Rack shadows and their invariants (2009)
A rack shadow is a set $X$ with a rack action by a rack $R$, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants...
Enhancements of counting invariants: the column group (2009)
The column group is a subgroup of the symmetric group on the elements of a finite rack (or quandle) which is invariant under rack (or quandle) isomorphism. We use this group to define enhancements of...
Semiquandles and flat virtual knots (2009)
We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined...
We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that $ns^at^a$-quandles are not...
Link invariants from finite Coxeter racks (2008)
We study Coxeter racks over $\mathbb{Z}_n$ and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples...
Link invariants from finite racks (2008)
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...
The 2-generalized knot group determines the knot (2008)
Nelson, Sam, Neumann, Walter D.
Generalized knot groups $G_n(K)$ were introduced independently by Kelly (1991) and Wada (1992). We prove that $G_2(K)$ determines the unoriented knot type and sketch a proof of the same for $G_n(K)$...
Generalized quandle polynomials (2008)
We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We...
Virtual Yang-Baxter cocycle invariants (2007)
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These...
Nelson, Sam, Rische, Jacquelyn L.
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.
Navas, Esteban Adam, Nelson, Sam
We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite...
A polynomial invariant of finite quandles (2007)
We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial...
Symbolic computation with finite biquandles (2006)
A method of computing a basis for the second Yang-Baxter cohomology of a finite biquandle with coefficients in Q and Z_p from a matrix presentation of the finite biquandle is described. We also...
An isomorphism theorem for Alexander biquandles (2006)
We show that two Alexander biquandles M and M' are isomorphic iff there is an isomorphism of Z[s,1/s,t,1/t]-modules h:(1-st)M --> (1-st)M' and a bijection g:O_s(A) --> O_s(A') between the s-orbits of...
Quandles and Linking Number (2006)
We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component...
Matrices and Finite Biquandles (2006)
We describe a way of representing finite biquandles with n elements as 2n x 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting...
Non-classicality and quandle difference invariants (2006)
Non-classical virtual knots may have non-isomorphic upper and lower quandles. We exploit this property to define the quandle difference invariant, which can detect non-classicality by comparing the...
Matrices and finite Alexander quandles (2005)
Murillo, Gabriel, Nelson, Sam, Thompson, Anthony
We describe an algorithm for determining whether a finite quandle is isomorphic to an Alexander quandle by finding all possible Alexander presentations of the quandle. We give an implementation of...
Symbolic computation with finite quandles (2005)
Henderson, Richard, Macedo, Todd, Nelson, Sam
Algorithms are described and Maple implementations are provided for finding all quandles of order $n$, as well as computing all homomorphisms between two finite quandles or from a finitely presented...
On the Orbit Decomposition of Finite Quandles (2005)
We study the structure of finite quandles in terms of subquandles. Every finite quandle $Q$ decomposes in a natural way as a union of disjoint $Q$-complemented subquandles; this decomposition...
Matrices and finite quandles (2005)
Finite quandles with n elements can be represented as n x n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as...
Matrices and Finite Quandles (2004)
Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well...
Alexander quandles of order 16 (2004)
Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of known distinct connected finite Alexander...
Signed ordered knotlike quandle presentations (2004)
We define enhanced presentations of quandles via generators and relations with additional information comprising signed operations and an order structure on the set of generators. Such a presentation...
On Generalized Knot Groups (2004)
Generalized knot groups G_n(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G_1(K) in this series of finitely presented groups. For each natural...
Virtual Crossing Realization (2003)
We study virtual isotopy sequences with classical initial and final diagrams, asking when such a sequence can be changed into a classical isotopy sequence by replacing virtual crossings with...
Classification of Finite Alexander Quandles (2002)
Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander...
The Betti numbers of some finite racks (2001)
Litherland, R. A., Nelson, Sam
We show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by...
Unknotting virtual knots with Gauss diagram forbidden moves (2000)
The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in...