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 CIRCULAR EMBEDDINGS OF 2-COMPONENT LINKS AND SPATIAL
GRAPHSTOSHIFUMI TANAKA
Abstract. A circular spatial embedding of a graph into the 3-dimensional Euclidean space R3 is anembedding which consists of round arcs. We deﬁne the circular number of a spatial graph as theminimal number of round arcs in R3 among such embeddings of the spatial graph. In this paper, weinvestigate the circular number for 2-component links, theta-curves and handcuﬀ graphs.

Key words: Knots, Links, Spatial graphs, Circular number.

1. Introduction
A graph G is a 1-dimensional simplicial complex. We call an embedding f : G −→ R3 of a graphG in R3 (= {(x, y, z) | x, y, z ∈ R}) a spatial embedding of G and a graph f (G) embedded in R3 aspatial graph. A cycle C in a graph G is a cyclic sequence {v0 , · · · , vn−1 } of distinct vertices of G suchthat any two consecutive vertices vi and vi+1 , i taken modulo n, are joined by an edge vi vi+1 . If one
embeds G in R3 by a spatial embedding f : G −→ R3 , then the image of any cycle in G is a knot, andany disjoint union of cycles in G forms a link in R3 . A spatial embedding of a graph which maps eachedge to a round arc in R3 is called a circular spatial embedding. For a spatial graph Ĝ, we deﬁne thecircular number Circ(Ĝ) as the minimal number n such that Ĝ has a circular spatial embedding withn round arcs. Clearly, the circular number is an invariant of a spatial graph. It is known that everygraph has a circular spatial embedding and a knot has circular number three if and only if the knot
is a trefoil knot, and the ﬁgure-eight knot has circular number four [1]. A spatial graph is trivial if it
has an embedding in the plane.
Theorem 1.1.
(1) A 2-component link L is either the trivial link or the Hopf link if and only if Circ(L) = 2.
(2) There is no 2-component link L with Circ(L) = 3.
(3) If a 2-component link L has circular number four, then each component of L is either a trefoilknot or the trivial knot.
A theta-graph is a (plane) graph with two vertices and three edges where each edge connects thetwo vertices. A theta-curve is a spatial embedding of a theta-graph.
Proposition 1.2.
(1) A theta-curve T is trivial if and only if Circ(T ) = 2.
(2) There exists a (non-trivial) theta-curve T with Circ(T ) = 3.
A handcuﬀ graph is a spatial embedding of a (plane) graph which is formed by two disjoint loops
connected by an edge.
Proposition 1.3.
(1) If H is a handcuﬀ graph with Circ(H) = 3, then H is the trivial link or the Hopf link with an
unknotted edge connecting the two components.
(2) There exists a non-trivial Handcuﬀ graph H with Circ(H) = 3.
This paper is organized as follows. In Section 2, we shall prove Theorems 1.1 and Propositions 1.2
and 1.3. In Section 3, we shall give some examples.
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TOSHIFUMI TANAKA

2. Proofs
Deﬁnition 2.1. (1) Let G be a graph. A spatial embedding f : G −→ R3 is called a circular spatialembedding if there exists a ﬁnite number of round circles 1 , · · · , n in R3 such that f (G) ⊂ ∪ni=1 i .
(2) For a spatial graph Ĝ, let Circ(Ĝ) = min{n ∈ N | Ĝ consists of n round arcs}. We call Circ(Ĝ)
the circular number of Ĝ. (We assume that a round circle is also a round arc.)
Proof of Theorem 1.1. First, we show the statement (1). It follows from the deﬁnition of the circularnumber that if L is the trivial link or the Hopf link, then Circ(L) = 2. Suppose Circ(L) = 2. Thenby the assumption, L can be composed with two round circles in R3 . We denote the two disks which
is bounded by the two circles by D1 and D2 . Then we may assume that D1 and D2 intersect eachother as in Figure 1. (The lines in the disks describe the intersection.)

(a)

(b)
Figure 1

If one of the disks has intersection as in Figure 1(a), then the intersection is canceled by an isotopyof R3 and we have the trivial link. If both disks have intersection as in Figure 1(b), then the link is theHopf link. Next we show the statement (2). Suppose that L is a 2-component link with Circ(L) = 3.
Then one of the components of L consists of a round circle and the other one is composed with tworound arcs. We denote them by C and C  respectively. Note that C  bounds a disk D as in Figure 2.

Figure 2
We may assume that D meets C in a ﬁnite number of points. In fact, we know that D ∩ C is either
empty or consists of n points where 1 ≤ n ≤ 4. Then it is easily seen that we have the trivial link orthe Hopf link. However their circular numbers are less than three. This is contradiction. Finally, weshow the statement (3). If Circ(L) = 4 then both components of L consists of two round arcs or oneof the components consists of a round circle and the other consists of three round arcs. In the former
case, all components are trivial. In the latter case, we have the trivial knot and either a trefoil knotor the trivial knot as the components by a result in [1].
Proof of Proposition 1.2. If a theta curve T is trivial, then it is easily seen that Circ(T ) = 2 sincewe need at least two round arcs to yield a theta curve. Conversely, if Circ(T ) = 2, then T consists oftwo round arcs and one of them must be a round circle. Then we know that the circle with a round           

CIRCULAR EMBEDDINGS OF 2-COMPONENT LINKS AND SPATIAL GRAPHS

3

arc can be embedded in the plane generically. Thus the statement (1) holds. To show the statement(2), we consider a theta-curve T as in Figure 3(a). It has a circular embedding as shown in Figure 3(b). Then we know that Circ(T ) ≤ 3. By the statement (1), we also know that Circ(T ) ≥ 3.
zTWos1

Sys2x(a)(b)
Figure 3

Proof of Proposition 1.3. First we show the statement (1), if Circ(H) = 3, then H has two roundcircles and a round arc. The two circles realize either the trivial link or the Hopf link where the twocircles are connected by a round arc. To show the statement (2), we consider the Handcuﬀ graph Ĥ
as shown in Figure 4.

HFigure 4
By the deﬁnition of the circular number, we know that Circ(Ĥ) ≤ 3. We need at least three roundarcs to compose a Handcuﬀ graph since it has trivalent vertices. So we know that Circ(Ĥ) ≥ 3. Since
it has the Hopf link as its cycles, we know that Ĥ is non-trivial. This completes the proof.
3. Examples
Example 1. Since the ﬁgure eight knot have circular numbers four [1], the theta curves as in Figure5(a) are four. In fact, it has a circular embedding with four round arcs as shown in Figure 5(b).
Example 2. The circular numbers of the spacial handcuﬀ graph as shown in Figure 6(a) and (b) areless than or equal to four. However we do not know if the circular numbers are equal to four or not.

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TOSHIFUMI TANAKA

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(a)

(b)
Figure 5(a)(b)
Figure 6References[1] K. Kobata and T. Tanaka, A circular embedding of a graph in Euclidean 3-space, Topology Appl. 157 (2010), no. 1,213–219.
Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193,
Japan.
E-mail address: tanakat@gifu-u.ac.jp