| Abstract |
Weinstein domains are an important class of symplectic 4-manifolds with contact boundary that include disk cotangent bundles, and complex affine varieties. They are Liouville domains with a compatible Morse function whose symplectic topology is described by a corresponding handle body decomposition. A Weinstein handlebody decomposition can be encoded by a collection of Legendrian links in the boundary of the connected sum of $S^1times S^2$ with the standard contact structure. Thus, for Weinstein 4-manifolds with explicit handlebody decompositions, the study of their symplectic topology can be reduced to the study of Legendrian links.
We will consider the 4-dimensional Milnor fibers of $T_{p,q,r}$ singularities, construct an explicit Weinstein handlebody decomposition (encoded by a Legendrian link $Lambda$), and find infinitely many exact Lagrangian tori using fillings of $Lambda$. A filling of a Legendrian link $Lambda$ is an exact Lagrangian surface in the standard symplectic 4-ball whose intersection with the boundary contact 3-sphere is $Lambda$.
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