Maciej Niebrzydowski

THE SECOND QUANDLE HOMOLOGY OF THE TAKASAKI QUANDLE OF AN ODD ABELIAN GROUP IS AN EXTERIOR SQUARE OF THE GROUP

We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology of the Takasaki quandle of G to the exterior square of G. In particular, for G=Z_k^n, k odd, we obtain Z_k^{n(n-1)/2}. Nontrivial second homology allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants.

The quandle of the trefoil knot as the Dehn quandle of the torus

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the sy mplectic space Z Z. The last quandle can be identified with the Dehn quandle of the tor us and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.

ON COLORED QUANDLE LONGITUDES AND ITS APPLICATIONS TO TANGLE EMBEDDINGS AND VIRTUAL KNOTS

Given a long knot diagram D and a finite quandle Q, we consider the set of all quandle colorings of D with a fixed color q of its initial arc. Using this set we define the family

COLORING INVARIANTS OF SPATIAL GRAPHS

\Phi_{Q}^{q}(K)

BIQUANDLE LONGITUDE INVARIANT OF LONG VIRTUAL KNOTS

of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum

Entropic magmas, their homology and related invariants of links and graphs

S_{\Phi}^{x}(K) = \sum_{\phi}\phi(x)

CATEGORIES OF DIAGRAMS WITH IRREVERSIBLE MOVES

, taken over all

Homology of dihedral quandles

\phi \in \Phi_{Q}^{q}

Homology operations on homology of quandles

. Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

On some ternary operations in knot theory

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.