Seminar on low-dimensional mathematics Moscow--Petersburg
September 9, 2005

V.Chernov (Dartmouth)
Relative framings of transverse knots

It is well-known that a knot in a contact manifold $(M,C)$ transverse to a
trivialized contact structure possesses the natural framing given by the  
first of the trivialization vectors along the knot. If the Euler class    
$e_C\in H^2(M)$ of $C$ is nonzero, then $C$ is nontrivializable and the   
natural framing of transverse knots does not exist.

We construct a new framing-type invariant of transverse knots called
relative framing. It is defined for all tight $C$, all closed irreducible
atoroidal $M$, and many other cases when the contact structure is not
trivializable and the classical framing of transverse knots is not defined.
We show that the relative framing distinguishes many transverse knots that 
are isotopic as unframed knots.

Our recent result is that the groups of Vassiliev-Goussarov invariants of
transverse and of framed knots are canonically isomorphic, when $C$ is   
trivialized and transverse knots have the natural framing. We show that the
same result is true whenever the relative framing is well-defined.

As a useful tool, we show that $|e_C(\alpha_*([F^2]))|\leq \max\{0, - \chi
(F^2)\}$ for a tight $(M,C)$ and a continuous mapping $\alpha:F^2\to M$ of a
closed oriented $F^2$. This generalizes the Eliashberg's Bennequin-type
inequality for embeddings $\alpha$.

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