BRIDGE TRISECTIONS AND SEIFERT SOLIDS JASON JOSEPH JEFFREY MEIER MAGGIE MILLER AND ALEXANDER ZUPAN Abstract. We adapt Seiferts algorithm for classical knots and links to the

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BRIDGE TRISECTIONS AND SEIFERT SOLIDS

JASON JOSEPH, JEFFREY MEIER, MAGGIE MILLER, AND ALEXANDER ZUPAN

Abstract. We adapt Seifert’s algorithm for classical knots and links to the

setting of tri-plane diagrams for bridge trisected surfaces in the 4–sphere.

Our approach allows for the construction of a Seifert solid that is described

by a Heegaard diagram. The Seifert solids produced can be assumed to have

exteriors that can be built without 3–handles; in contrast, we give examples

of Seifert solids (not coming from our construction) whose exteriors require

arbitrarily many 3–handles. We conclude with two classiﬁcation results.

The ﬁrst shows that surfaces admitting doubly-standard shadow diagrams

are unknotted. The second says that a b–bridge trisection in which some

sector contains at least b−1 patches is completely decomposable, thus the

corresponding surface is unknotted. This settles aﬃrmatively a conjecture

of the second and fourth authors.

1. Introduction

Abridge trisection of a surface Sin S4is a certain decomposition of (S4,S)

into three trivial disk systems (B4

1,D1),(B4

2,D2),(B4

3,D3) that can be encoded

diagrammatically either as a triple of tangles called a tri-plane diagram or as

a corresponding shadow diagram. In this paper, we show how topological

information about the surfaces Scan be recovered from these diagrammatic

representations.

In Section 3, we give a version of Seifert’s algorithm for bridge-trisected

surfaces, showing how a tri-plane diagram can be used to produce a 3–manifold

bounded by a connected surface Swith normal Euler number zero.

Theorem 3.4. If Sis connected and e(S) = 0, then there is a procedure to

produce a Seifert solid for Sthat takes as input a tri-plane diagram for S.

In Subsection 3.2, we give an explicit procedure for constructing a Heegaard

diagram for such a 3–manifold when S∼

=S2. As a corollary of the work in

building Seifert solids, we recover a combinatorial proof of the existence of

Seifert solids. We also show that certain bridge trisected surfaces are unknot-

ted.

Theorem 3.3. If a surface Shas a doubly-standard shadow diagram, then S

is unknotted.

arXiv:2210.09669v1 [math.GT] 18 Oct 2022

2 JOSEPH, MEIER, MILLER, AND ZUPAN

In Section 4, we give 4–dimensional analogs to the 3–dimensional concepts

of free Seifert surfaces and canonical Seifert surfaces. We call a Seifert solid

canonical if it is obtained from the procedure presented in Section 3, and we

call a Seifert solid spinal if its exterior in S4can be built without 3–handles.

We prove the following two results relating (and distinguishing) these concepts.

Theorem 4.1. If a surface-knot Sadmits a Seifert solid, then it admits a

canonical Seifert solid that is spinal.

In fact, modulo some additional, easily satisﬁed connectivity conditions,

every canonical Seifert solid is spinal. The next result shows that some Seifert

solids (in contrast to canonical Seifert solids and many standard examples) are

“far” from being spinal.

Theorem 4.2. Given any n∈N, there exists a 2–knot Kthat bounds a Seifert

solid Yhomeomorphic to (S1×S2)◦such that S4\ν(Y)requires at least n

4–dimensional 3–handles.

Finally, in Section 5 we prove the following standardness result, aﬃrmatively

settling Conjecture 4.3 of [MZ17].

Theorem 5.2. Let Tbe a (b;c)–bridge trisection with ci=bi−1for some

i∈Z3. Then, Tis completely decomposable, and the underlying surface-link

is either the unlink of min{ci}2–spheres or the unlink of min{ci}2–spheres

and one projective plane, depending on whether |ci−1−ci+1|= 1 or 0.

The proof relies on theorems of Scharlemann and Bleiler-Scharlemann re-

garding planar surfaces in 3–manifolds [BS88, Sch85]. The second and fourth

authors previously handled this case when ci=bfor some i∈Z3[MZ17,

Proposition 4.1].

Acknowledgements. This paper began following discussions at the work-

shop Unifying 4–Dimensional Knot Theory, which was hosted by the Banﬀ

International Research Station in November 2019, and the authors would like

to thank BIRS for providing an ideal space for sparking collaboration. We are

grateful to Masahico Saito for sharing his interest in adapting Seifert’s algo-

rithm to bridge trisections and motivating this paper. The authors would like

to thank Rom´an Aranda, Scott Carter, and Peter Lambert-Cole for helpful

conversations. JJ was supported by MPIM during part of this project, as well

as NSF grants DMS-1664567 and DMS-1745670. JM was supported by NSF

grants DMS-1933019 and DMS-2006029. MM was supported by MPIM during

part of this project, as well as NSF grants DGE-1656466 (at Princeton) and

DMS-2001675 (at MIT) and a research fellowship from the Clay Mathemat-

ics Institute (at Stanford). AZ was supported by MPIM during part of this

project, as well as NSF grants DMS-1664578 and DMS-2005518.

BRIDGE TRISECTIONS AND SEIFERT SOLIDS 3

2. Preliminaries

We work in the smooth category. This section includes an abbreviated

introduction to the concepts relevant to this paper, but the interested reader

is encouraged to consult the reference [GK16] for further information about

4–manifold trisections and the references [MZ17] and [JMMZ22, Section 2]

for more detailed discussions of bridge trisections. We limit our work here

to surfaces in S4, but there is also a theory of bridge trisections in arbitrary

4–manifolds; see [MZ18].

2.1. Bridge trisections. Let Sbe an embedded surface in S4, let bbe a

positive integer, and let c= (c1, c2, c3) be a triple of positive integers. A

(b;c)–bridge trisection of (S4,S) is a decomposition

(S4,S)=(X1,D1)∪(X2,D2)∪(X3,D3)

such that

(1) Each Diis a collection of ciboundary-parallel disks in the 4–ball Xi;

(2) Each intersection Ti=Di−1∩ Dia boundary-parallel tangle in the

3–ball Hi=Xi−1∩Xi(with indices considered mod 3);

(3) The triple intersection D1∩ D2∩ D3is a collection of bpoints in the

2–sphere Σ = X1∩X2∩X3.

In [MZ17], it was proved that every surface Sadmits a (b;c)–bridge trisec-

tion for some (b;c). We choose orientations so that ∂(Xi,Di) = (Hi,Ti)∪

(Hi+1,Ti+1). When we wish to be succinct, we use Tto represent a bridge

trisection, with components labeled as above.

2.2. Diagrams for bridge trisections. The existence of bridge trisections

gives rise to a new diagrammatic theory for surfaces in S4, using an object

called a tri-plane diagram, a triple D= (D1,D2,D3) of trivial planar diagrams

with the additional condition that each Di∪Di+1 is a classical diagram for

an unlink. In [MZ17], it was shown that every tri-plane diagram determines

a bridge trisection T. Conversely, given a bridge trisection Tof (S4,S), we

can choose a triple of disks Ei⊂Hiwith common boundary and project

the tangles Tionto Eito obtain a tri-plane diagram. Of course, the choices

of disks and projections are not unique, but any two tri-plane diagrams cor-

responding the same bridge trisection Tare related by a ﬁnite collection of

interior Reidemeister moves and mutual braid transpositions, while any two

bridge trisections Tand T0for the same surface Sare related by perturbation

and deperturbation moves.

In addition, bridge trisections yield another type of diagram: Each trivial

tangle Tican be isotoped rel-boundary into the surface Σ, yielding a triple

(A, B, C) of pairwise disjoint collections of arcs called a shadow diagram, which

4 JOSEPH, MEIER, MILLER, AND ZUPAN

has the property that ∂A =∂B =∂C, and the pairwise unions of any two of

the tangles TA,TB,TCdetermined by the arcs are unlinks. As with tri-plane

diagrams, any shadow diagram determines a bridge trisection. Further details

about shadow diagrams can be found in [MTZ20].

Here we consider special types of shadow diagrams. We say that a pair

of collections of arcs in a shadow diagram is standard if their union is em-

bedded. Any bridge trisection admits a shadow diagram (A, B, C) in which

one of the pairs is standard. If two or three pairs of shadows in a shadow

diagram (A, B, C) are standard, then we say that (A, B, C) is doubly-standard

or triply-standard, respectively. Theorem 3.3 says that doubly-standard (and

thus triply-standard) diagrams always describe unknotted surfaces.

2.3. Unknotted surfaces. In this subsection, we review standard notions

of unknottedness for surfaces in S4. A closed, connected, surface Sin S4is

unknotted if it bounds an embedded 3–dimensional handlebody H⊂S4. For

nonorientable surfaces, the deﬁnition is slightly more involved. We deﬁne the

two unknotted projective planes, P±, to be the two standard projective planes

in S4, pictured via their tri-plane diagrams in Figure 1, where e(P±) = ±2.

Figure 1. Tri-plane diagrams for P+and P−.

In general, for a nonorientable surface S, we say that Sis unknotted if

Sis isotopic to a connected sum of some number of copies of P+and P−.

See [JMMZ22, Remark 2.6] for a detailed discussion of the orientation conven-

tions used here.

3. Seifert solids

Classical results of Gluck [Glu62] (resp., Gordon-Litherland [GL78]) assert

that every orientable surface S(resp., surface Swith e(S) = 0) in S4bounds

an embedded 3–manifold, called a Seifert solid in the orientable case. In the

setting of broken surface diagrams, Carter and Saito provided a procedure that

in many respects mimics Seifert’s algorithm for classical knots [CS97]. In this

section, we describe an extension of Seifert’s algorithm that takes an oriented

tri-plane diagram Dand produces a Seifert solid whose intersection with ∂Xi

agrees with the classical Seifert’s algorithm performed on the oriented unlink

diagram Di∪Di+1. We also obtain alternative proofs of the theorems of Gluck

and Gordon-Litherland mentioned above.

BRIDGE TRISECTIONS AND SEIFERT SOLIDS 5

3.1. Existence of Seifert solids. Given a spanning surface Ffor an unlink

U, we deﬁne the cap-oﬀ Fof Fto be the closed surface F ⊂ S4obtained by

gluing a collection of trivial disks in B4

−to Falong U. (There is a unique such

choice of disks up to isotopy rel-boundary in B4

−by e.g. [KSS82] or [Liv82].)

Let F+⊂S3denote the M¨obius band bounded by the unknot so that F+

contains a positive half-twist and has boundary slope +2, and let F−⊂S3

denote the M¨obius band bounded by the unknot with a negative half-twist and

boundary slope −2. For n > 0, let Fnbe the connected surface obtained by

attaching n−1 trivial bands to the split union of ncopies of F+; that is, Fnis

obtained by taking the boundary connected sum of ncopies of F+. For n < 0,

let Fnbe obtain by taking the boundary connected sum of (−n) copies of F−.

Finally, let F0be the disk bounded by the unknot in S3. Additionally, let Fn

be the cap-oﬀ of Fn. In Figure 1, the negative M¨obius band is shown to cap oﬀ

into B4

+to obtain P+. (See also [JMMZ22, Figure 2].) Here, we are capping

oﬀ into B4

−, so that by deﬁnition the cap-oﬀ F−1of the negative M¨obius band

F−is P−. In contrast, the cap-oﬀ F1of the positive M¨obius band F+is P+.

(Recall that P+and P−denote the two unknotted projective planes in S4; see

Subsection 2.3.) It follows that

Fn=









a connected sum of ncopies of P+,if n > 0

a connected sum of −ncopies of P−if n < 0

an unknotted 2–sphere if n= 0

The intent of the cap-oﬀ notation is the emphasize the way in which Fncan

be obtained from a speciﬁc surface in S3, which will be useful in the rest of

this section – especially given the following lemma.

Lemma 3.1. Every incompressible spanning surface Ffor the unknot is iso-

topic to Fnfor some n∈Z.

Proof. First, we argue that Fnis incompressible for all n. This follows from

[Tsa92], but we include a proof here. Certainly, F0and F±1are incompressible,

since a compression increases Euler characteristic by two. Suppose now that

Fnis compressible for some n > 1, and let F0

nbe the component of the surface

obtained by compressing Fnsuch that ∂F 0

n=∂Fn. In addition, let F0

n⊂S4

be the cap-oﬀ of F0

n. Then the embedded surface Fncan be obtained by

from F0

nby a 1–handle attachment, and thus e(F0

n) = e(Fn)=2n. However,

since the nonorientable genus of F0

nis strictly less than n, this contradicts the

Whitney–Massey Theorem (see discussion in [JMMZ22]). We conclude that

Fnis incompressible.

On the other hand, suppose that Fis an arbitrary incompressible spanning

surface for the unknot U. The exterior of Uis a solid torus V, and every simple

closed curve c⊂∂V is homotopic to a (p, q)–curve, where a (0,1)–curve is the

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BRIDGETRISECTIONSANDSEIFERTSOLIDSJASONJOSEPH,JEFFREYMEIER,MAGGIEMILLER,ANDALEXANDERZUPANAbstract.WeadaptSeifert'salgorithmforclassicalknotsandlinkstothesettingoftri-planediagramsforbridgetrisectedsurfacesinthe4{sphere.OurapproachallowsfortheconstructionofaSeifertsolidthatisdescribedbyaHeegaarddiagra...

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