Kanako Oshiro

HOMOLOGY GROUPS OF SYMMETRIC QUANDLES AND COCYCLE INVARIANTS OF LINKS AND SURFACE-LINKS

We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orientable. The invariants can be used in order to estimate the minimal triple point numbers of non-orientable surface-links.

SYMMETRIC QUANDLE COLORINGS FOR SPATIAL GRAPHS AND HANDLEBODY-LINKS

In this paper, colorings by symmetric quandles for spatial graphs and handlebody-links are introduced. We also introduce colorings by LH-quandles for LH-links. LH-links are handlebody-links, some of whose circle components are specified, which are related to Heegaard splittings of link exteriors. We also discuss quandle (co)homology groups and cocycle invariants.

ON LINEAR n-COLORINGS FOR KNOTS

If a knot has the Alexander polynomial not equivalent to 1, then it is linearly n-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e. the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in detail.

HOMOLOGY GROUPS OF TRIVIAL QUANDLES WITH GOOD INVOLUTIONS AND TRIPLE LINKING NUMBERS OF SURFACE-LINKS

The purpose of this paper is to determine the homology groups of trivial quandles with good involutions. We also show that the quandle cocycle invariants of surface-links obtained from trivial quandles with good involutions are equivalent to the triple linking numbers.

Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

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′ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.