee An (Department of Mathematics Education, Kyungpook National University, Daegu, South Korea; and Center for Geometry and Physics, Institute for Basic Science, Pohang, South Korea)

Youngjin Bae (Department of Mathematics, Incheon National University, Incheon, South Korea)

Tamás Kálmán (Department of Mathematics, Tokyo Institute of Technology, Tokyo, Japan)


We define ruling invariants for even-valent Legendrian graphs in standard contact three-space. We prove that rulings exist if and only if the DGA of the graph, introduced by the first two authors, has an augmentation. We set up the usual ruling polynomials for various notions of gradedness and prove that if the graph is fourvalent, then the ungraded ruling polynomial appears in Kauffman–Vogel’s graph version of the Kauffman polynomial. Our ruling invariants are compatible with certain vertex-identifying operations as well as vertical cuts and gluings of front diagrams. We also show that Leverson’s definition of a ruling of a Legendrian link in a connected sum of $S^1 \times S^2$’s can be seen as a special case of ours.

Received 31 December 2019

Accepted 17 April 2021

Published 21 October 2022