23

数理解析研究所講究録
第2004巻 2016年 23-29

Morse‐Novikov numbers of 2‐knots and surface‐links
Hisaaki Endo and Andrei Pajitnov
1. INTRODUCTION
1.1. A brief overview of the article. In this paper we give a short presentation of
results on the Morse‐Novikov theory for 2‐knots and surface‐links (see the articles

our\mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:1502.06352 and \mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:1605.04532 for more details and full proofs.)

Let N^{k}\subset S^{k+2} be a closed oriented submanifold, let C(N)=S^{k+2}\backslash N be its comple‐

ment. The orientation of N determines a cohomology class$\xi$\in H^{1}(C(N))\approx[C(N), S^{1}].

We say that N is fibred if there is a Morse map f : C(N)\rightarrow S^{1} homotopic to $\xi$ which isregular nearby N (see Definition 1.1) and has no critical points. In general a Morse map

C(N)\rightarrow S^{1} has some critical points, the minimal number of these critical points will becalled the Morse‐Novikov number of N and denoted by \mathcal{M}\mathcal{N}(C(N)).

In the first part of this paper we study this invariant in relation with constructionsof spinning. The classical Artins spinning construction [2] associates to each classicalknot K\subset S^{3} a2‐knot

S(K)\subset S^{4}.

A twisted version of this construction is due to

E.C. Zeeman [12]. In [10] D. Roseman introduced a frame spinning construction, and G.
Friedman [3] gave a generalization of D. Rosemans construction to include twisting. Let
M be a framed closed submanifold of the (m+k) ‐dimensional sphere, K be an m ‐knot

and $\lambda$:

M\rightarrow S^{1} a C^{\infty} map. The twist spinning construction associates to these data

an n ‐knot$\sigma$(M, K, $\lambda$) (where n=k+m ). In Section 2 we give an upper bound for the

Morse‐Novikov number of the twist spun knot in terms of Morse‐Novikov invariants of M
and K.

Section 3 is about Morse‐Novikov theory for surface‐links. In Subsection 3.1 we intro‐
duce a related invariant of surface‐links, namely the saddle number sd(F) (Definition 3.1)
and prove the formula(1)\mathcal{M}\mathcal{N}(C(F))\leq 2sd(F)+ $\chi$(F)-2.

In Subsection 3.2 we discuss the case of spun knots. In subsection 3.3 we determine theMorse‐Novikov numbers of certain surface‐links.
1.2. Basic definitions. We start with the definition of a regular Morse map.

Definition 1.1. Let

N^{k}\subset S^{k+2} be a closed oriented submanifold.

Denoteby $\xi$\in\mathrm{A}cohomologyH^{1}(C(N))\approx[C(N), S^{1}]
Morse map f : C(N)\rightarrow S^{1} is said to be regular if there is an orientation preserving C^{\infty}
class dual to the orientation class of N.

thetrivialisation(2)$\Phi$ :T(N)\rightarrow N\times B^{2} ( 0, č)of a tubular neighbourhood T(N) of N such that the restriction

f\mathrm{o}$\Phi$^{-1}(x, z)=z/|z|.

f|(T(N)\backslash N) satisfies

24

An f ‐gradientvof a regular Morse map

f

C(N)\rightarrow S^{1} will be called regular if:

there is a C^{\infty} trivialisation (2) such that $\Phi$^{*}(v) equals (0, v_{0}) where v_{0} is the Riemannian

gradient of the function z\mapsto z/|z|.

If f is a Morse map of a manifold to \mathrm{R} or to S^{1} , then we denote by m_{p}(f) the numberof critical points of f of index p

m(f)

.

The number of all critical points of f is denoted by.

Definition 1.2. The minimal number m(f) where f:

C(N)\rightarrow S^{1} is a regular Morse

map is called the Morse‐Novikov number of N and denoted by \mathcal{M}\mathcal{N}(C(N))

To obtain lower bounds for numbers m_{p}(f)\mathbb{Z}[t, t^{-1}]; denote by\hat{L}=\mathbb{Z}((t))

andone uses\hat{L}_{\mathbb{Q} =\mathbb{Q}( t).

the Novikov homology. Let L=

the rings of all series in one variable twith integer (respectively rational) coeffcients and finite negative part. Recall that \hat{L} is a

PID, and \hat{L}_{\mathb {Q} is a field. Consider the infinite cyclic covering \overline{C(N)}\rightarrow C(N) ; the Novikov

homology of C(N) is defined as follows:\displaystyle \hat{H}_{*}(C(N) =H_{*}(\overline{C(N)})\bigotimes_{L}\hat{L}.
The rank and torsion number of the \hat{L} ‐module \hat{H}_{k}(C(N)) will be denoted by \hat{b}_{k}(C(N))
respectively \hat{q}_{k}(C(N)) For any regular Morse function f there is a Novikov complex \mathcal{N}_{*}
over \hat{L} generated in degree k by critical points of f of index k and such that H_{*}(\mathcal{N}_{*})\approx\hat{H}_{*}(C(N)) (see [8]). Therefore we have the Novikov inequalities,.

\displaystyle \sum_{k}(\hat{b}_{k}(C(N) +\hat{q}_{k}(C(N) +\hat{q}_{k-1}(C(N) )\leq \mathcal{M}\mathcal{N}(C(N).

These inequalities, which are far from being exact in general, are however very useful inthe case of surface‐links (see Section 3).
2. SPINNING AND RELATED CONSTRUCTIONS

the construction. In this subsection we recall the

2.1. Frame twist spun knots:

Artin‐Zeeman‐Roseman‐Fkiedman frame twist spinning construction. The input data forthis construction is:(TFSI) A closed manifold M^{k}\subset S^{m+k} with trivial (and framed) normal bundle.
(TFS2) An m ‐knot K^{m}\subset S^{m+2}.
(TFS3) A C^{\infty} map $\lambda$ : M\rightarrow S^{1}.
When $\lambda$ is a
To these data one associates an n ‐knot $\sigma$(M, K, $\lambda$) where n=k+m
constant map we denote this knot by $\sigma$(M, K) ; this is the Rosemans frame spun knot.
Let a\in K^{m} Removing a small open disk D(a) from S^{m+2} we obtain an embedded(knotted) disk K_{0} in the disk D^{m+2}\approx S^{m+2}\backslash D(a) We identify D^{m+2} with the standard,.

.

Euclidean disk of radius 1 and center 0 in \mathbb{R}^{m+2} then \partial D^{m+2}=S^{m+1},diffeomorphism$\chi$ :S^{m+1}\times]0 1 ] \rightarrow^{\approx}D^{m+2}\backslash \{0\},,(x, t)\mapsto tx.

.

We have the usual

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can assume that K_{0}\cap\partial D^{m+2} is an equatorial sphere $\dagger$ S^{m-1} in \partial D^{m+2}=S^{m+1}.
Moreover, we can assume that the intersection of K_{0} with a neighbourhood of \partial D^{m+2} isalso standard, that is,We

K_{0}\cap $\chi$(S^{m+1}\times[1- $\epsilon$, 1])= $\chi$(S^{m-1}\times[1- $\epsilon$, 1]).

We have a framing of M in S^{n} (recall that n=m+k ); combining this with the standard
framing of S^{n} in S^{n+2} we obtain a diffeomorphism$\Phi$ :N(M, S^{n+2})\rightarrow^{\approx}M\times D^{m}\times D^{2}

where N(M, S^{n+2}) is a regular neighbourhood of M in S^{n+2}restriction of $\Phi$ to.

We can assume that theN(M, S^{n}) is a diffeomorphism

N(M, S^{n})\rightarrow^{\approx}M\times D^{m}\times\{0\}$\Phi$ :induced by the given framing of M.

The Euclidean disc D^{m+2} is a subset of D^{m}\times D^{2} , sothat K_{0}\subset D^{m}\times D^{2}.

For $\theta$\in S^{1} denote by R_{ $\theta$} the rotation of D^{2} around its center. The disc D^{m+2}\subset D^{m}\times D^{2}

is invariant with respect to this rotation as well as the intersection of K_{0} withneighbourhood of \partial D^{m+2} We have $\Phi$(S^{n}\cap N(M, S^{n+2}))=M\times D^{m}\times\{0\}
.

.

a

small

Let

Z=\{(x, y, z)|(y, z)\in R_{ $\lambda$(x)}(K0)\}.
This is an n ‐dimensional submanifold of M\times D^{m}\times D^{2}

We define $\sigma$(M, K, $\lambda$) as follows.

$\sigma$(M, K, $\lambda$)=(S^{n+2}\backslash N(M, S^{n+2}) \cup$\Phi$^{-1}(Z).

This is the image of an embedded n ‐sphere, knotted in general.

Examples and particular cases.

1)

Let \dim M=0 ,n ‐knotsoThen thethat M is a finite set; denote by p its cardinality.

$\sigma$(M, K, $\lambda$) is equivalent to the connected sum of p copies of K.

2) If M is the equatorial circle of the sphere S^{2} which is in turn considered as an
equatorial sphere of S^{4} and $\lambda$(x)=1 for all x we obtain the classical Artins,,,construction. If $\lambda$ : S^{1}\rightarrow S^{1} is a map of degree d , we obtain the Zeemans twist‐

spinning construction [12].
3) If $\lambda$(x)=1 for all x\in M we obtain the Rosemans construction of spinning aroundthe manifold M[10] In this case we will denote $\sigma$(M, K, $\lambda$) by $\sigma$(M, K).

.

2.2. Morse‐Novikov numbers of twist spun knots.

Theorem 2.1.

\mathcal{M}\mathcal{N}(C( $\sigma$(M, K, $\lambda$ \leq \mathcal{M}\mathcal{N}(C(K) \cdot \mathcal{M}\mathcal{N}(M, [ $\lambda$]).

(where [ $\lambda$]\in H^{1}(M, \mathbb{Z})\approx[M, S^{1}] is the homotopy class of $\lambda$ ).
\upar ow\mathrm{B}\mathrm{y} equatorial sphere in S^{N}\subset \mathbb{R}^{N+1} we mean the intersection of a linear subspace L\subset \mathbb{R}^{N+1} with S^{N} ; thisintersection is a Euclidean sphere of dimension \dim L-1.

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Corollary 2.2. Let K\subset S^{3} be a classical knot,

denoteThenby S(K)\mathcal{M}\mathcal{N}(C(S(K)))\leq 2\mathcal{M}\mathcal{N}(C(K))(3)
Proof.

In this case M=S^{1} and[A]follows.

=O.

the spun knot of K.

.

We have \mathcal{M}\mathcal{N}(S^{1},0)=2 and the result

The classical theorems concerning fibrations of spun knots follow from Theorem 2.1:

If K is fibred, then $\sigma$(M, K) is fibred.

Corollary 2.3. (D. Roseman [10])Proof.

Since \mathcal{M}\mathcal{N}(C(K))=0 Theorem 2.1 implies \mathcal{M}\mathcal{N}(C( $\sigma$(M, K,

The d ‐twist spun knot of any classical knot K is

Corollary 2.4. (E. C. Zeeman [12])fibred for d\geq 1.
Proof.

=0.

Let $\Sigma$ be an equatorial circle in S^{2}

The d‐twist spun knot of K is by definition.

the 2‐knot $\sigma$( $\Sigma$, K, $\lambda$) in S^{4} where $\lambda$ : $\Sigma$\rightarrow $\Sigma$ is a map of degree d

The assertion follows,.

since \mathcal{M}\mathcal{N}(S^{1}, [ $\lambda$])=0.

Remark 2.5. The Zeemans theorem above
statement: If

generalizes immediately to the following\mathcal{M}\mathcal{N}(M, [ $\lambda$])=0 then the knot $\sigma$(M, K, $\lambda$) is fibred for any knot K.
,

2.3. Rotation. In this subsection we present one more geometric construction related tospinning techinques. Let $\Sigma$ be an equatorial n ‐sphere of S^{n+1} We can view the sphereS^{n+1} as the union of two (n+1) ‐dimensional discs D_{+}\cup D_{-} intersecting by $\Sigma$ Consider
S^{n+1} as the equatorial sphere of S^{n+2} The sphere S^{n+2} can be considered as the result ofrotation of the disc D_{+} around its boundary $\Sigma$ We have the (linear orthogonal) actionof S^{1} on S^{n+2} such that $\Sigma$ is the fixed point set of the action, and the action is free on
the rest of the sphere S^{n+2} Let K^{n-1} be an (n-1) ‐knot in S^{n+1} We can assume thatK^{n-1}\subset Int D_{+} Rotation of K^{n-1} around $\Sigma$ gives a submanifold R(K) of codimension 2
in S^{n+2} The manifold R(K) is diffeomorphic to S^{1}\times K We call this constrution rotation.
.

.

.

.

,

.

.

.

.

.

When \dim K=1 , the manifold R(K) is sometimes called the spun torus of K.

In thissection we relate the Morse‐Novikov numbers of R(K) with those of K.

Theorem 2.6.

\mathcal{M}\mathcal{N}(C(R(K)))\leq 2\mathcal{M}\mathcal{N}(C(K))+2.
3. MORSE NOVIKOV NUMBERS OF SURFACE‐LINKS

In this section we develop circle‐valued Morse theory for surface‐links.
3.1. Motion pictures and saddle numbers. Let F be a surface‐link, that is, a closedoriented 2‐dimensional C^{\infty} submanifold of S^{4}.

We can assume F\subset \mathbb{R}^{4}.

Choose a projection p of \mathbb{R}^{4} onto a line. Assume that the critical points of the function
p|F are non‐degenerate. Denote by sdl(F) the minimal number of saddle points of p|Fover

all the projections p.

Definition 3.1. A saddle number sd(F) is the minimum of numbers sdl(F') where F'

ranges over all surface‐links F' ambiently isotopic to F.

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The invariant sd(F) is closely related to the ch‐index

by K. Yoshikawa in [11].

In particular,

we

of F

,

have sd(F)\leq ch(F)introduced and studied.

In order to relate thenumber sd(F) to \mathcal{M}\mathcal{N}(S^{4}\backslash F) we will reformulate the definition of the saddle number.
Let F\subset S^{4} be a surface‐link.

The equatorial 3‐sphere $\Sigma$^{3} of the standard Euclidean

sphere S^{4} divides S^{4} into two parts:S^{4}=D_{+}^{4}\cup D_{-}^{4}

with

,

D_{+}^{4}\cap D_{-}^{4}=$\Sigma$^{3}.

We assume that F is included in Int (D) and F does not contain the centre of D_{-}^{4} Per‐turbing the embedding F\subset D^{\underline{4} if necessary, we can assume that the restriction $\rho$=r|_{F} of
the radius function r:D^{\underline{4}}\rightarrow[0 1 ] is a Morse function. The family \{(r^{-1}(t), $\rho$^{-1}(t))\}_{t\in[0,1]}
of possibly singular links can be drawn as a motion picture (see [5], Chapter 8). Eachsingularity of a link in the family corresponds to a critical point of $\rho$ A critical point
of $\rho$ of index 0 (1, 2, respectively) is called minimal point (saddle point, maximal point,respectively) of $\rho$ which is represented by a minimal band (saddle band, maximal band,
.

,

.

,

respectively) in (a modification of) the motion picture.

It is clear that the minimal number of the saddle points for all such Morse functions $\rho$

and all surface‐links ambiently isotopic to F is equal to sd(F)Theorem 3.2.

.

\mathcal{M}\mathcal{N}(C(F))\leq 2sd(F)+ $\chi$(F)-2.

Corollary 3.3. Let K\subset S^{4} be a 2‐knot.

Then \mathcal{M}\mathcal{N}(C(K))\leq 2sd(K)Proposition 3.4. Let F\subset S^{4} be the trivial k ‐component surface‐link.

\mathcal{M}\mathcal{N}(C(F))=4k-2- $\chi$(F)

Then

.

It is not diffcult to show that

Proof.

.

\hat{b}_{1}(C(F))\geq k-1, \hat{b}_{3}(C(F))\geq k-1 Therefore.

for every regular Morse map f : C(F)\rightarrow S^{1} we have m_{1}(f)+m_{3}(f)\geq 2(k-1).

Assumingm_{0}(f)=m_{4}(f)=0 we have m_{1}(f)-m_{2}(f)+m_{3}(f)=2- $\chi$(F) and \mathcal{M}\mathcal{N}(C(F))\geq
4k-2- $\chi$(F) ; this lower bound coincides with the upper bound derived from Theorem
,

3.2.

3.2.

Spun knots. Let K be a classical knot in S^{3} ; denote by S(K) the corresponding

spun knot.

Proposition 3.5. If K is a non‐fibered knot of tunnel number 1, then \mathcal{M}\mathcal{N}(S^{4}\backslash S(K))=
4.

Recall that \mathcal{M}\mathcal{N}(S^{4}\backslash S(K))\leq 2\mathcal{M}\mathcal{N}(K)(Corollary 2.2). In the paper [7] of\mathcal{M}\mathcal{N}(C(K))\leq 2t(K) hence \mathcal{M}\mathcal{N}(C(S(K)))\leq 4
by Corollary 2.2. Put G=$\pi$_{1}(S^{3}\backslash K) then $\pi$_{1}(S^{4}\backslash S(K))\approx G ; let H=[G, G] Let
f : S^{4}\backslash S(K)\rightarrow S^{1} be a regular Morse map without minima and maxima. If m_{1}(f)=0,
then a standard Morse‐theoretic argument applied to the infinite cyclic cover of S^{4}\backslash S(K)Proof.

the second author it is shown that

,

,

.

implies that H is finitely generated, which is impossible, since K is not fibred. Thereforem_{1}(f)\geq 1 and similarly, m_{3}(f)\geq 1 hence m_{2}(f)\geq 2 and the proposition is proved. \square,,

3.3. Surface‐links of Yoshikawas table. A. Kawauchi, T. Shibuya and S. Suzuki [6]developed a method of representing surface‐links by diagrams. Based on this method K.

Yoshikawa [11] introduced a numerical invariant ch(F) of surface‐links F and enumerated
all the (weakly prime) surface‐links F with ch(F)\leq 10.

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FIGURE 1

It is clear from the definition of the invariant ch(F) that we have sd(F)\leq ch(F).

In

the rest of this section we assume that the reader is familiar with Yoshikawas work, andwith his terminology. There are 6 two‐knots in Yoshikawas table, namely

0_{1}, 8_{1}, 9_{1}, 10_{1}, 10_{2}, 10_{3}.
The trivial 2‐knot 0_{1} is obviously fibred. The knots 8_{1} and 10_{1} are spun knots of thetrefoil knot and respectively of the figure 8 knot, thus both 8_{1} and 10_{1} are fibred by [1].

The case of 9_{1} is more complicated. The saddle number of this 2‐knot is 2. Therefore\mathcal{M}\mathcal{N}(9_{1})\leq 4 Using the presentation of the fundamental group of the complement to 9_{1}(see [11]) and Poincaré duality properties it is easy to compute the Novikov numbers of
.

9_{1}

.

Namely we have \hat{q}_{1}=1, \hat{q}_{2}=\hat{q}_{3}=0 Therefore.

2\leq \mathcal{M}\mathcal{N}(9_{1})\leq 4.
The 2‐knot 10_{2} is the 2‐twist spun knot of the trefoil knot, hence fibered by Zeemanstheorem [12]. Similarly, 10_{3} is fibered, being the 3‐twist spun of the trefoil knot.

6_{1}^{0,1} is the result of spinning of the Hopf link which is fibred (see the\mathcal{M}\mathcal{N}(6_{1}^{0_{)}1})=0.
The surface‐link 8_{1}^{1,1} is the spun torus of the Hopf link. Applying Theorem 2.6
getthe upper bound \mathcal{M}\mathcal{N}(8_{1}^{1,1})\leq 2 Computing the Euler charcateristic implis the inverse
inequality,\mathcal{M}\mathcal{N}(8_{1}^{1,1})=2.
The surface‐link

left of Figure 2) therefore

we

.

so

The same argument applies to the surface‐link 10_{1}^{1} , which is the spun torus of the trefoil

knot, see the figure 2 (middle), so that \mathcal{M}\mathcal{N}(10_{1}^{1})=2.
The surface‐link 10_{1}^{0,1} is the result of spinning of the link 4_{1}^{2} which is fibred, therefore\mathcal{M}\mathcal{N}(10_{1}^{0,1})=0.

The case of the surface‐linkcomplicated. Applying
F=10_{1}^{0_{)}0,1} is\mathcal{M}\mathcal{N}(10_{1}^{0,0,1})=2.
more

ageneralisationof spinning constructions we prove that

4. ACKNOWLEDGEMENTS

This work was accomplished when the second author was visiting the Tokyo Institute ofTechnology in 2016 with the support of the JSPS fellowship. The first author was partially
supported by JSPS KAKENHI Grant Numbers 25400082, 16\mathrm{K}05142 The second author
thanks the Tokyo Institute of Technology for support and warm hospitality.
.

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Soc., 153 (1971), 415‐426.
[2] E. Artin, Zur Isotopie zweidimensionalen Flächen im R_{4} Abh. Math. Sem. Univ. Hamburg
4 (1926), 174‐177.
[3] G. Friedman, Alexander polynomials of non‐locally‐flat knots, Indiana Univ. Math. J. 52(2003), 1479—1578.
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Friedman, Knot Spinning, Handbook of Knot Theory, Elsevier, 2005, ch.4.
[4][5] S. Kamada, Braid and knot theory in dimension four, Math. Surveys Monogr. 95, Amer.
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Math. Soc., Providence, RI, 2002.

[6] A. Kawauchi, T. Shibuya and S. Suzuki, Descriptions on surfaces in four‐space, Normal
forms, Math. Sem. Notes Kobe Univ. 10 (1982), 75-125.
A.
Pajitnov, On the tunnel number and the Morse‐Novikov number of knots, Algebraic &[7]
Geometric Topology 10 (2010) 627—635.
[8] A. Pajitnov, Circle‐Valued Morse Theory (de Gruyter Studies in Mathematics 32).
[9] A.V. Pajitnov, C. Weber, L. Rudolph, Morse‐Novikov number for knots and links, Algebra \mathrm{i}Analiz, 13, no.3 (2001), (in Russian), English translation: Sankt‐Petersbourg Mathematical
Journal. 13, no.3 (2002), p. 417— 426.
[10] D. Roseman, Spinning knots about submanifolds; spinning knots about projections of knots,
Topology and Appl. 31 (1989), 225—241.
[11] K. Yoshikawa, An enumeration of surfaces in four‐space, Osaka J. Math. 31 (1994), 497‐522.

[12] E. C. Zeeman, Twisting spun knots Trans. Amer. Math. Soc. 115 (1965), 471—495.

Tokyo Institute of Technology
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JAPAN\mathrm{E}‐mail address:

endo@math.titech.ac.jp\ovalbox{\t \smal REJECT}\overline{\mathrm{F}_{\backslash} \mathrm{I}\ovalbox{\t \smal REJECT}\star^{\backslash}\neq^{\backslash}\mathrm{I}\ovalbox{\t \smal REJECT}^{\backslash}\not\equiv^{\backslash}$\beta$_{$\pi$}^{B})$\Sigma$\backslash\rightar ow\ovalbox{\t \smal REJECT}$\lambda$)\ovalbox{\t \smal REJECT},

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Faculté des Sciences,2, rue de la Houssinière,
44072, Nantes, Cedex\mathrm{E}‐mail address:

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