数理解析研究所講究録 1222 巻 2001 年 138-144

138
A bracket representation of the monoid of links
片海 直樹 (Naoki Kataumi)
稲田 勇 (Isamu Inata)
小林 ゆう治 (Yuji Kobayashi)
Department of Information Science, Toho University

東邦大学理学部

1Introduction

Aknot is an oriented simple closed curve embedded in the -dimensional Euclidian space . A
We say that two h.nk are \eta uiv&nt if one can be
mutually disjoint union of knots is called aare equivalent, then weandtransformed to the other by some homeomorphism in . If two links. By , we denote the set of equivalence classes of Inks. We usually identify alink withwrite
an equivalence class of Inks, but it is simply called aits equivalence class. So an element of islink. To handle Inks, we usually project them on the 2- imensional plane (see Fig.l). Alink projectedof it. The link diagram of aInk is denoted byplane is called alinkon the$D(L)$ . Remark that there are infinitely many link diagrams of alink$\mathrm{E}^{3}$$b_{l}.A$$\mathrm{B}^{3}$$L_{1}\sim L_{2}$$L_{1}$$L_{2}$$\mathcal{L}$$\mathcal{L}$$\mathrm{a}\theta \mathrm{u}\mathrm{a}\mathbb{I}\mathrm{y}$$L$$d_{\dot{1}ag}1nm$$\mathrm{a}\mathrm{d}\mathrm{i}.\mathrm{e}\mathrm{n}\cdot \mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$ffivial knot

Fig.l

The following defonnations of apart of aInk diagram are called the(Type 1)$\sim$$\sim$(Type 2)$\sim$$\sim$(Type 3)

Wet moves.

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In the Ink (or knot) theory, the following result is fundamental.
Result 1.1 Two links are equivalent $f$ and only there is a finite sequence of the $R\dot{\alpha}demeskr$ moves
such that a link diagram of one of them can be deformed to a link diagram of the other one.
$|.f$

In this paper, we assume that for each link one fixed component is chosen and afixd point on thiscomponent is given. For each link , afixed point on achosen component of is called amarked point of. Aproduct of two links and , denoted by Li#L2, is defined as follows. Let and a2 be marked
points ofand , respectively. If the orientations of $D(L_{1})$ atand $D(L_{2})$ at a2 are opposite, then,remove asmall arc from $D(L_{1})$ containing and asmall arc from $D(L_{2})$ containing . Let and $d$(resp.
and ) be two end points of $D(L_{1})$ (resp. $D(L_{2})$ ) removed the small arc. Then attach $D(L_{1})$and $D(L_{2})$ at these points identifying with 4andwith .
$L$

$L$

$L_{1}$$L_{1}$$L$$L_{2}$$a_{1}$$L_{2}$$a_{1}$$a_{2}$$a_{1}$$\beta$$\alpha$$\beta’$$\alpha$$\alpha’$$\beta’$Fig.2

If the orientations of $D(L_{1})$ atand $D(L_{2})$ at a2 agree, apply the deformation (Type 1) to $D(L_{1})$ator $D(L_{2})$ at a2, then the orientations of $D(L_{1})$ at and $D(L_{2})$ at a2 become opposite. And thenattach $D(L_{1})$ and $D(L_{2})$ as in the above way (see Fig.3).
$a_{1}$$a_{1}$$a_{1}$Fig.3

Remark that the above product does not depend on the choice of the marked points, but depends on-defind, thatthe choice of components and orientations. It is easy to see that the above product is, then,is, for any Inks ,. Further it satisfies
andand , if
the associative law and the commutative law and the trivial knot acts as the identity element in . Thus
forms acommutative monoid.
$\mathrm{w}\mathrm{e}\mathrm{U}$$L_{1}$$L_{2}$$L_{S}$$L_{4}$$L_{1}\sim L_{2}$$L_{3}\sim L_{4}$$L_{1}\# L_{3}\sim L_{2}\# L_{4}$$\mathcal{L}$$C$

In this PaPer, we represent links as words over the four letters $(, )$ , [and 1, and consider relations
between them. This method is given by Manturov in [2] and we reformulate his result. For more
information about links (or knots), we refer to [1]1402Regular bibradcets and chord diagrams
Let be an alphabet consisting of the four letters $(, )$ , $[, ]$ . Aword over is called abibrackets A) if the number of opening and closing parentheses (resp.
(resp.
bibracket is calledsquare brackets) are equal in , and in every proper prefix of , the number of opening parentheses (resp.
square brackets) is larger than or equal to the number of closing parentheses (resp. square brackets).
. Aproduct of regular bibrackets isregular andAbibracket is caUd regular if it is bothdefined by aconcatenation of words over C.
$A$$\Sigma$$A$$(,)-[] \mathrm{w}ular$$\Sigma$$[,]-\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{h}\mathrm{r}$

$A$

$A$

$(,)$

$[,]-\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$To construct aregular bibradcet from alink, we need the following notion. Achord diagram is drawn
on the plane as follows. First draw circle and give afixed point on the circumference, then draw straight(resp. curves) inside (resp. outside) the cirde connecting two points not equal to the fixed point on
the circumference in such away that each pair of straight lnes and curves do not intersect (see Fig.4).
For achord diagram , straight Ones inside (resp. curves outside) the circle are called inner (resp. outer)chords of $C$.
$\underline{[j\mathrm{n}}\mathrm{a}\mathrm{e}$

$C$

Fig.4
the circle ofWith each chord diagram $C$, we associate aregular bibracket as bUows. Westarting&0m the fixed point in the counterdodcwise direction. If we encounter the first end pointof an inner (resp. outer) chord, then we rite an opening parenthesis (resp. square bracket) and, if weencounter the second end of it, we write adoeing parenthesis (resp. square bracket). We proceed in thismy and when we reach the start point, we speU out abibracket (see Fig.5) and it is easy to verify that
it is actually aregular bibracket. Conversely, for agiven regular bibracket , we can easily construct achord diagram $C$ with which associates.
$A$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$

$C$

$A$

$A$

$A$

–

$[$$(((][)))]$

Fig.5

Next we construct achord diagram&0m agiven Ink . We may suppose
plane. Split every crcming in by the rule givenv in Fig.6.
$L$

$L$

Fig.4

$L$is alink diagram on the141We assume that arcs a and c are parts of acircle of achord diagram and b is an (inner or outer) chord.
If we get only one circle when we split all the crossings, then we can get achord diagram (see Fig .

Fig.7

But the above case is exceptional, that is, when we split all the crossings, many
general (see Fig.8).

dae may occur in$\mathrm{c}\dot{\mathrm{u}}$$arrow$

Fig.8
In this case, applying the deformation of TyPe 2, we can obtain achord diagram as seen in Fig.9.

Fig.9
Thus, we obtain achord diagram and aregular bibracket associated with it&0m alink diagram.
Conversely, applying the above process in areverse way, we can construct aInk diagram from agiven

regular bibradcet. Remark that the Ink diagram constructed from aregular bibracket is unique, butthere are many different regular bibradcets, which are constructed from aInk diagram. For each regularbibracket $A$ , the Ink uniquely constructed from $A$ is denoted by $L(A)$ and is called the Ink represente

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3Representation of the monoid of links) can be represented by some(or linkAs we have seen in the previous section, every
regular bibradcet. In this section, we shall find a relationship between regular bibradcets which represent
equivalent Inks. To this end, due to Result 1.1, it is sufficient to find relations between regular bibradcets
which are transformed by the Reidemeister moves to each other. But since the Reidemeister moves are
local deformations of links, we must consider the global situations where the Reidemeister moves apply.
$\underline{[j\mathrm{n}\mathrm{k}}$$\mathrm{d}\mathrm{i}\mathrm{a}\Psi \mathrm{m}$The global Reidemeister moves can be obtained by closing end points of the Reidemeister moves withsome parts of Inks as shown in Fig.lO. These parts are denoted by $A$ , $B$ and $C$ .

$\sim$

Fig.lOAs described before, to obtain aregular bibracket from alink diagram, we may have to aPply thedeformation Type 2, but the way of the application is not unique. Accordingly we must consider thefollowing three situations.

Fig.llIn Fig.ll, $A$, $B$ and $C$ are parts of Inks, but without loss of generality, we may consider that , $B$ and$C$ are bibrackets though they may not be regular. Then, from the leftmost equivalence of the Inks inFig.11, we obtain the relation
$A$

$A$

through the process described in Fig$[$$(]B)=A(B[)]$.

$arrow \mathrm{A}$

Fig.lO$[$($]\mathrm{B}$$)=\mathrm{A}(\mathrm{B} [ )]$143
Similarly we obtain other relations between regular bibrackets from the other equivalences in Fig.ll.
Next, we see the relations corresponding to the other types of the global Reidemeister moves. Forexample, in Fig.13, the equivalence at the top is the global Reidemeister moves of Type 3. At the second
and third levels, the Inks are transformed to the chord diagrams. Finally, at the bottom, we have arelation between the regular bibrackets which associate with these Inks. In the same my, we can obtainall the relations from the other global Reidemeister moves, but details are omitted.

$\sim$$\sim$$$\mathrm{f}$$\sim$A($\mathrm{B}$A((C) [)[)1]$(\mathrm{B} (\mathrm{C} [ )[[ )]]$

$1$

Fig.13

To describe the defining relations of the regular bibracket monoid, we need the following notions. For
is aregular bibracket, which is obtained from by changing all opening
any regular bibradcet $A$,
parentheses to opening square bradcets, closing parentheses to closing square bradcets and vice versa. It.
is easy to verify that links determined by and Iare equivalent. Thus we have arelation$A$$\overline{A}$$A=\mathrm{Z}$$A$

Next, for any regular bibradcet of the form AB, where both $A$ and $B$ are bibradcets, the regularbibradcet $B’A’$ is obtained in the following my. If $BA$ is aregular bibracket, then $B’A’$ is equal to $BA$ .
If $BA$ is not aregular bibradcet, then there exists an opening parenthesis or an opening square bradcet

insuch that acorresponding closing parenthesis or square bradcet is in $B$ . If there is an opening
parenthesis (resp. square bradcet) in such that the corresponding closing parenthesis (resp. squareto the closingbracket) is in $B$ , then we change this opening parenthesis (resp. square bracket) inparenthesis (resp. square bracket) and the closing parenthesis (resp. square bracket) in $B$ to the opening
$A$

$A$

$A$

au

such pairs of parentheses andparenthesis (resp. square bradcet). Applying the above replacement tosquare brackets in $BA$ , $BA$ is transformed to aregular bibradcet, which is denoted by $B’A’$ . Since theproduct of Inks does not depend on the choice of amarked point, the Inks represented by AB and $B’A’$

are equivalent. Thus we include the relation $AB=B’A’$ .

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Now we have our main theorem.
b.bm&g. Then truo links $L(A)$ and $L(B)$ are equivalent if
Theorem 3.1 Let and $B$ be two
and only if one $ofA$ and can $k$ obtained from the other one by applying the following relations finitely
many times.
, other A is regular,
1. A
$A$

$|\mathrm{W}\mathrm{u}1\mathrm{a}r$$B$$=\mathrm{Z}$. $AB=B’A’$, where ,
$A$

$t$

$B$

are bibrackets and AB is regular,( $]B)=A(B [ )]$ , where $A$ , $B$ are $(,)-[] wular$ and AB is regular,S.

$A$$[$4.

$A$$((B) C [)$ $]=A(B(C) [ )]$ , where

5.

$A$

$(BC [)]=A(C’B’ [ )]$ , where , $BC$ are $(,)-[]\eta ular_{1}B$ , $AC$ are $[,]-[]\eta ula$’and $ABC$ is$A$

,

$B$

,$C$

are$(, )-[] \mathrm{w}\mathrm{u}\mathrm{l}\mathrm{a}r$

and $ABC$ is $[] wular$,$A$,$[] w[] br$

5. $()=1$ where 1 is the empty word,,are

7.

$A$

$(B[[ )]$ $(])=AB$ , where

8.

$A$

$( B ((C) [)[)$ $]]=A(B(C [)[[ )]$ $]]$ , where ,$A$

$B$$(,)-[] \mathrm{w}\mathrm{u}lar$ated AB is regular,$A$$B$, $Ca|$ $(,)$ -ngular and f( is,$|\mathrm{W}^{\mathrm{u}lar}$

9. A $([ (B] C))=A$ $[ ([ B]]C)$ , where $A_{1}BC$ are,,$B$$B$,$(,)-[] \mathrm{w}ular$, $AC$ are$[,]-[] \mathrm{w}ular$

and $ABC$$\dot{\mathrm{u}}|\eta ular$

1 . $A$ $(B$ ( $C$ ( $[[$ ) [)]][regular,$\theta$

11.
and$A$$)$]] $=A$$[$(($]$$B)C)$ , where$A$,$Car\epsilon$

and $ABC$ is$(,)-[] \mathrm{w}ular$( $[ (B]$ $C[ [ )]]=A[([$ $B((]$ )) $]C$ ), $whm$ $A$, $BCa|\mathrm{e}(,)- m2kr$, $B$ , $AC$ are $[,]$ regular,$ABC\dot{u}[] \mathfrak{B}ular$

12. A$[,]-|\mathrm{w}ular$$($[ ( $B$ ] $C[[$ )]] $)=A$ $[ ([ B]((]))C)$ , where $A$, $BC$ are,$(, )-[]\eta \mathrm{u}lar$$B$, $AC$ areand $ABC$ is $[]\eta uhr$.

where 3, 4and 5come from the equivalences in Fig.ll, 6is from the Reidemeister moves of Type 1, 7is&0m Type 2and 8-12 are&0m Type 3.

References[1] C. C. Adams, The Knot Book -an elementary introduction to the mathe matical theory of knots -,W.H.FREEMAN AND COuMP 憶$\backslash \mathrm{N}\mathrm{Y}$, New Yorb 1994.

[2] V. O. Manturov, The Bracket Semigroup of Knots, Mathematical Notes 67, N0.4 (2000) 468478.