Volume 6, issue 2 (2006)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Generating family invariants for Legendrian links of unknots

Jill Jordan and Lisa Traynor

Algebraic & Geometric Topology 6 (2006) 895–933

arXiv: 0904.2690


Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in 3. It is shown that the unknot with maximal Thurston–Bennequin invariant of 1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for 2–component Legendrian links where each component is a maximal unknot. Techniques are developed to compute these polynomials, and computations are done for two families of Legendrian links: rational links and twist links. The polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links.

Legendrian links, generating functions, generating families, DGA
Mathematical Subject Classification 2000
Primary: 53D10
Secondary: 57M25
Forward citations
Received: 28 March 2006
Accepted: 25 April 2006
Published: 24 July 2006
Jill Jordan
Department of Mathematics
Bryn Mawr College
Bryn Mawr PA 19010
Lisa Traynor
Department of Mathematics
Bryn Mawr College
Bryn Mawr PA 19010