A NOTE ON SURFACES IN CP2AND CP2CP2 MARCO MARENGON ALLISON N. MILLER ARUNIMA RAY AND ANDR AS I. STIPSICZ Abstract. In this brief note we investigate the CP2-genus of knots i.e. the least genus of a smooth

2025-04-30 0 0 418.79KB 9 页 10玖币

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A NOTE ON SURFACES IN CP2AND CP2#CP2

MARCO MARENGON, ALLISON N. MILLER, ARUNIMA RAY, AND ANDR ´

AS I. STIPSICZ

Abstract. In this brief note, we investigate the CP2-genus of knots, i.e. the least genus of a smooth,

compact, orientable surface in CP2∖˚

B4bounded by a knot in S3. We show that this quantity is

unbounded, unlike its topological counterpart. We also investigate the CP2-genus of torus knots. We

apply these results to improve the minimal genus bound for some homology classes in CP2#CP2.

1. Introduction

The genus function GXof a smooth 4-manifold X, as a function H2(X;Z)→Z≥0, is deﬁned for

α∈H2(X;Z) as

GX(α) = min{g(F)|i:F→X, i∗([F]) = α},(1.1)

where the minimum is taken over all smooth embeddings iof smooth, closed, oriented surfaces F. A

triumph of modern gauge theory consists of Kronheimer and Mrowka’s determination of GCP2, i.e. the

solution to the minimal genus problem in CP2, also called the Thom conjecture [KM94]. They showed

that if h∈H2(CP2;Z)∼

=Zis a generator and d̸= 0 is an integer, then

GCP2(d·h) = (|d| − 1)(|d| − 2)

and GCP2(0) = 0. In this paper we will consider two generalisations of this: in one direction we will

study a relative version of the genus bound, for surfaces with boundary, embedded in a punctured CP2

(Subsection 1.1); in another direction, we will examine the classical minimal smooth genus problem in

the connected sum of two copies of CP2(Subsection 1.3).

1.1. Relative minimal genus problem in CP2.The CP2-genus of a knot K⊆S3, denoted by

gCP2(K), is the least genus of a smooth, compact, connected, orientable, properly embedded surface

F⊆CP2∖˚

B4=: (CP2)×bounded by K⊆∂(CP2)×∼

=S3. Similarly, there is the topological CP2-genus,

denoted by gtop

CP2(K), the least genus of a surface Fas above which is only locally ﬂat and embedded. A

knot Kis said to be slice in CP2if gCP2(K) = 0, and topologically slice in CP2if gtop

CP2(K) = 0.

It was shown in [KPRT22, Corollary 1.12 (2)] that gtop

CP2(K)≤1 for every knot K. By contrast, we

show that gCP2is unbounded.

Theorem 1.1. Let n≥0. There exists a topologically slice knot Kwith gCP2(K)≥n.

This result answers [AN09, Question 2.1], which asked for a knot which is topologically slice in CP2but

not smoothly slice. The largest value of gCP2previously in the literature was gCP2(T2,17) = 7 from [AN09,

Theorem 1.2]. Our examples can be taken to be connected sums of the untwisted, negative clasped

Whitehead double of the left-handed trefoil knot. Since these are topologically slice in B4, they have

trivial gtop

CP2.

On the constructive side, we also give a method to ﬁnd knots with trivial gtop

CP2, that are not necessarily

topologically slice in B4.

Theorem 1.2. Let K⊆S3be a knot. If Arf(K) = 0 then gtop

CP2(K)=0.

Note that the above result does not give a complete characterization of knots which are topologically

slice in CP2, since for example the right-handed trefoil Tsatisﬁes gCP2(T) = gtop

CP2(T) = 0, as an unknotting

number one knot, but has Arf(T) = 1. The proof of Theorem 1.2 uses a result of Freedman and Quinn on

when an immersed disk with an algebraically dual sphere is homotopic to a locally ﬂat embedding [FQ90,

Theorem 10.5 (1)] (see Theorem 2.6).

2020 Mathematics Subject Classiﬁcation. 57R40, 57K10, 57N70,

arXiv:2210.12486v2 [math.GT] 19 Mar 2024

2 M. MARENGON, A. N. MILLER, A. RAY, AND A. I. STIPSICZ

The CP2-genus was previously studied by [Yas91,Yas92,AN09,Pic19]. Speciﬁcally, Yasuhara [Yas91,

Yas92] studied the sliceness of certain torus knots in CP2. Pichelmeyer [Pic19] studied the CP2-genus for

low crossing knots and alternating knots. Ait Nouh showed in [AN09] that for 3 ≤q≤17,

gCP2(T2,q) = g4(T2,q )−1 = q−3

and asked whether the equality

gCP2(Tp,q) = g4(Tp,q )−1 = (p−1)(q−1)

2−1

holds for all p, q > 0. Theorem 2.8 shows that this is not the case, and yields the following corollary.

Corollary 1.3. For every n≥1,

gCP2(Tn,n−1)≤(g4(Tn,n−1)−n−2

2if n≡0 mod 2

g4(Tn,n−1)−n−1

2if n≡1 mod 2.

Indeed the above inequality holds for the least genus of null-homologous surfaces bounded by Tn,n−1,

as we indicate in the proof.

1.2. Relative minimal genus problems in other 4-manifolds. Given a closed, smooth 4-manifold

M, let M×denote the punctured manifold M∖˚

B4. The M-genus of a knot K⊆S3, denoted by gM(K),

generalizes the deﬁnition of Equation (1.1) and is deﬁned as

gM(K) = min{g(Σ) |i: Σ →M×, i(∂Σ) = K},

where the minimum is taken over all smooth, compact, orientable, properly embedded surface Σ ⊆M×

bounded by K⊆∂M×∼

=S3. A knot Kis said to be slice in Mif gM(K) = 0. One may also consider

the topological M-genus, denoted by gtop

M(K), the least genus of a surface Fas above which is only locally

ﬂat and embedded. Note that gM(K)≤gS4(K) and gtop

M(K)≤gtop

S4(K) for all M.

The smooth and topological S4-genera correspond to the usual smooth and topological slice genera

of knots and have been studied extensively. In particular, there exist inﬁnitely many knots with trivial

topological S4-genus and nontrivial smooth S4-genus [Gom86,End95]. Any such knot can be used to

produce a nonstandard smooth structure on R4[Gom85] and in general slicing knots in S4is connected to

major open questions in 4-manifold topology, such as the smooth Poincar´e conjecture and the topological

surgery conjecture [FGMW10,CF84] (see also [KOPR21]). Slicing knots in more general 4-manifolds

has also been fruitful, e.g. in revealing structure within the knot concordance group [COT03,COT04,

CT07,CHL09,CHL11], and in distinguishing between smooth concordance classes of topologically slice

knots [CHH13,CH15,CK21]. In [MMP20] it was shown that the set of knots which bound smooth, null-

homologous disks in a 4-manifold Mcan distinguish between smooth structures on M, that is, there are

examples of homeomorphic smooth 4-manifolds M1, M2and a knot K⊂S3which bounds a smooth,

null-homologous disk in M×

1, but does not bound such a disk in M×

2. It is an open question whether the

set of slice knots can distinguish between smooth structures on a manifold.

It was shown in [KPRT22, Corollary 1.12] using Freedman’s disk embedding theorem [Fre82,FQ90]

that gtop

M(K) = 0 for every closed, simply connected 4-manifold Mother than S4,CP2,CP2,∗CP2,

and ∗CP2. Here Mdenotes the 4-manifold Mwith its orientation reversed, and ∗CP2is the topological

4-manifold homotopy equivalent, but not homeomorphic, to CP2, constructed by Freedman in [Fre82].

As mentioned before, gtop

CP2(K)≤1 for every knot K. Moreover, gtop

∗CP2(K)≤1 for every knot Kas well.

Further, gtop

CP2(K) = gtop

CP2(−K) for every knot K, where −Kis the mirror image. Therefore, the topological

problem for closed, simply connected 4-manifolds with positive second Betti number is better understood.

By contrast, many open questions remain in the smooth setting. It is a straightforward consequence of the

Norman trick that all knots in S3are slice in both S2×S2and S2e

×S2∼

=CP2#CP2[Nor69, Corollary 3 and

Remark; Suz69, Theorem 1] (see also [Ohy94,Liv21]). By Theorem 1.1 the CP2-genus can be arbitrarily

large. It is open whether there is a knot that is not slice in the K3-surface – if such a knot exists, its

unknotting number must be more than 21 [MM22].

1.3. Minimal genus problem in CP2#CP2.Above we provided a possible extension of the Thom

conjecture from CP2to (CP2)×, by replacing closed surfaces with surfaces having boundary knotted in

S3=∂(CP2)×. Another extension of Kronheimer-Mrowka’s seminal result on GCP2would be the deter-

mination of the corresponding function for other closed 4-manifolds, for example GCP2#CP2of CP2#CP2.

(The cases of CP2#CP2and S2×S2were resolved by Ruberman in [Rub96].) Gauge-theoretic methods

are less eﬀective for this 4-manifold – the lower bounds, resting on variants of Furuta’s 10

8-theorem, from

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ANOTEONSURFACESINCP2ANDCP2#CP2MARCOMARENGON,ALLISONN.MILLER,ARUNIMARAY,ANDANDR´ASI.STIPSICZAbstract.Inthisbriefnote,weinvestigatetheCP2-genusofknots,i.e.theleastgenusofasmooth,compact,orientablesurfaceinCP2∖˚B4boundedbyaknotinS3.Weshowthatthisquantityisunbounded,unlikeitstopologicalcounterpart.Weals...

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A NOTE ON SURFACES IN CP2AND CP2CP2 MARCO MARENGON ALLISON N. MILLER ARUNIMA RAY AND ANDR AS I. STIPSICZ Abstract. In this brief note we investigate the CP2-genus of knots i.e. the least genus of a smooth

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