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

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.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Erratum : a presentation for Hilden's subgroup of the Braid group

After publication of [1] Allen Hatcher found a gap in the proof that the complex Xn is simply connected. This complex is defined in terms of isotopy classes of discs, but the argument uses representatives of the isotopy classes. There was an implicit assumption that for an edge path in the complex there exists sufficiently nice representatives of each isotopy class. In definition 2 of this paper the properties of these representatives will be made explicit. It is clear that such representatives exist for a path, the problem is that for a loop it is not obvious that the representative at the beginning and end can be chosen to coincide. This problem is addressed in Lemma 3. Section 2 of this paper contains the complete proof that Xn is simply connected, incorporating all of the necessary changes

A presentation for Hilden's subgroup of the braid group

Consider the unit ball, B = Dx[0,1], containing n unknotted arcs a(1),..., a(n) such that the boundary of each a(i) lies in D x {0}. We give a finite presentation for the mapping class group of B fixing the arcs {a(1),..., a(n)} setwise and fixing D x {1} pointwise. This presentation is calculated using the action of this group on a simply-connected complex

A presentation for the pure Hilden group

Consider the half ball, B3 +, containing n unknotted and unlinked arcs a1, a2, . . . , an such that the boundary of each ai lies in the plane. The Hilden (or Wicket) group is the mapping class group of B3 + fixing the arcs a1 an setwise and fixing the half sphere S2 + pointwise. This group can be considered as a subgroup of the braid group on 2n strands. The pure Hilden group is defined to be the intersection of the Hilden group and the pure braid group. In a previous paper, we computed a presentation for the Hilden group using an action of the group on a cellular complex. This paper uses the same action and complex to calculate a finite presentation for the pure Hilden group. The framed braid group acts on the pure Hilden group by conjugation and this action is used to reduce the number of cases

Normal forms of random braids

Analysing statistical properties of the normal forms of random braids, we observe that, except for an initial and a final region whose lengths are uniformly bounded (that is, the bound is independent of the length of the braid), the distributions of the factors of the normal form of sufficiently long random braids depend neither on the position in the normal form nor on the lengths of the random braids. Moreover, when multiplying a braid on the right, the expected number of factors in its normal form that are modified, called the expected penetration distance, is uniformly bounded. We explain these observations by analysing the growth rates of two regular languages associated to normal forms of elements of Garside groups, respectively to the modification of a normal form by right multiplication. A universal bound on the expected penetration distance in a Garside group yields in particular an algorithm for computing normal forms that has linear expected running time