アブストラクト:
Viro氏により導入されたsurface braidの理論は,
鎌田氏により基本的な部分が整備されてきた.
surface braid (の同値類)は, 以下の二種類の対象と
一対一対応が付けられており, それらを用いて様々な研究がなされている.
一つ目はbraid system (のslide同値類)と呼ばれる対象で,
１次元braid群の元をいくつか並べたものである.
二つ目はchart (のC-moveによる同値類)と呼ばれる対象で,
２次元円板上のグラフである.
 
本講演では, braid systemの標準形を定義し,
任意のbraid systemはslide同値の下で標準形に変形できる事を示す.
これまでの研究では, braid systemかchartのどちらか一方を使って
研究される事が多かったが, braid systemの言葉で書かれた標準形を
chartに書き直す事により, 次の応用が得られた:
任意のsurface braidはcrossing changeと呼ばれる操作で
unknotなsurface braidに変形できる.
(このcrossing changeに関する主張は, 広島大学の岩切氏によって
別証明が与えられている. )
 

Abstract:
The notion of a surface braid was defined by Viro
and extensively studied by Kamada.
There exists a one-to-one correspondence between
the set of (equivalence classes of) surface braids and
each of the following two sets,
and many results are obtained by using the correspondences.
One is the set of (slide equivalence classes of) braid systems,
where a braid system is a sequence of elements of
the one-dimensional braid group.
The other is the set of (C-move equivalence classes of) charts,
where a chart is a graph in a two-dimensional disk.

In this talk, we define a canonical form of braid systems,
and prove that any braid system can be deformed into
a canonical form up to slide equivalence.
Though either of braid systems or charts were used
in many of previous studies, we obtain the following as an application
by interpreting the canonical form of braid systems in terms of charts:
Any surface braid can be deformed into an unknotted one
by doing some operations, called crossing changes.
(Iwakiri has a different proof of the above application.)