Blackboard biracks and their counting invariants

A blackboard 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 describe a family of blackboard biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(\tau,\sigma,\rho)$-biracks. We define a counting invariant of unframed classical and virtual knots and links using labelings of link diagrams by finite blackboard biracks, and we give enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.

[1]  Sam Nelson,et al.  (t,s)-racks and their link invariants , 2010, 1011.5455.

[2]  R. Fenn,et al.  Update of "Biquandles of Small Size and some Invariants of Virtual and Welded Knots" , 2010, 1001.5127.

[3]  A. Henrich,et al.  Semiquandles and flat virtual knots , 2009, 0901.4315.

[4]  Sam Nelson,et al.  Link invariants from finite Coxeter racks , 2008, 0808.1584.

[5]  Sam Nelson.  Link invariants from finite racks , 2008, 0808.0029.

[6]  Sam Nelson,et al.  Generalized Quandle Polynomials , 2008, Canadian Mathematical Bulletin.

[7]  Sam Nelson,et al.  On symplectic quandles , 2007, math/0703727.

[8]  Sam Nelson,et al.  An isomorphism theorem for Alexander biquandles , 2006, math/0611887.

[9]  V. Turaev,et al.  Weyl algebras and knots , 2006, math/0610481.

[10]  R. Fenn,et al.  QUATERNION ALGEBRAS AND INVARIANTS OF VIRTUAL KNOTS AND LINKS II: THE HYPERBOLIC CASE , 2006, math/0610483.

[11]  J. Vo,et al.  Matrices and finite biquandles , 2006, math/0601145.

[12]  L. Kauffman,et al.  Biquandles and virtual links , 2004 .

[13]  Nicolas Andruskiewitsch,et al.  From racks to pointed Hopf algebras , 2002, math/0202084.

[14]  L. Kauffman,et al.  Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links , 2001, math/0112280.

[15]  Masahico Saito,et al.  Quandle cohomology and state-sum invariants of knotted curves and surfaces , 1999, math/9903135.

[16]  Roger Fenn,et al.  Trunks and classifying spaces , 1995, Appl. Categorical Struct..

[17]  Roger Fenn,et al.  RACKS AND LINKS IN CODIMENSION TWO , 1992 .

[18]  S. Matveev.  DISTRIBUTIVE GROUPOIDS IN KNOT THEORY , 1984 .

[19]  R. Fenn,et al.  The equation [B,(A-1)(A,B)] = 0 and virtual knots and links , 2004 .

[20]  David Joyce,et al.  A classifying invariant of knots, the knot quandle , 1982 .