arXiv subjects
Publications and source records attributed to Yoshiro Yaguchi.
We study cabling operations on braids, and characterize the matrices that can be realized as crossing matrices of reducible pure braids. We also use cabling to construct enhanced conjugacy invariants of braids.
We show that the first derivative of the Burau matrix at $t=1$ coincides with the Laplacian of the crossing matrix for pure braids. We also discuss conjugacy invariants derived from these matrices.
We study the crossing matrix of a braid and introduce a polynomial invariant for braid systems that is invariant under Hurwitz equivalence. As an application to the study of surface braids and surface links, we also define an invariant that can be used as an indicator of the necessity of Euler fusion or fission between braid systems.
We compare two crossed homomorphisms on a braid group, one defined diagrammatically and the other defined algebraically. We show that these crossed homomorphisms are essentially the same, and compute them in detail for simple braids, namely elements conjugate to the standard generators of the braid group or to their inverses.
The CN matrix of an $n$-braid projection $B$ is an $n \times n$ matrix such that each $(i,j)$ entry indicates the number of crossings between $i^{th}$ and $j^{th}$ strands of $B$. In this paper, several patterns of an $n \times n$ matrix to be a CN matrix are discussed, and the CN matrix of a pure 6-braid projection is characterized. As an application, the OU matrix of a pure 6-braid diagram and the crossing matrix of a positive pure 6-braid are also characterized.
The OU matrix of a braid diagram is a square matrix that represents the number of over/under crossings of each pair of strands. In this paper, the OU matrix of a pure braid diagram is characterized for up to 5 strands. As an application, the crossing matrix of a positive pure braid is also characterized for up to 5 strands. Moreover, a standard form of the OU matrix is given and characterized for general braids of up to 5 strands.
The set of all virtual or classical braid diagrams forms a monoid and gives a natural monoid action on a direct product of ${\mathbb Z}$ called the up-down action. In this paper, we determine the orbit of every tuple of ${\mathbb Z}$ under the up-down action of virtual or classical braid diagrams. Moreover, we determine the orbit for irreducible braid diagrams. We also consider the isotropy submonoid and give a condition for a braid diagram to admit an up-down coloring to its closure.
In this paper, we define the OU matrix of a braid diagram and discuss how the OU matrix reflects the warping degree or the layeredness of the braid diagram, and show that the determinant of the OU matrix of a layered braid diagram is the product of the determinants of the layers. We also introduce invariants of positive braids which are derived from the OU matrix.
We introduce an infinite family of quantum enhancements of the biquandle counting invariant we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring $R$ using virtual crossings as smoothings, these invariants take the form of multisets of elements of $R$ and can be written in a "polynomial" form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander-Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in previous work as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.
In this paper it is shown that a complete graph with $n$ vertices has an optimal diagram, i.e., a diagram whose crossing number equals the value of Guy's formula, with a free maximal linear tree and without free hamiltonian cycles for any odd integer $n \ge 7$.
An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.
We introduce an up-down coloring of a virtual-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 diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of type II for a pair of diagrams of the trivial virtual-knot. This implies 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.