Oshiro Kanako

In this paper, we study Dehn colorings for spatial graphs, and give a family of spatial graph invariants that are called vertex-weight invariants. We give some examples of spatial graphs that can be distinguished by a vertex-weight invariant, whereas distinguished by neither their constituent links nor the number of Dehn colorings. (C) 2021 Elsevier B.V. All rights reserved.

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for [Formula: see text]-oriented handlebody-links using [Formula: see text]-cocycles. When a multiple conjugation biquandle [Formula: see text] is obtained from a biquandle [Formula: see text] using [Formula: see text]-parallel operations, we provide a [Formula: see text]-cocycle (or [Formula: see text]-cocycle) of the multiple conjugation biquandle [Formula: see text] from a [Formula: see text]-cocycle (or [Formula: see text]-cocycle) of the biquandle [Formula: see text] equipped with an [Formula: see text]-set [Formula: see text].

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.

The Roseman moves are seven types of local modifications for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of any surface-link, we construct a new diagram of the same surface-link such that any sequence of Roseman moves between them must contain moves involving triple points (and the number of triple points of the two diagrams are the same). Moreover, we find a pair of diagrams of an S-2-knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.

Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rack colorings are invariants of S-2-knots. We also prove that rack colorings for S2-knots can be interpreted in terms of quandles, and discuss a relationship with regular-equivalences of surface-knot diagrams. (C) 2015 Elsevier B.V. All rights reserved.

It is known that any 7-colorable knot in 3-space is presented by a diagram assigned by four of the seven colors. In this paper, we prove the existence of a 7-colorable 2-knot in 4-space such that any non-trivial 7-coloring requires at least six of the seven colors.

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.

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.

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements. (C) 2011 Elsevier B.V. All rights reserved.

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.

We Introduce the notion of a quandle with a good involution and its homology groups Cartel 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 mutable The invariants can be used in order to estimate the minimal triple point numbers of non-orientable surface-links

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example (R) over bar (3) of order 6 that is an extension of the three-element dihedral quandle R(3), various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given. (C) 2009 Elsevier B.V. All rights reserved.

For any positive integer n, we give a 2-component surface-link F = F(1) boolean OR F(2) such that F(1) is orientable, F(2) is non-orientable and the triple point number of F is equal to 2n. To give lower bounds of the triple point numbers, we use symmetric quandle cocycle invariants.