Atsushi Ishii

Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies

We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.

A TABLE OF GENUS TWO HANDLEBODY-KNOTS UP TO SIX CROSSINGS

A handlebody-knot is a handlebody embedded in the 3-sphere. We enumerate all genus two handlebody-knots up to six crossings.

Infinitely many two-variable generalisations of the Alexander-Conway polynomial

We show that the Alexander-Conway polynomialis obtain- able via a particular one-variable reduction of each two-variable Links- Gould invariant LG m,1 , where m is a positive integer. Thus there exist infinitely many two-variable generalisations of �. This result is not obvi- ous since in the reduction, the representation of the braid group generator used to define LG m,1 does not satisfy a second-order characteristic identity unless m = 1. To demonstrate that the one-variable reduction of LG m,1 satisfies the defining skein relation of �, we evaluate the kernel of a quan- tum trace. AMS Classification 57M25, 57M27; 17B37, 17B81

The Markov theorems for spatial graphs and handlebody-knots with Y-orientations

We establish the Markov theorems for spatial graphs and handlebody-knots. We introduce an IH-labeled spatial trivalent graph and develop a theory on it, since both a spatial graph and a handlebody-knot can be realized as the IH-equivalence classes of IH-labeled spatial trivalent graphs. We show that any two orientations of a graph without sources and sinks are related by finite sequence of local orientation changes preserving the condition that the graph has no sources and no sinks. This leads us to define two kinds of orientations for IH-labeled spatial trivalent graphs, which fit a closed braid, and is used for the proof of the Markov theorem. We give an enhanced Alexander theorem for orientated tangles, which is also used for the proof.

Algebraic links and skein relations of the Links-Gould invariant

We give certain skein relations of the Links-Gould invariant. We also show that the relations lead us to recursive calculation of the invariant for algebraic links. As an application we give a formula for the Links-Gould invariant of 2-bridge links.

THE POLE DIAGRAM AND THE MIYAZAWA POLYNOMIAL

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram. Furthermore, we construct infinitely many virtual link diagrams which attain the minimal real and virtual crossing numbers together.

THE LINKS–GOULD INVARIANT OF CLOSED 3-BRAIDS

In this paper, we study the Links–Gould invariant of closed 3-braids. We consider the algebra generated by the image of the 3-string braid group by the linear representation which yields the Links–Gould invariant. We find fundamental linear relations among natural generators of the algebra, and we obtain a basis of the algebra. The relations allow us to evaluate the invariant of closed 3-braids recursively. As an application we give a computer program to calculate the invariant for closed 3-braids.

Handlebody-knot invariants derived from unimodular Hopf algebras

A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category

Homology for quandles with partial group operations

\mathcal{T}

THE VIRTUAL MAGNETIC KAUFFMAN BRACKET SKEIN MODULE AND SKEIN RELATIONS FOR THE f-POLYNOMIAL

of handlebody-tangles and present it by generators and relations. The result tells us that every functor on