Narrow Results

We introduce (co)homology theory for multiple group racks and construct cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles, where a multiple group rack is a rack consisting of a disjoint union of groups. We provide a method to find new rack cocycles and multiple group rack cocycles from existing rack cocycles and quandle cocycles. Evaluating cocycle invariants, we determine several symmetry types, caused by chirality and invertibility, of compact oriented surfaces in the 3-sphere.

Given a tangle diagram of a classical knot K, we construct a surface diagram of any twist-spun 2-knot of K. From the crossings of the tangle diagram, we can get information of the corresponding triple points of the surface diagram, which are used to compute cocycle invariants of twist-spun 2-knots.

We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle R_p and all of whose cocycle invariants derived from ℤ_p-valued 3-cocycles on R_p take value in ℤ ⊂ ℤ[ℤ_p] for any odd prime integer p.

Fox's shadow p-colorings of a knot K define two kinds of Laurent polynomials, Φ_p(K) and , as invariants of K. We prove that the equality holds for any knot K. Also we prove that, if the p-coloring number of K is equal to p², then has the form for some N ∈ ℤ_p.

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two $2$-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram $D$, there exists a diagram $D^{\prime }$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

We characterize the para-associative ternary quasi-groups (flocks) applicable to knot theory, and show which of these structures are isomorphic. We enumerate them up to order 64. We note that the operation used in knot-theoretic flocks has its non-associative version in extra loops. We use a group action on the set of flock colorings to improve the cocycle invariant associated with the knot-theoretic flock (co)homology.