数理解析研究所講究録
第 1716 巻 2010 年 37-4237

MULTIPLICITY DISTANCE OF KNOTS
KOUKI TANIYAMAABSTRACT. We define multiplicity and multiplicity distance in acategory. We define neighbourhood category of knots and discuss
its multiplicity distance.

\S 1. Multiplicity in a category
Let $C=(\mathcal{O}, \{Hom(X, Y)\}_{X,Y\in O}, 0)$ be a category. Namely$\mathcal{O}$is a setof objects and $Hom(X, Y)$ is the set of morphisms from an object $X$
to an object$Y$.

$\forall_{\varphi}\in Hom(X, Y)^{\forall}\psi\in Hom(Y, Z),$$\psi\circ\varphi\in Hom(X, Z)$is definedsuch that the following (1) (2) and (3) hold.
$($Notation: $\varphi\in Hom(X,$(1)$\forall_{\varphi}$(2)$\forall x\in \mathcal{O},$: $Xarrow Y,$$\forall\psi$$Y)\Leftrightarrow\varphi$: $Yarrow z,$$\text{ョ_{}id_{X}}:Xarrow X$$\forall\psi:Yarrow X,$ $id_{X}o\psi=\psi$: $Xarrow Y)$$\forall_{\mathcal{T}}$: $Zarrow W,$s.t.

$\forall\varphi$$(\varphi\circ\psi)\circ\tau=\varphi\circ(\psi\circ\tau)$: $Xarrow Y,$$\varphi\circ id_{X}=\varphi$,(3) $Hom(X, Y)\cap Hom(Z, W)\neq\emptyset$$\Rightarrow X=Z$

and $Y=W$ .

Definition 1.

$m: \bigcup_{(X,Y)\in \mathcal{O}\cross \mathcal{O}}Hom(X, Y)arrow \mathbb{R}_{\geq 1}\cup\{\infty\}$

is a multiplicity ones (1)(2)$\forall x\in \mathcal{O},$$\forall_{\varphi}$

$m$

$C$

(id ) $=1$ , and: $Xarrow Y,$$x$$\forall\psi$: $Yarrow Z,$$m(\psi\circ\varphi)\leq m(\varphi)m(\psi)$

Key words and phrases. multiplicity, category, knot.

.

, and38
Here we think$\infty\cdot\infty=$$\forall x\in \mathbb{R}_{\geq 1},$$x\leq\infty,$$x\cdot\infty=\infty\cdot x=\infty,$$\infty\leq\infty$

andoo.

Definition 2.

$X,Y\in \mathcal{O}$$m(X:Y)= \inf\{m(\varphi)|\varphi\in Hom(X,Y)\}$$m(X : Y)$ :multiplicity of $X$ over$Y$

Proposition 3. $m$ : multiplicity on(1)$X\in \mathcal{O}_{f}m(X:X)=1$(2)$X,$ $Y,$ $Z\in$m$with respect to $m$ .
$C$,\mathcal{O}_{f}m(X:Z)\leq m(X:Y)m(Y:Z)$.

has finite multiplicity property $(FMP)$$m(X:Y)<\infty$ .

$\Leftrightarrow^{def}\forall X,$$Y\in \mathcal{O},$Definition 4.
$d_{m}(X, Y)=\log_{e}(m(X:Y)m(Y:X))$$d_{m}(X, Y)$: m-distance of $X$ and$Y$

Proposition 5. $m$ : multiplicity on with $FMP$
is a pseudo metric space $i.e$ .
$C$$(\mathcal{O}, d_{m})$(Dl’) $d_{m}(X, Y)\geq 0,$ $d_{m}(X, X)=0$ ,(D2) $d_{m}(X, Y)=d_{m}(Y, X)$ ,(D3) $d_{m}(X, Z)\leq d_{m}(X, Y)+d_{m}(Y, Z)$ .

Example 6.
$X,$ $Y\in \mathcal{O},$$C$: subcategory of the category of sets and maps (SET)$f:Xarrow Y$$m(f)= \sup\{|f^{-1}(y)||y\in Y\}$$m$

: map multiplicity

Proposition 7. map multiplicity is a multiplicity, $i.e$ .
$m(id_{X})=1$ , and(1)$X\in \mathcal{O},$(2)$f:Xarrow Y,$ $\forall g:Yarrow Z_{f}m(g\circ f)\leq m(f)m(g)$ .

39
Example 8.
$r(M)$ :$f$

$R$

: PID, $M,$ $N$ : R-modules finitely generated overthe minimal number of generators of $M$ over

$R$

$R$.

.

: $Marrow N$ : R-linear map$m_{ker}(f)=e^{r(kerf)}$$m_{coker}(f)=e^{r(cokerf)}$$($coker $f=N/f(M))$

Proposition 9. (1)(2)$m_{coker}$When
$R$

$R$

$m_{ker}$

is a multiplicity.

is a multiplicity.

is a field, $M$ and $N$ are finite dimensional vector spaces over, and$d_{m_{ker}}(M, N)=d_{m_{coker}}(M, N)=|\dim M-\dim N|$.

\S 2. Neighbourhood category of knots$\mathcal{K}$: unoriented knot types in$f=(V_{2}, k_{1})$$\Leftrightarrow^{def}V_{2}\subset S^{3}$$k_{1}\subset$

int$S^{3},$

is a morphism from

$K_{1},$$K_{2}\in \mathcal{K}$$K_{1}$

to.

$K_{2}$: knotted solid torus with knot type
$V_{2}$: knot in$(V, k)=(V^{f}, k’)\Leftrightarrow^{def}$$S^{3}$

with knot type

ョ $h:S^{3}arrow S^{3}$$K_{1}$$K_{2}$,.

ori. preserving homeomorphism s.t.

$h(V)=V’$ and $h(k)=k^{f}$$Hom(K_{1}, K_{2})$: the set of morphisms from

$K_{1}$

to$K_{2}$.

$f:K_{1}arrow K_{2)}g:K_{2}arrow K_{3},$ $f=(V_{2}, k_{1}),$ $g=(V_{3}, k_{2})$40: regular neighbourhood of$V_{2}’$st.

$(V_{2}’, k_{1}’)=(V_{2}, k_{1})$$k_{2}$

with$V_{2}’\subset$

int$V_{3},$$k_{1}’$: knot in$V_{2}’$.

We define $gof=(V_{3}, k_{1}’)$ .
Proposition 10.
$C_{\mathcal{K}}$$id_{K}$$C_{\mathcal{K}}=(\mathcal{K}, \{Hom(K, J)\}_{K,J\in \mathcal{K}}, \circ)$

is a category.

: neighbourhood category of Knots: $Karrow K$ is given by $id_{K}=(V, k)$ wherebourhood of a knot

$k$$V$

is a regular neigh-with knot type $K$ .

\S 3. Multiplicity distance of knots
Definition 11. $f:K_{1}arrow K_{2}$ : morphism, $f=(V_{2}, k_{1})$ ,$h:V_{2}arrow S^{1}\cross D^{2}$: homeomorphism,

$\pi$

:

$S^{1}\cross D^{2}arrow S^{1}$: naturalprojection.
$h$is generic$\Leftrightarrow^{def}\pi oh|_{k_{1}}$:$k_{1}arrow S^{1}$

is a Morse map,

i.e. it has only finitely many critical points
in different levels.

$m(h)= \max\{|(\pi oh|_{k_{1}})^{-1}(y)||y\in S^{1}\}$$m(f)= \min$ $m(h)|h:V_{2}arrow S^{1}\cross D^{2}$ generic homeomorphism}{

Proposition 12. $m$ is a multiplicity on

Proposition 13. (1)$f$$C_{\mathcal{K}}$.

: $K_{1}arrow K_{2_{f}}m(f)=1\Rightarrow K_{1}=K_{2}$ .

(2) $m(K_{1}:K_{2})=1\Leftrightarrow K_{1}=K_{2}$ .

Proposition 14.
$d_{m}$$(\mathcal{K}, d_{m})$is an unbounded metric space.

: multiplicity distance of knots

Definition 15. (Ozawa [1])$k$: knot in $S^{3}-\{(0,0,0,1), (0,0,0, -1)\}$ ,41$\pi$

: $S^{3}-\{(0,0,0,1), (0,0,0, -1)\}\cong S^{2}\cross \mathbb{R}arrow \mathbb{R}$: natural projection,$\pi|_{k}:karrow \mathbb{R}$: Morse function.

trunk $(k)= \max\{|\pi^{-1}(y)||y\in \mathbb{R}\}$

trunk $(K)= \min$ {trunk $(k)|$ knot type of
Proposition 16. (Ozawa [1])Proposition 17. (1) $K,$$\forall_{n}\in \mathbb{N},$$J\in \mathcal{K},$$J$$\Rightarrow m(K:J)=$

is $K$ }ョ $K\in \mathcal{K}s.t$. trunk $(K)\geq n$ .

$J\in \mathcal{K}_{\rangle}$$\frac{trunk(K)}{trunk(J)}\leq m(K:J)\leq$(2) $K,$$k$trunk $(K)$ .

is not a companion of $K$ ,trunk $(K)$ .

Proposition 18. $K,$

$J\in \mathcal{K}_{f}$(1) $m(K:0_{1})\leq$ braid $(K)$ ,(2) $m(K : 0_{1})\leq 2bridge(K)-1$ ,(3) $m(K : J)=2\Rightarrow K$ is a 2, $p)-$ cable of$($Corollary 19.

$K\in \mathcal{K}$$J_{\rangle}$or $K=0_{1}$ .

,$m(K:0_{1}) \geq\frac{trunk(K)}{trunk(0_{1})}=\frac{trunk(K)}{2}$.

$K\neq 0_{1\prime}$$d_{m}(K, 0_{1}) \geq\log_{e}\frac{trunk(K)}{2}\cdot 2=\log_{e}trunk(K)$$\{d_{m}(K, 0_{1})|K\in \mathcal{K}\}$

Definition 20.
$m(K)$ :is unbounded.

$K\in \mathcal{K},$$m(K)=m(K:0_{1})$ .

multiplicity of $K,$ $m(K)\in$ N.

Remark 21. $m(K)\leq n\Leftrightarrow d_{m}(K, 0_{1})\leq\log_{e}2n$ .
Proposition 22.

$K\in \mathcal{K}_{f}$(1) $m(K)=1\Leftrightarrow K=0_{1}$ ,(2) $m(K)=2\Leftrightarrow K$ is a 2, $p)$ -torus knot;$($.

42(3) $m(K)=3\Leftrightarrow K\neq 0_{1},$ $K$ is not a 2, $p)$ -torus knot, and$($$(a)K$ is a closed 3-braid, $or$$(b)K$ is a connected sumof some 2-bridge knots.

Proposition 23. K:Montesinos $knot\Rightarrow m(K)\leq 4$ .

The detail will appear in [2].

REFERENCES[1] M. Ozawa, Waist and trunk of knots, to appear in Geometriae Dedicata.
[2] K. Taniyama, Multiplicity of a space over another space, in preparation.

DEPARTMENT OF MATHEMATICS, SCHOOL OF EDUCATION, WASEDA UNIVERSITY, NISHI-WASEDA 1-6-1, SHINJUKU-KU, TOKYO, 169-8050, JAPANE-mail address: $taniyama\Phi waseda$ . jp