coefficient of $z^{2n} - Project Euclid

TOKYO J. MATH. VOL. 13, No. 1, 1990

On the Triviality Index of Knots Yoshiyuki OHYAMA and Yasuko OGUSHI Waseda University S. Suzuki)

(Communicated by

\S 1.

Introduction.

In [6] the first author derived a new numerical invariant, denoted by $0(K)$ , of knots from their diagrams and showed that if the Conway polynomial of a knot $K$ is not one, then $0(K)$ is finite ([6] Corollary 2.4). In this paper, we call $0(K)$ the triviality index of $K$. It arises a problem as to whether or not there exists a knot $K$ such that $0(K)=n$ for any natural number . In this paper, we show the following theorems. $n$

THEOREM A.

coefficient of

$z^{2n}$

If a knot

$K$

of the Conway

has a

$2n$

polynomial

THEOREM B.

-trivial diagram of $K$ is even.

For any natural number infinitely many knots $K’ s$ with $0(K)=n$ .

$n$

with

$n>1$ ,

$(n>1)$ ,

the

there exist

Moreover in the case $0(K)=3$ we show the following. THEOREM C. Let $f(z)=1+\sum_{i=2}^{l}a_{2i}z^{2\ell}$ , where are integers. is odd, there is a knot $K$ such that $0(K)=3$ and the Conway If polynomial of $K$ is $f(z)$ . $a_{2t}(2\leqq i\leqq l)$

$a_{4}$

Throughout this paper, we work in PL-category and refer to Burde and Zieschang [1] and Rolfsen [8] for the standard definitions and results

of knots and links.

\S 2. Definitions and facts. The Conway polynomial ([2]) and the Jones polynomial ([3]) are invariants of the isotopy type of an oriented knot or link in a 3-sphere . The Conway polynomial is defined by the following formulas: $\nabla_{L}(z)$

$S^{8}$

Received August 23, 1989

$V_{L}(t)$

YOSHIYUKI OHYAMA AND YASUKO OGUSHI

180

$\nabla_{U}(z)=1$

for the trivial knot

$U$

,

.

$\nabla_{L+}-\nabla_{L-}=z\nabla_{L_{0}}$

And the Jones polynomial is defined by the followings: $V_{\sigma}(t)=1$

for the trivial knot

$U$

,

$t^{-1}V_{L-}(t)-tV_{L+}(t)=(t^{1/2}-t^{-1/2})V_{L_{0}}(t)$

and where as in Fig. 2-1. $L_{+},$

$L_{-}$

,

are identical except near one point where they are

$L_{0}$

$L-$

$L+$

$Lo$

FIGURE 2-1

We defined the following number in [6].

a diagram of $L$ with the set of NOTATION. Let $L$ be a link, and of . For a subset crossing points $L$ by changing the the diagram obtained from , we denote by crossing at all points of $D$ . $\tilde{L}$

$D(\tilde{L})=\{c_{1}, c_{2}, \cdots, c_{n}\}$ $\tilde{L}_{D}$

$D(\tilde{L})$

a diagram of $K$ with the 8et be a knot and be nonempty subsets of . Let . For any nonempty subfamily for , we denote the set of say is an n-trivial diagram of $K$ with that for convenience. We

DEFINITION. Let of crossing points with

$\tilde{K}$

$K$

$D(\tilde{K})$

$D(\tilde{K})$

$ A_{i}\cap A_{j}=\emptyset$

$\{A_{j}, A_{j_{2}}, \cdots, A_{j_{l}}\}$ $1$

by

$ D=\{c_{k_{1}}, c_{k_{2}}, \cdots, c_{k_{n}}\}\sim$

$A_{1},$

$A_{2},$

$\cdots,$

$A_{n}$

$\mathscr{A}=$

$i\neq j$

$A_{\dot{g}_{1}}\cup A_{\dot{J}2}\cup\cdots\cup A_{i_{l}}$

$\{A_{1}, A_{2}, \cdots, A_{n}\}$

$\tilde{K}$

$\ovalbox{\tt\small REJECT}$

respect to subfamily

If a diagrams,

if for any nonempty (not necessarily proper) is a diagram of the trivial knot. of knot $K$ has an n-trivial diagram and has no $(n+1)$ -trivial by $0(K)$ , and call it the triviality we denote the number $K$ . If a knot $K$ has an n-trivial diagram for any natural $\{A_{1}, A_{2}, \cdots, A_{n}\}$

$\mathscr{A}$

$\{A_{1}, A_{2}, \cdots, A_{n}\},\tilde{K}_{\vee}$

$n$

index of number , we define $n$

$ 0(K)=\infty$

.

In our notation, Lemma 2 of Yamamoto [9] is stated as follows. PROPOSITION.

For any knot

$K,$

$O(K)\geqq 2$

.

In [6], we showed the following theorem and corollary.

THEOREM 2.

If a knot

$K$

has an n-trivial diagram, then the Conway

181

TRIVIALITY INDEX OF KNOTS

polynomial

$\nabla_{K}(z)$

if

(1)

$n$

of

$K$

is of the following form;

is odd, then ,

$\nabla_{K}(z)=1+a_{n+1}z^{n+1}+a_{n+3}z^{n+3}+\cdots$

and

if

(2)

$n$

is even, then

.

$\nabla_{K}(z)=1+a_{n}z^{n}+a_{n+2}z^{n+2}+\cdots$

COROLLARY. is

finite.

If the

Conway polynomial

of

$K$

is not one, then

$0(K)$

Theorem 2 gives an upper bound of $0(K)$ for a knot $K$ , but it makes no difference between the knot $K$ with $0(K)=2m-1$ and the knot with $0(K’)=2m$ . It arises a problem as to whether or not there exists with $n>1$ . a knot $K$ with $0(K)=n$ for any natural number At first we show Theorem A to distinguish between the knot $K$ with $0(K)=2m-1$ and the knot $K$ with $0(K)=2m$ . $K^{\prime}$

$n$

\S 3. Proof of Theorem A.

a We define the following model. Let $K$ be a knot, $K$ $K$ , and has no associated to , i.e. the projection of $C=\{c_{1}, c_{2}, \cdots, c_{2n}\}$ crossings. be And let information of over and under a subset of the set of crossing points . Since is a knot projection, in there is an immersion $f$ of such that $f(S^{1})=\hat{K}$ . By , we de. Let and of note also a point of associated to . We have the model as shown in Fig. 3-1. Step 1. diagram of

$\tilde{K}$

$\hat{K}$

$\tilde{K}$

$\hat{K}$

$D(\tilde{K})$

$S^{1}$

$R^{2}$

$c_{i}$

$\hat{K}$

$\tilde{K}$

$f^{-1}(c_{l})=\{d_{i}, d_{i}^{\prime}\}$

$c_{i}$

$S^{1}=\sigma=\partial D^{2}$

$\hat{K}$

$\sigma$

$d_{1}^{\prime}$

FIGURE 3-1

Let

$\delta_{i},$

$(1\leqq i\leqq 2n)$

3-2.

.

$\delta_{i}^{\prime}$

be regular neighborhoods of Let be a band and $B_{t}$

$d_{i},$

$d_{i}^{\prime}$

in

$\sigma$

and mutually di8joint as shown in Fig.

$\partial B_{i}=\alpha_{i}\cup\alpha_{i}^{\prime}\cup\beta_{i}\cup\beta_{i}^{\prime}$

182

YOSHIYUKI OHYAMA AND YASUKO OGUSHI $\alpha_{1}$

$\mathfrak{g}_{i}$

$6_{i}^{\prime}$

$B1$

FIGURE 3-2

We make full twisted and attach then we have an orientable surface $B_{i}$

$\beta_{l}$

and

$S=(\bigcup_{i=1}^{2n}B_{i})\cup D^{2}$

to

, in and as shown in Fig. 3-3.

$\beta_{l}^{\prime}$

$D^{2}$

$\delta_{i}$

$\delta_{l}$

$\epsilon_{1}$

$\delta_{2}$

$6_{1}^{\prime}$

FIGURE 3-8

. Then $L$ is a link or a knot. We call $L$ a band And let with respect to . Let be a diagram of $L$ and model of one of two crossing points of the boundary of the full-twisted band in . $C$ For any 8ubset $C’=\{x_{1}, x_{2}, \cdots, x_{q}\}$ of , we denote the link diagram and smoothing at the points of by also link type obtained from components the number of of the link $L$ or and denote by . Then we have Proposition 3.1. $\partial S=L$

$\tilde{K}$

$C$

$\tilde{L}$

$a_{i}$

$\tilde{L}$

$B_{l}$

$\tilde{K}$

$C^{\prime}$

$\tilde{K}(C^{\prime})$

$\tilde{K}(x_{1}, x_{2}, \cdots, x_{q})$

$\mu L$

PROPOSITION 3.1. Let $M=\{1,2, \cdots, 2n\}$ and $N$ be a subset For a knot $K$ and the band model $L$ of with respect to we have $\tilde{K}$

$\mu\tilde{L}(\{a_{l}|i\in M-N\})=\mu\tilde{K}(\{c_{i}|i\in N\})$

of

$M$

$C=\{c_{1}, c_{2}, \cdots, c_{2n}\}$

. ,

.

For a set $X$ , we denote the number of elements of $X$ by $\# X$ , Let be a knot diagram with the set of crossing points . We show the following lemma. and $C=\{c_{1}, c_{2}, \cdots, c_{2n}\}$ a subset of

.

Step 2.

$\tilde{K}$

$D(\tilde{K})$

$D(\tilde{K})$

183

TRIVIALITY INDEX OF KNOTS

LEMMA 3.2.

Let

$M=\{1,2, \cdots, 2n\}$ ,

$\# M_{i}=2(i=1,2, \cdots, n),$

that

for

$\mu\tilde{K}(C)=1$

any

$\bigcup_{i=1}^{n}M_{i}=M$

$\nu=\{\{M_{1}, M_{2}, \cdots, M_{n}\}|M_{i}\subset M$

And let

}.

$\kappa_{c}$

,

such be a subset of , then we have that $\nu$

$i(1\leqq i\leqq n)\mu K(\{\{c_{j}, c_{j^{\prime}}\}|M_{i}=\{j, j^{\prime}\}\})=1$

if and

only

if

is odd.

$\#\kappa_{c}$

We prove Lemma 3.2 by the induction on . In the case $n=1,$ $C=\{c_{1}, c_{2}\},$ $M=\{1,2\}$ and $\nu=\{\{M\}\}$ . If is a knot, we have since . Then . If since is a link, we have Lemma 3.2. Let $n>1$ and where $\# C’=2m(n>m)$ . It is be a subset of supposed that if and only if is odd. We consider the $L$ $K$ band model of with respect to as defined in Step 1. Let B. be an outermost band in into , namely when we separate two parts , and one of $\sigma_{i}(i=1,2)$ where does not contain both for any $j(j\neq i, j=1,2, \cdots, 2n)$ . Let and be a part of satisfying the above condition as shown in Fig. 3-4. PROOF.

$n$

$\tilde{K}(C)$

$\#\kappa_{c}=1$

$\tilde{K}(C)$

$\kappa_{c}=\{\{M\}\}$

$\#\kappa_{c}=0$

$\kappa_{c}=\emptyset$

$C$

$C^{\prime}$

$\mu\tilde{K}(C^{\prime})=1$

$\#\kappa_{C^{\prime}}$

$C$

$B_{1},$

$\sigma_{1},$

$B_{2},$

$\cdots,$

$\sigma_{1}\cup\sigma_{2}=\sigma,$

$\sigma_{2}$

$d_{j}$

$B_{2n}$

$\sigma$

$\sigma_{1}\cap\sigma_{2}=\{d_{\tau}, d_{i}^{\prime}\}$

$d_{j}^{\prime}$

$\sigma_{1}$

$\sigma$

$d_{1}^{\prime}$

$FI_{\overline{J}}\backslash URE3-4$

or on the }. Since there is $k\in M-N-\{i\}$ by Proposition 3.1, any element of has $C(j)=\{c_{q}\}(q\in M-\{i, j\})$ , then we have an element. Let Let $N=$ { $j\in

M|$

$d_{j}$

$\mu\tilde{K}(c_{i}, c_{k})=3$

$d_{j}^{\prime}$

$\sigma_{1}$

$\kappa_{c}$

(3.1)

$\#\kappa_{c}=\sum_{jeN}\#\kappa_{C(j)}$

$\{i, i\}(jeN)$

for

as

.

By the hypothesis of induction, we have $\mu\tilde{K}(C(j))=1$ if and only if by considering is odd. Then we show the relation between and $\# N$ . We . By Proposition 3.1, we have and . We note that, consider two cases on the number of components of $\#\kappa_{C(j)}$

$\#\kappa_{c}$

$\mu\tilde{K}(C(j))$

$\mu\tilde{K}(C)$

$\mu\tilde{K}(C(j))=\mu\tilde{L}(a_{i}, a_{j})$ $\tilde{L}(a_{i})$

$1$

184

YOSHIYUKI OHYAMA AND YASUKO OGUSHI

since Case 1.

$\mu\tilde{L}(a_{i})=\mu\tilde{K}(\{c_{k}|k\in M-\{i\}\}),$

$\mu\tilde{K}(C(j))\geqq 3$

for any $jeN$, we have By the hypothesis of induction, is even. By (3.1), , we have Therefore is even. And since $\mu\tilde{L}(a_{i})\geqq 4$

.

Since

$\mu\tilde{L}(a_{i}, a_{\dot{f}})\geqq 3$

$\#\kappa_{C(j)}$

we have we have that $\#\kappa_{c}$

.

is even.

$\mu\tilde{L}(a_{i})$

$\mu\tilde{K}(C)\geqq 3$

$\mu\tilde{L}\geqq 3$

is even. is a link and . Let $N’=\{jeN|\alpha_{j}$ and Case 2. ferent components on }. We have by (3.1) $\tilde{K}(C)$

.

$\#\kappa_{c}$

$\mu\tilde{L}(a_{i})=2$

$\alpha_{\acute{\dot{f}}}$

are contained in dif-

$\tilde{L}(a_{i})$

(3.2)

$\#\kappa_{c}=\sum_{jeN}\#\kappa_{G(j)}$

$=\sum_{\dot{g}eN^{\prime}}\#\kappa_{C(j)}+\sum_{jeN-N^{\prime}}\#\kappa_{C(j)}$

.

for any $jeN’$ , we have $\mu\tilde{K}(C(J))=1$ and by the hyfor any $jeN-N’$ , pothesis of induction is odd. Since is even. Therefore we have by (3.2) and we have

Since

$\mu\tilde{L}(a_{i}, a_{\dot{f}})=1$

$\mu\tilde{L}(a_{i}, a_{\dot{f}})\geqq 3$

$\#\kappa_{ctj)}$

$\mu\tilde{K}(C(j))\geqq 3$

$\#\kappa_{Ctj)}$

(3.3)

$\#\kappa_{c}\equiv\sum_{\dot{g}eN^{\prime}}1+\sum_{\dot{g}eN-N^{\prime}}0$

$\equiv\# N^{\prime}$

$(mod 2)$

.

on the is a knot, considering there is two points In the case components . Moreover of and are contained in different components $jeN,$ are contained in different and for ’ . Therefore $\# N$ is odd when $\mu\tilde{K}(C)=1$ . In the same way when of are contained in the 8ame component and is a link and ’ $\# N$ is even when i8 a link. By (3.3), . Therefore we have in is a link is odd, and when is a knot we have when is even. is a knot is By Ca8e 1 and Case 2, we have that when odd, and when is even. Thi8 completes the proof of is a link Lemma 3.2. $d_{l},$

$\tilde{K}(C)$

$\tilde{L}(a_{i}),$

$d$

$d_{i}^{\prime}$

$\tilde{L}(a)$

$d^{\prime}$

$\alpha_{\dot{f}}^{\prime}(\alpha_{j}, \alpha_{\dot{f}}^{\prime}e\partial B_{\dot{f}})$

$a_{j}$

$\tilde{L}(a_{i})$

$\mu\tilde{L}=3,$

$\tilde{K}(C)$

$d$

$d^{\prime}$

$\tilde{K}(C)$

$\tilde{L}(a_{i})$

$\tilde{K}(C)$

$\tilde{K}(C)$

$\#\kappa_{c}$

$\tilde{K}(C)$

$\tilde{K}(C)$

$\#\kappa_{c}$

$\#\kappa_{c}$

$\#\kappa_{C}$

In this Step, we complete the proof of Theorem A by making use of Lemma 3.2 and the following Lemma 3.3. . be an n-trivial diagram of $K$ with respect to Let Step 3. $\tilde{K}$

$\{A_{1}, A_{2}, \cdots, A_{n}\}$

$+$

1

1 FIGURE 3-5

185

TRIVIALITY INDEX OF KNOTS

Let 3.5

$A_{i}=\{c_{i1}, c_{i2}, \cdots, c_{t\alpha(i)}\}$

$(i=1,2, \cdots, n)$

By

.

$K\left(\begin{array}{llll}1 & 2 & \cdots & k\\i_{1}i_{2} & \cdots & \cdots & i_{k}\end{array}\right)$

, and

the sign of

$c_{ij}$

defined as shown in Fig.

, we denote the link which is obtained from

changing the crossing at

and smoothing at lemma. $c_{ki_{k}-1}$

$c_{11},$

$c_{1i_{1}},$

$c_{12},$

$c_{2i_{2}},$

$\cdots,$

$\cdots,$

$c_{1i_{1}-1},$

$c_{ki_{k}}$

.

$c_{21},$

$\cdots,$

$c_{22},$

$\cdots,$

$c_{2:_{2}-1},$

$c_{k1},$

$K$

$c_{k2},$

by $\cdots$

,

In [6], we showed the following

knot $K$ has an n-trivial diagram with respect to , then the Conway polynomial of $K$ is of the following

LEMMA 3.3. $\{A_{1}, A_{2}, \cdots, A_{n}\}$

$\epsilon_{ij}$

If a

$\nabla_{K}(z)$

form. (3.4)

$\nabla_{K}(z)=1+Z^{n}\sum_{1\leqq i_{\dot{f}}\leq\alpha.(\dot{g})\dot{g}=1.2,\cdot,n}.\epsilon_{1i_{1}}\epsilon_{2i_{2}}\cdots\epsilon_{nt_{n}}\nabla_{K(i_{1}i_{2}\cdots i_{\hslash})^{(z)}}12\cdots n$

.

Let be a $2n$ -trivial diagram with respect to $\{A_{1}, A_{2}, \cdots, A_{2n}\}$ of $K$ , and $A_{i}=\{c_{i1}, c_{i2}, \cdots, c_{i\alpha(t)}\}(i=1,2, \cdots, 2n)$ . is a 2We note that $k(j1)$ ,

we have

$a_{2}=0$

. we have for

, then

$k(j