Cabling, contact structures and mapping class monoids

Some remarks on cabling, contact structures, and complex curves

We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a genus constraint. This generalizes the well-known classification of links of plane curve singularities.

Contact structures and reducible surgeries

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston-Bennequin numbers of cables.

Contact monoids and Stein cobordisms

Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},ξ_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, we construct a Stein cobordism from the contact connected sum (M_{S,h},ξ_{S,h}) # (M_{S,g},ξ_{S,g}) to (M_{S,hg},ξ_{S,hg}), for any two boundary-fixing diffeomorphisms h and g. This cobordism accounts for the comultiplication map on Heegaard Floer homology discovered in an earlier paper by the author, and it illuminates several geometrically interesting monoids in the map…

Stein可填充性与映射类群中的正性

Stein-fillability and positivity in the mapping class group

本文构造非正开书并证明：带边界紧致曲面的映射类群中，支撑Stein可填充结构的幺半群等于正幺半群，当且仅当曲面为平面。

平移接触栈的导出勒让德范畴

A Derived Legendrian Category for Shifted Contact Stacks

本文为n-平移接触导出阿廷栈X构造了导出勒让德范畴F_c(X)及勒让德对应的(∞,2)-范畴Leg_n，并应用于模理论。