The Circular Braid Group and Its Relationship to the Standard Braid Group

In this paper, we will define the circular braid group on n strings and examine its relationship to the standard braid group on n strings. A braid on n strings has n strings attached to two bars, where one end of each string is attached to each bar. The strings cross as they please. A circular braid on n strings has n strings attached to two circular bars and the strings may also cross around as they please. As all of the braids on n strings form a group, Bn, it will also be shown that the circular braids on n strings form a group, which we will denote n. We will then continue on to define a function from n to Bn+1 and examine it. It will be shown that the function is not an isomorphism between the two groups, but that it is likely that it is an isomorphism between n and a subgroup of Bn+1. In conclusion, further opportunities for research concerning the above function, the presentation of n and suggestions for finding an isomorphism between n and Bn+1 will be made