Monodromy, vanishing cycles, knots and the adjoint quotient

Floer同调与右转向单值

Floer homology and right-veering monodromy

本文证明纽结Floer复形可检测纤维化纽结的单值是否右转向，从而给出紧接触结构的纯Floer理论刻画，并应用于Dehn手术与紧叶状结构。

Contact homology and one parameter families of Legendrian knots

We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S^1,R^3) of Legendrian knots, although it is contractible in the space Emb(S^1,R^3…

Knot Floer homology of fibred knots and Floer homology of surface diffeomorphisms

We prove that the Knot Floer homology group of a fibred knot of genus g in the Alexander grading 1-g is isomorphic to a version of the fixed point Floer homology of an area-preserving representative of the monodromy.

Knot Floer homology and fixed points

If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$ fixed points. This generalizes earlier work of Baldwin--Hu--Sivek and Ni. We also clarify a misleading formula in Cotton-Clay's computation of the symplectic Floer homology of mapping classes of surfaces.

Isotopies of Legendrian 1-knots and Legendrian 2-tori

We construct a Legendrian 2-torus in the 1-jet space of $S^1\times\R$ (or of $\R^2$) from a loop of Legendrian knots in the 1-jet space of $\R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit…

Legendrian tori的非收缩环来自结的族

Non-contractible loops of Legendrian tori from families of knots

本文通过结的弦代数，提出一种拓扑方法计算Legendrian接触同调中的monodromy，并构造了一个无限族非收缩的Legendrian环。