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Link invariants from finite biracks

100%
Nelson S.
Banach Center Publications
|
2014
|
tom 100
|
nr 1
197-212
EN
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, (t,s)-racks, Alexander biquandles and Silver-Williams switches, known as (τ,σ,ρ)-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.
2
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Link invariants from finite racks

100%
Nelson S.
Fundamenta Mathematicae
|
2014
|
tom 225
|
nr 1
243-258
EN
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
3
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On bilinear biquandles

63%
Nelson S., Rische J.
Colloquium Mathematicum
|
2008
|
tom 112
|
nr 2
279-289
EN
We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples.
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