We start by giving a description of some generalised braids. For the cases of singular and virtual braids we give the algebraic definition and the geometric interpretation. Then we use combinatorial methods to give a proof that the singular braid group is torsion free. Motivated by the formation of labels for the edges of diagrams of virtual knots we give the abstract definiton for biracks and biquandles. Examples of these new algebraic objects are given and some properties are proved as well. We show how biracks provide representations of virtual braids and biquandles provide invariants of virtual knots. Finally, we give some applications of this theory using elements from the Alexander birack as labels for diagrams of virtual knots.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:249106 |
| Date | January 2002 |
| Creators | Jordan-Santana, Maria Mercedes |
| Publisher | University of Sussex |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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