Mathematics Colloquia and Seminars

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Description

This is an account of a joint work in progress with Dan Rutherford. A front diagram of a Legendrian knot is a closed curve in the (x,z)-plane without vertical (parallel to the z-axis) tangents and without self-tangencies, but with cusps. A Morse family of functions $F_t: R^n\to R$ such that $F_t(x_1,..., s_n)=x_n$ for for $(x_1,...,x_n;t)\notin K$ where $K\subset R^n\times R$ is called a generating family of functions for a front diagram L if the following holds: $(t,z)\in L$ if and only if $z$ is a critical value of $F_t$. Following P. Pushkar, we consider the manifold

$$ W=\{(x,y,t)\in R^n\times R^n\times R| F_t(y)-F_t(x)> \epsilon\} $$

where $\epsilon>0$ is small. Our main result, so far, states that the $Z_2$-homology of $W$ is equal, up to a dimension shift, to the homology of the Chekanov--Eliashberg DGA linearized by an augmentation. Absolutely all necessary definitions from the Legendrian knot theory will be provided and explained in the talk.