Half grid diagrams and Thompson links

We define half grid diagrams and prove every link is half grid presentable by constructing a canonical half grid pair (which gives rise to a grid diagram of some special type) associated with an element in the oriented Thompson group. We show that this half grid construction is equivalent to Jones' construction of oriented Thompson links. Using this equivalence, we relate the (oriented) Thompson index to several classical topological link invariants, and give both the lower and upper bounds of the maximal Thurston-Bennequin number of a knot in terms of the oriented Thompson index. Moreover, we give a one-to-one correspondence between half grid diagrams and elements in symmetric groups and give a new description of link group using two elements in a symmetric group.