Plat closure of braids

Given a braid b ∈ B2n we can produce a link by joining consecutive pairs of strings at the top, forming caps, and at the bottom, forming cups. This link is called the plat closure of b. The set of all braids that fix the caps form a subgroup H2n and the plat closure of a braid is unchanged after multiplying on the left or on the right by elements of H2n. So plat closure gives a map from the double cosets H2n\B2n/H2n to the set of isotopy classes of non-empty links. As well moving within a double coset there is a stabilisation move which leaves the plat closure unchanged but increases the braid index by two and multiplies on the right by σ2n. Birman [2] has shown that any two braid with isotopic plat closures can be related by a sequence of double coset and stabilisation moves. In Chapter 1 we show that if we change the way we draw the cups then we can use twisted cabling as the stabilisation move. Moreover, we show that any two braids with equal plat closure can be stabilised until they lie in the same double coset. If we restrict to even braids then we can give the plat closure a well defined orientation. In this case we show that untwisted cabling can be used as the stabilisation move. Assuming an oriented version of Birman’s result we construct a groupoid G and two subgroupoids H+ and H− which satisfy the following. All the even braid groups embed in G. There is a plat closure map on G which takes the same value on the embedded even braid group. This plat closure is constant on the double cosets H+\G/H− and induces a bijection between double cosets and isotopy classes of non-empty oriented links. In Chapter 2 we compute a presentation for H2n. To do this we construct a 2-complex Xn on which H2n acts. Then we show that this complex is simply connected, the action is transitive on the vertex set and the the number of edge and face orbits is finite. We get generators from each edge orbit and relations from the edge and face orbits. In the final chapter we compute a presentation for the intersection of H2n and the pure braid group