PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 2, Pages 613–616 S 0002-9939(04)07522-7 Article electronically published on August 30, 2004

NON-ADDITIVITY FOR TRIPLE POINT NUMBERS ON THE CONNECTED SUM OF SURFACE-KNOTS

SHIN SATOH

(Communicated by Ronald A.Fintushel)

Abstract. Any surface-knot F in 4-space can be projected into 3-space with a finite number of triple points, and its triple point number, t(F), is defined similarly to the crossing number of a classical knot. By definition, we have t(F_{1}#F_{2}) ≤ t(F_{1}) + t(F_{2}) for the connected sum. In this paper, we give infinitely many pairs of surface-knots for which this equality does not hold.

A surface-knot F is an (orientable or non-orientable) connected, closed surface smoothly embedded in Euclidean 4-space ℝ^{4}. Two surface-knots F and F′ are equivalent, denoted by F ≅ F′, if there is an ambient isotopy of ℝ^{4} that maps F to F′. For a fixed projection π : ℝ^{4} → ℝ^{3}, we can isotope F slightly so that the projection π|_{F} into ℝ^{3} is a generic map (cf. [4]). The set of triple points of such a generic map is discrete. The triple point number of F, denoted by t(F), is the minimal number of triple points for all possible generic projections of F. There are several studies on triple point numbers: [9, 10, 12, 13, 14, 16], for example.

The triple point number has an analogy to the crossing number c(K) of a classical knot K. The connected sum K_{1}#K_{2} of classical knots K_{1} and K_{2} satisfies

(1)  c(K_{1}#K_{2}) ≤ c(K_{1}) + c(K_{2}),

and it is still an open problem whether the equality in (1) holds for any K_{1} and K_{2}. Similarly, for the connected sum F_{1}#F_{2} of surface-knots F_{1} and F_{2}, we have the following by definition:

(2)  t(F_{1}#F_{2}) ≤ t(F_{1}) + t(F_{2}).

Hence, it is natural to ask whether the equality in (2) holds for any F_{1} and F_{2}. The aim of this paper is to give a negative answer to this question.

Let τ^{n}K denote the n-twist-spin of a classical knot K (cf. [17]). Also, let P_{g}(e) denote the non-orientable trivial surface-knots of genus g specified with the normal Euler number e (cf. [8]).

Theorem 1. Assume that K is a 2-bridge knot and n ≥ 2. Then we have

(3)  τ^{n}K#P_{3}(±2) ≅ \{ \begin{matrix}τ^{0}K#P_{3}(±2) & if n is even, \\ P_{3}(±2) & if n is odd. \end{matrix}

It follows that the equality in (2) does not hold for any pair {F_{1}, F_{2}} = {τ^{n}K, P_{3}(±2)}.

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Received by the editors July 27, 2003 and, in revised form, August 29, 2003.

2000 Mathematics Subject Classification. Primary 57Q45; Secondary 57Q35.

Key words and phrases. Surface-knot, connected sum, triple point, twist-spun knot.

Let σ^{n}K = τ^{n}K + h_{1} denote the surface-knot of a torus obtained from τ^{n}K by surgery along a 1-handle h_{1} contained in the axis-plane of twisting [2].

Lemma 2. For any classical knot K, we have

(4)  σ^{n}K#P_{1}(±2) ≅ σ^{n+2}K#P_{1}(±2).

Proof. Recall that a projection of τ^{n}K is constructed by (i) making n writhes on the embedded sphere in ℝ^{3}, (ii) taking a simple closed curve L on the sphere that travels around all the writhes, and (iii) replacing the neighborhood of L by the product of L and a tangle diagram of K. Here, a writhe is regarded as a pile of motions representing Reidemeister moves I. Refer to [1] for more details. Hence, σ^{n}K has a projection as shown in Figure 1.

Figure 1

We will eliminate the writhes from the diagram as follows. First, we move the attaching region of the handle h_{1} close to the feet of the tangle diagram, and then expand and extend the inside of h_{1} along L (the left of Figure 2). Next, we push the tube formed by the knot diagram of K out of the writhes, and then eliminate them by an ambient isotopy of ℝ^{4} (the right of Figure 2). Hence, σ^{n}K is equivalent to the surface-knot obtained from a surface-link S_{0} ∪_{n} T^{n}K by surgery along the 1-handle h_{2}, where S_{0} is the trivial sphere-knot (cf. [6]), T^{n}K is the n-turned torus-knot of K defined by Boyle [3], and ∪_{n} means that T^{n}K links S_{0} by n times. By taking the connected sum of P_{1}(±2) with S_{0}, we can regard σ^{n}K#P_{1}(±2) as (P_{1}(±2) ∪_{n} T^{n}K) + h_{2}.

Figure 2

Since P_{1}(±2) has the fundamental group ℤ_{2} of the complement in ℝ^{4}, the linking number between P_{1}(±2) and T^{n}K is changeable up to the parity and relative to the handle h_{2}. It follows that

(P_{1}(±2) ∪_{n} T^{n}K) + h_{2} ≅ (P_{1}(±2) ∪_{n+2} T^{n}K) + h_{2}.

This equivalence is similar to Viro’s work in [15]. Since T^{n}K ≅ T^{n+2}K relative to S_{0} and h_{2} (cf. [3]), we have the equivalence (4). □

Proof of Theorem 1. Let h_{0} be the trivial 1-handle on τ^{n}K. Then it is easy to see that τ^{n}K#P_{3}(±2) ≅ (τ^{n}K + h_{0})#P_{1}(±2). On the other hand, Boyle [2] proved that if K is a 2-bridge knot, then h_{0} and h_{1} are equivalent relative to τ^{n}K. It follows that τ^{n}K + h_{0} ≅ τ^{n}K + h_{1}, and hence, τ^{n}K#P_{3}(±2) ≅ σ^{n}K#P_{1}(±2). Since Zeeman [17] proved that τ^{1}K ≅ S_{0}, we have the equivalence (3) by Lemma 2.

The latter assertion is proved as follows. Since τ^{n}K is not of ribbon-type for n ≥ 2 [5], we have t(τ^{n}K) > 0 [16]; see also [7]. (This result has been strengthened to t(τ^{n}K) ≥ 4 in [11].) On the other hand, it follows that t(τ^{0}K#P_{3}(±2)) = t(P_{3}(±2)) = 0 by definition (cf. [4]). Note that t(τ^{0}K) = 0 for any K. Hence, we have t(τ^{n}K#P_{3}(±2)) = 0 < t(τ^{n}K) + t(P_{3}(±2)) by the equivalence (3). □

Remark 3. It is still an open problem whether τ^{n}K#P_{1}(±2) ≅ P_{1}(±2) for any classical knot K and odd integer n > 1. Note that they have the same fundamental group ℤ_{2} of the complement in ℝ^{4}.

Acknowledgments

The author is partially supported by JSPS Postdoctoral Fellowships for Research Abroad, and expresses his gratitude for the hospitality of the University of South Florida and Professor Masahico Saito.

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Graduate School of Science and Technology, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan

   E-mail address: satoh@math.s.chiba-u.ac.jp