数理解析研究所講究録
第 1911 巻 2014 年 48-56

48

On the arc index of knots and links
Hwa Jeong Lee
Department of Mathematical Sciences, KAIST

1Introduction

A knot is a one-dimensional circle embedded in three-dimensional space and a link is adisjoint union of knots which may be tangled up together. An arc presentation of a knot
or a link is a special way of presenting . It is an ambient isotopic image of containedin the union of finitely many half planes, called pages, with a common boundary line,
called binding axis, in such a way that each half plane contains a properly embedded
single simple arc as in Figure 1. It is an essential condition that meets each page ina single arc. If the requirement is removed then can be embedded in at most threepages [9]. It is known that every knot or link has an arc presentation [5, 6]. So for a, as the minimumlink we can define an invariant, called arc index and denoted bynumber of pages among all arc presentations of
$L$

$L$

$L$

$L$

$L$

$L$

$\alpha(L)$$L.$

Figure 1: An arc presentation of the trefoil knot

In this paper, we present a small survey and introduce recent results on arc index. Arcpresentations were originally described by Brunn [5] more than 100 years ago, when heproved that any link has a diagram with only one multiple point. Birman and Menasco
used arc-presentations of companion knots to find braid presentations for some satellites [4]. Cromwell adapted Birman-Menasco’s method and used the term (arc index” asan invariant and established some of its basic properties.
Proposition 1.1 (Cromwell [6]) Every link has an arc-presentation.

Theorem 1.2 (Cromwell [6]) For two nonalternating links$\alpha(L_{1}\sqcup L_{2})=\alpha(L_{1})+\alpha(L_{2})$,$\alpha(L_{1}\# L_{2})=\alpha(L_{1})+\alpha(L_{2})-2.$$L_{1}$

and$L_{2}$, we have

49
The theory of arc-presentations was developed by Dynnikov [8]. He proved that any
arc presentation of the unknot admits a monotonic simplification by elementary moves.
He also showed that the problem of recognizing split links and of factorizing a composite
link can be solved in a similar manner.

Theorem 1.3 (Dynnikov [8]) The decomposition problem of arc-presentations is solvable by monotonic simplification.

2Arc index and other knot invariants

Arc presentations can be represented in various ways. Figure 2 depicts diﬀerent ways todescribe an arc presentation. All arcs named and all integers correspond to each other inthe figure.

(a)

(b)

(c)

Figure 2: Diﬀerent ways to describe the arc presentation of Figure 1

Cromwell and Dynnikov used the arc presentation called grid diagram to prove Proposition 1.1 and Theorem 1.3, respectively. A grid diagram is a finite union of vertical line
segments and the same number of horizontal line segments with the properties that atevery crossing the vertical strand crosses over the horizontal strand and no two horizontal
segments are collinear and no two vertical segments are collinear as in Figure 2(a). A grid
diagram can be converted easily to an arc presentation with the number of arcs which isequal to the number of vertical line segments and vice versa. If we consider an oriented
grid diagram of a link, we can get a braid form of the link by cutting open horizontal arcs
of same orientation. The fact yields the followings:
Proposition 2.1 (Cromwell [6]) Let$\beta(L)$denote the braid index of a link L. Then$\alpha(L)\geq 2\beta(L)$.

It is well-known that grid diagrams are closely related to front projections of its Legendrian imbedding in contact geometry. Grid diagrams are also used to compute Heegaard
Floer homology and Khovanov homology. Due to the connections and other nice properties, arc presentations became very popular in recent years. Matsuda described a relation
between arc index
and the maximal Thurston-Bennequinn numbers of a knot $K$
and its mirror, denoted byand.
$\alpha(K)$$K^{*}$$\overline{tb}(K)$$\overline{tb}(K^{*})$

50
Theorem 2.2 (Matsuda [24])$-\alpha(K)\leq\overline{tb}(K)+\overline{tb}(K^{*})$.

In [1] Bae and Park presented an algorithm for constructing arc presentations of alink which is given by edge contractions on a link diagram. The resulting diagram iscalled wheel diagram. The projection of an arc presentation of a knot or link into theplane perpendicular to the binding axis is of this shape. See Figure 2(b). Unorderedpairs of integers in the figure indicate -levels of the end point of the corresponding arcs
in Figure 1, They showed that the algorithm leads an upper bound on the arc index interms of the crossing number, $c(L)$ , of a nonsplit link
$z$

$L.$

Theorem 2.3 (Bae-Park [1]) Let$L$

be any prime non-split link. Then$\alpha(L)\leq c(L)+2.$

By refining Bae-Park’s algorithm, Beltrami constructed minimal arc presentations ofn-semi-alternating links and Jin and Park obtained an inequality sharper than the one in
Theorem 2.3 for non-alternating prime links.
Theorem 2.4 (Beltrami [2]) Let$L$

be an n-semi-alternating link. Then$\alpha(L)=c(L)-2(n-2)$ .

Theorem 2.5 (Jin-Park [15]) A prime link$L$

is non-alternating if and only if$\alpha(L)\leq c(L)$.

In [25] Morton and Beltrami gave an explicit lower bound for the arc index of a linkin terms of the Laurent degree of the Kauﬀman polynomial $F_{L}(a, z)$ . In [18] the reader
will find details of the Kauﬀman polynomial.
$L$

Theorem 2.6 (Morton-Beltrami [25]) For every link$spread_{a}(F_{L})+2\leq\alpha(L)$$L$

we have.

Combing Theorem 2.3, Theorem 2.6 and an observation of Thistlethwaite [29] on theKauﬀman polynomial of alternating links, we have the following equality:

Corollary 2.7 For a non-split alternating link $L,$$\alpha(L)=c(L)+2.$

admits an adequate diagram, the lower
By Thistlethwaite’s work [30], if a link$spread_{a}(F_{L})$a graph theoretical viewpoint. Since nfrom
calculated
can bebound ofsemi-alternating links are adequate, from Corollary 1.1 in [30], Beltrami got $spread_{a}(F_{L})\geq$$c(L)-2(n-1)$ for an n-semi-alternating link . These permit that the equality of arc
index holds for n-semi-alternating links.
$L$

$L$

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The arc index of some knots and links

3

No one can doubt that the arc index of the unknot is 2. Table 1 gives all list of links witharc index up to 5. Beltrami [2] and Ng [26] determined arc index for prime knots up to 10and 11 crossings, respectively. Nutt [27], Jin et al. [11] and Jin and Park [14] identified
all prime knots up to arc index 9, 10 and 11, respectively. In [13] the author with Jinshowed that the existence of certain local diagrams indicates that the arc index is strictly
less than the crossing number. They also determined arc index for new 364 knots with 13crossings and 15 knots with 14 crossings. Recently, Jin and Kim [12] identified all primeknots with arc index 12 up to 16 crossings.
Table 1: All links with arc index up to 5

Matsuda determined the arc index for torus knots.
Theorem 3.1 (Matsuda [24]) Let$T_{(p,q)}$

be a torus knot of type $(p, q)$ . Then$\alpha(T_{(p,q)})=|p|+|q|.$

The author determined the arc index of some of Pretzel knots of type $(-p, q, r)$ (withJin) and Montesinos links of type, denoted by $P(-p, q, r)$ and $M(-r_{1}, r_{2}, r_{3})$ ,respectively. $P(p, q, r)$ particularly satisfies the following properties for nonzero integers$p,q$ , and:$(-r_{1_{\rangle}}r_{2}, r_{3})$$r$$\bullet$$\bullet$

The link type of $P(p, q, r)$ is independent of the order of$p,$ $q,$ $r,$$P(p, q, r)$ is a knot if and only if at most one of $p,$ $q,$ $r$ is an even number.

Since we consider Pretzel knots of type $(-p, q, r)^{1}$ , we may assume that$p,$ $q,$ $r\geq

2$

and$r\geq q.$$T$heorem 3.2 (Lee-Jin [20]) Let$p,$ $q,$ $r$

be integers with$p,$ $q\geq 2$and $r\geq q.$(1) If $K=P(-2, q, r)$ is a knot with $q\geq 3$ , then $\alpha(K)\leq c(K)-1.$(2) If $K=P(-p, 2, r)$ is a knot with $p\geq 3$ , then $\alpha(K)=c(K)$ .

(3) If $K=P(-p, 3, r)$ is a knot with $p\geq 3$ , then $\alpha(K)=c(K)-1.$(4) If $K=P(-p, 4, r)$ is a knot with $p\geq 5$ , then $\alpha(K)=c(K)-2.$(5) If $K=P(-3,4, r)$ is a knot with $r\geq 7$ , then $c(K)-4\leq\alpha(K)\leq c(K)-2.$
lBy Lickorish Thistlethwaite’s work [23], it is known that $c(P(-p, q, r))=p+q+r$ . We also know reduced Montesinos
links admit minimal crossing diagrams.

52
Theorem 3.3 (Lee [19])(1) Let be a reduced Montesinos link $M(-r_{1}, r_{2}, r_{3})$ for all positive irreducible rational
numbers . If $r_{1}>1,$ $r_{2}>2$ and $r_{3}>2$ , then$L$$r_{i}$$\alpha(L)\leq c(L)-1.$

be a positive integer greater than 1 andbe all positive irreducible rational
numbers. Let be a reduced Montesinos link $M(-n, r_{2}, r_{3})$ . If $r_{2}>3$ and has acontinued fraction
with, thenand(2) Let$n$$r_{2},$$r_{3}$$L$$r_{3}$$(a_{1}, a_{2}, \ldots, a_{m})$$a_{1}\geq 3$$a_{2}\geq 2$$\alpha(L)\leq c(L)-2.$(3) Let $n,$ $m$ be positive integers and let$L$

be a reduced Montesinos link$M(-n, \frac{m}{2}, \frac{m}{2})$.

(a) If $n>1$ and $m=3$ , then $\alpha(L)=c(L)$ .
(b) If $n>2$ and $m=5$ , then $\alpha(L)=c(L)-1.$(c) If $n>3$ and $m=7$ , then $\alpha(L)=c(L)-2.$(4) Let $n,$ $m$ be positive integers and let$L$

be a reduced Montesinos link $M(-n, m, \frac{17}{5})$ .

(a) If $m=2$ , then $\alpha(L)=c(L)$ .
(b) If $m=3$ , then $\alpha(L)=c(L)-1.$(c) If $m=4$ , then $\alpha(L)=c(L)-2.$

In [16, 17] Kanenobu introduced an infinite family of knots, denoted by $K(p, q)$ , thatis composed of infinite classes of knots which have the same HOMFLY-PT and Jones
polynomials which are hyperbolic, fibered, ribbon, of genus 2 and 3-bridge, but with
distinct Alexander module structures. Since $K(p, q)\approx K(q,p)$ , $K(p, q)^{*}\approx K(-p, -q)\approx$$K(-q, -p)$ and $\alpha(L)=\alpha(L^{*})$ for a link , it is suﬀcient to consider $K(p, q)$ with $|p|\leq q$
in order to determine the arc index of $K(p, q)$ .
$L$

Theorem 3.4 (Lee-Takioka [21])(1) $Let1\leq p\leq qandpq\geq 3$ . Then a $(K(p, q)$ ) $=p+q+6.$(2) $Letq\geq 3.$ $Thenq+6\leq\alpha(K(0, q))\leq q+7.$(3) $Letq\geq 3$ . Then $q+5\leq\alpha(K(-1, q))\leq q+7.$(4) $Letq\geq 3.$$Thenq+4\leq\alpha(K(-2, q))\leq q+7.$

The author with Takioka found some examples to show that the bounds of (2) and (3) inTheorem 3.4 are best possible.

To prove Theorem 3.2, 3.3, 3.4, the author with Jin and Takioka in the proper paper
used the way of finding arc presentations on knot or link diagrams as depicted in Figure 2(c). The idea was introduced by Cromwell and Nutt [7] first. The definition is asfollows:53
Let $D$ be a diagram of a knot or alink . Suppose that there is a simple closed curve
meeting in distinct points which divide into arcs,...,
with the following
$L$

$D$

$k$

$C$

$D$

$k$

$\alpha_{1},$$\alpha_{2}$$\alpha_{k}$properties:

1. Each$\alpha_{i}$has no self-crossing.

crosses overat a crossing inresp.
, then $i>j($ resp. $i<j)$ and it
crosses over at any other crossings with , respectively. Here,andis theinner and the outer region divided by , respectively.2

2. If$\alpha_{i}$$R_{O})$$R_{I}($$\alpha_{j}$$\alpha_{j}$$R_{O}$$R_{I}$$\alpha_{j}$$C$

3. For each , there exists an embedded disk$i$$d_{i}$

such that$\partial d_{i}=C$

and$\alpha_{i}\subset d_{i}.$4. $d_{i}\cap d_{j}=C$ , for distinct andThen the pair $(D, C)$ is called an arc presentation of with arcs, and is called thebinding circle of the arc presentation. Figure 2(c) shows an arc presentation of the trefoilknot.
$i$

$j.$

$L$

$C$

$k$

Finally, we consider satellite knots. The class of satellite knots contains basic familiesof composite knots, cable knots and Whitehead doubles. The arc index of composite
knots was determined by Cromwell as stated in Theorem 1.2. The others were dealt within [22].

Letand be integers with $p>1$ . Given a knot $K$ , let
andbe the $(p, q)$ -cable link, the -twisted positive Whitehead double and the -twisted negative
Whitehead double of $K$ , respectively. Let ne (G) and se (G) denote the number of north-east
corners and south-east corners for a grid diagram of a knot, respectively.
$p,$ $q$$K^{(p,q)},$$t$$t$$K^{(+,t)}$$K^{(-t)}\rangle$

$t$

$G$

Theorem 3.5 (Lee-Takioka [22]) 3 Let
be a grid diagram of a knot $K$ andintegers with $p>1$ . Suppose that $n(G)=q-pw(G)$ .
$G$$p,$ $q$

be(1) If $n(G)\geq 0,$ $\exists!m(G)s.t.$ $pm(G)\leq n(G)<p(m(G)+1)$ . Then,$\alpha(K^{(p_{)}q)})\leq\{\begin{array}{ll}p\alpha(G) if ne (G)>m(G)p(\alpha(G)+tb(G^{*}))+q if ne (G)\leq m(G)\end{array}$(2) If $n(G)<0,$ $\exists!m’(G)s.t.$ $p(m’(G)-1)<n(G)\leq pm’(G)$ . Then,$\alpha(K^{(p,q)})\leq\{\begin{array}{ll}p\alpha(G)) if se (G)>-m’(G)p(\alpha(G)+tb(G))-q if se (G)\leq-m’(G)\end{array}$

Theorem 3.6 (Lee-Takioka [22]) Let
integer. Suppose that $n(G)=2t-2w(G)$ .

$G$

be a grid diagram of a knot $K$ and(1) If $n(G)\geq 0$ , then$\alpha(K^{(+,t)})\leq\{\begin{array}{ll}2\alpha(G)+1 if 2ne(G)>n(G)2(\alpha(G)+tb(G^{*})+t+1) if 2ne(G)\leq n(G)\end{array}$andare only inand $tb(G)=w(G)-se(G)$ .

$2For$ example, in Figure 2(c)$3_{w(G)}$is the writhe of$G$$\alpha_{4}$$\alpha_{5}$$R_{I}.$$t$

be an

54$\alpha(K^{(-,t)})\leq\{\begin{array}{ll}2\alpha(G)+1 if 2ne(G)\geq n(G)2(\alpha(G)+tb(G^{*})+t+1) if 2ne(G)<n(G)\end{array}$(2) If $n(G)<0_{f}$ then$\alpha(K^{(+,t)})\leq\{\begin{array}{ll}2\alpha(G)+1 if 2se(G)\geq-n(G)2(\alpha(G)+tb(G)-t)+1 if 2se(G)<-n(G)\end{array}$$\alpha(K^{(-t)}))\leq\{\begin{array}{ll}2\alpha(G)+1 if 2se(G)>-n(G)2(\alpha(G)+tb(G)-t)+1 if 2se(G)\leq-n(G)\end{array}$Using Theorem 3.5 and 3.6, the author and Takioka exactly determined the arc index
of infinite families of the (2, q)-cable link, the -twisted positive Whitehead double andthe -negative Whitehead double of all knots with up to 8 crossings.
$t$

$t$

Example. The table below gives the arc index of the (2, q)-cable link, the -twisted
positive Whitehead double and the -negative Whitehead double of . Here,is themirror image of the diagram of in Rolfsen’s tables [28].
$t$

$t$

$3_{1}^{*}$$3_{1}^{*}$$3_{1}$

Acknowledgments
This work was supported by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MEST) (No. 2011-0027989).

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1655-1672.

[16] T. Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer.
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[19] H. J. Lee, Arc index of Montesinos links of type of$(-r_{1}, r_{2_{\rangle}}r_{3})$, preprint.

[20] H. J. Lee and G. T. Jin, Arc index of pretzel knots type of (-p, q, r) , arXiv:1204.0597.
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[28] D. Rolfsen, Knots and links (AMS Chelsea Publishing, 2003).
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Kauﬀman polynomial and alternating links, Topology 27[30] M. B. Thistlethwaite, On the Kauﬀman polynomail of an adequate link, Invent. Math.
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Department of Mathematical Sciences
Korea Advanced Institute of Science and Technology
Daejeon 305-701
South KOREA-mail address: hjwith@kaist.ac.kr$E$