Fillability of small Seifert fibered spaces

TL;DR

This paper characterizes which zero-twisting contact structures on certain small Seifert fibered spaces are fillable, using monodromy factorizations of associated planar open books, extending understanding of fillability in contact topology.

Contribution

It provides a characterization of fillable zero-twisting contact structures on specific small Seifert fibered spaces, a case not fully understood before.

Findings

All tight structures on spaces with $e_0 eq -1,-2$ are Stein fillable.

Fillability of zero-twisting structures on spaces with $e_0=-1$ is characterized.

Analysis of monodromy factorizations is used to determine fillability.

Abstract

On small Seifert fibered spaces $M(e_0;r_1,r_2,r_3)$ with $e_0\neq-1,-2,$ all tight contact structures are Stein fillable. This is not the case for $e_0=-1$ or $-2$. However, for negative twisting structures it is expected that they are all symplectically fillable. Here, we characterize fillable structures among zero-twisting contact structures on small Seifert fibered spaces of the form $M(-1;r_1,r_2,r_3)$. The result is obtained by analyzing monodromy factorizations of associated planar open books.