Banded Unlink Diagrams in 4-Manifolds

• Banded Unlink Diagrams are a framework that records Kirby handle data and Morse level sets to represent smooth surface embeddings in 4-manifolds.
• The technique employs bands to modify unlinks via moves like band slides and cup/cap operations, preserving Morse-theoretic structures while achieving isotopy classification.
• This diagrammatic calculus unifies classical S⁴ approaches with bridge trisections and quandle presentations, providing actionable insights into surface isotopy and handle interactions.

Searching arXiv for the cited works on banded unlink diagrams and related developments. Banded unlink diagrams are a diagrammatic framework for studying smoothly embedded surfaces in smooth, oriented $4$-manifolds by combining Kirby calculus with Morse-theoretic level sets. In the Hughes–Kim–Miller formulation, a banded unlink diagram for a surface $S \subset X$ records, at the intermediate level $M_{3/2}$ of a self-indexing Morse function $h:X\to[0,4]$, an unlink together with attached bands inside the Kirby-diagram exterior $E(\mathcal K)$, so that the unlink caps below and its band resolution caps above. The framework yields a complete calculus for isotopy in arbitrary $4$-manifolds, extends the Swenton–Kearton–Kurlin calculus from $S^4$, and interacts directly with bridge trisections and applications to unit surfaces in $\mathbb{C}P^2$ (Hughes et al., 2018).

1. Definition in an arbitrary $4$-manifold

Fix a smooth, oriented, closed $4$-manifold $S \subset X$0 together with a self-indexing Morse function $S \subset X$1 with a single index-$S \subset X$2 and index-$S \subset X$3 critical point. Let $S \subset X$4 denote the subhandlebody consisting of handles of index $S \subset X$5, and write $S \subset X$6. In particular, $S \subset X$7 and $S \subset X$8. Let $S \subset X$9 be a Kirby diagram for $M_{3/2}$0: a dotted unlink $M_{3/2}$1 encoding $M_{3/2}$2-handles together with a framed link $M_{3/2}$3 encoding $M_{3/2}$4-handles. Set

$M_{3/2}$5

and identify $M_{3/2}$6 with the $M_{3/2}$7-surgery on $M_{3/2}$8 and $M_{3/2}$9 with the further surgery on $h:X\to[0,4]$0 (Hughes et al., 2018).

A band in a $h:X\to[0,4]$1-manifold $h:X\to[0,4]$2 is the image of an embedding $h:X\to[0,4]$3 with $h:X\to[0,4]$4 for some link $h:X\to[0,4]$5; the core is $h:X\to[0,4]$6. If $h:X\to[0,4]$7 is a finite, pairwise disjoint family of bands for $h:X\to[0,4]$8, resolving along $h:X\to[0,4]$9 produces the link $E(\mathcal K)$0 obtained by the usual band-surgery. A banded unlink diagram in $E(\mathcal K)$1 is then a triple $E(\mathcal K)$2 with $E(\mathcal K)$3 and $E(\mathcal K)$4 a finite family of disjoint bands in $E(\mathcal K)$5, such that $E(\mathcal K)$6 bounds a union of disjoint embedded disks in $E(\mathcal K)$7 and $E(\mathcal K)$8 bounds a union of disjoint embedded disks in $E(\mathcal K)$9 (Hughes et al., 2018).

The same definition can be stated more succinctly as follows: a banded unlink diagram consists of an unlink $4$0 together with a finite set of embedded bands $4$1, where the corresponding surface is obtained by attaching tubes along $4$2 and capping with disks after $4$3-handle attachments. This formulation emphasizes that the diagram records the $4$4-, $4$5-, and $4$6-handle data of the surface relative to the handle decomposition of the ambient $4$7-manifold (Hughes et al., 2018).

When $4$8, the Kirby diagram is empty. In that case the framework reduces to the classical banded unlink description in $4$9, and the only relevant moves are the Yoshikawa-type moves already known in the $S^4$0 setting (Hughes et al., 2018). A distinct earlier usage of the term occurs in link theory, where a banded unlink diagram means an unlink of disks with bands attached, viewed as a planar description of an unknotted flat banded surface whose boundary is a link (Kim et al., 2011). This suggests a historical continuity of terminology, but the $S^4$1-manifold theory adds explicit interaction with $S^4$2- and $S^4$3-handles.

2. Reconstruction and normal forms

Given a banded unlink diagram $S^4$4, the represented surface is reconstructed by viewing $S^4$5 in $S^4$6, disjoint from the descending manifolds of the $S^4$7-handles, equivalently inside $S^4$8. One vertically extends $S^4$9 down to $\mathbb{C}P^2$0 and caps with disks, extends $\mathbb{C}P^2$1 up to $\mathbb{C}P^2$2 and caps with disks, and takes the portions between $\mathbb{C}P^2$3, $\mathbb{C}P^2$4, and $\mathbb{C}P^2$5 to be vertical. The resulting embedded surface is denoted $\mathbb{C}P^2$6 (Hughes et al., 2018).

A central structural point is that arbitrary embedded surfaces can be isotoped into banded unlink position. In this position the surface is vertical between the levels $\mathbb{C}P^2$7 and $\mathbb{C}P^2$8, the intersection with $\mathbb{C}P^2$9 is a banded unlink disjoint from the $4$0-handle descending manifolds, and the intersections with $4$1 and $4$2 are unions of disks. If $4$3 is already in banded unlink position, the associated diagram $4$4 recovers $4$5 up to isotopy; this is the content of Lemma 2.7 (Hughes et al., 2018).

The proof of the isotopy calculus proceeds through a hierarchy of normal forms. The first is horizontal–vertical position, with minima below all saddles below maxima. From such a position, bands can be repositioned to $4$6 while avoiding the ascending and descending manifolds of the ambient handles; the choices made in this procedure differ by Morse-preserving band moves. For a generic surface, meaning $4$7 is Morse with distinct critical values, isotoping to horizontal–vertical form yields a well-defined banded unlink diagram up to Morse-preserving band moves (Hughes et al., 2018).

The Morse-theoretic viewpoint is essential. An $4$8-disjoint isotopy through generic surfaces preserves the diagram up to Morse-preserving moves, while the only singularity that introduces a non-Morse-preserving change is an $4$9 singularity, which produces exactly one cup or cap move. The decomposition of an arbitrary isotopy into segments separated by finitely many $4$0 and $4$1 singularities is the mechanism that upgrades local movie arguments into a complete global calculus (Hughes et al., 2018).

The framework is also algorithmic in a direct sense. To check that $4$2 is valid, one verifies that $4$3 bounds disjoint disks in $4$4, that $4$5 bounds disjoint disks in $4$6, and that $4$7. To compare two diagrams, one uses isotopy in $4$8, dotted circle slides to control interactions with $4$9-handles, band slides and swims to control band–band interactions, and $S \subset X$00-handle–band slides and swims to control interactions with $S \subset X$01-handles, with cup or cap moves added only when births or deaths occur (Hughes et al., 2018).

3. Local calculus and the completeness theorem

The local move set on banded unlink diagrams consists of ambient isotopies in $S \subset X$02 together with explicit elementary moves. Cup and cap moves create or cancel a local minimum or maximum of the surface by adding or removing a trivial disk to or from $S \subset X$03. Band slides move an endpoint of one band along the core of another. Band swims pass a band lengthwise through the interior of another. Three further families encode the ambient handle structure: $S \subset X$04-handle–band slides, dotted circle slides over $S \subset X$05-handles, and $S \subset X$06-handle–band swims (Hughes et al., 2018).

Move	Effect	Morse-preserving
Cup/Cap	Create or cancel a local minimum/maximum	No
Band slide / band swim	Change band attachments or pass bands through bands	Yes
$S \subset X$07-handle–band slide / $S \subset X$08-handle–band swim	Modify interaction with $S \subset X$09	Yes
Dotted circle slides	Modify interaction with $S \subset X$10	Yes
Isotopy in $S \subset X$11	Ambient simplification in the Kirby-diagram exterior	Yes

The terminology “Morse-preserving band moves” refers precisely to band slide, band swim, $S \subset X$12-handle–band slide, dotted circle slides, $S \subset X$13-handle–band swim, and isotopy in $S \subset X$14. Cup and cap moves are excluded because they alter the critical set of $S \subset X$15 (Hughes et al., 2018).

The central classification statement is Theorem 3.1: if $S \subset X$16 and $S \subset X$17 are embedded surfaces in $S \subset X$18, and $S \subset X$19 and $S \subset X$20 are banded unlink diagrams for them, then $S \subset X$21 is ambiently isotopic to $S \subset X$22 if and only if $S \subset X$23 can be transformed into $S \subset X$24 by a finite sequence of band moves (Hughes et al., 2018). This theorem generalizes the Swenton–Kearton–Kurlin calculus from $S \subset X$25 to arbitrary $S \subset X$26-manifolds.

The relation to the $S \subset X$27 case is exact. When $S \subset X$28, $S \subset X$29 is empty, so the general calculus reduces to cup/cap, band slides, band swims, and ambient isotopy in $S \subset X$30. The only genuinely new moves in the arbitrary-$S \subset X$31 setting are those encoding interaction with the $S \subset X$32- and $S \subset X$33-handles of the ambient manifold: dotted circle slides, $S \subset X$34-handle–band slides, and $S \subset X$35-handle–band swims (Hughes et al., 2018). This sharply delineates what changes when one passes from $S \subset X$36 to a general Kirby-presented $S \subset X$37-manifold.

A basic example appears in $S \subset X$38, where Figure 1.1 of the paper gives $S \subset X$39 with $S \subset X$40 a $S \subset X$41-component unlink in $S \subset X$42 and $S \subset X$43 consisting of four bands. The Euler characteristic is computed as $S \subset X$44, and the orientability is evident from the band attachments, so the surface is a torus. Since $S \subset X$45 bounds two capping disks in $S \subset X$46, the construction yields an embedded $S \subset X$47 in $S \subset X$48 (Hughes et al., 2018).

4. Bridge trisections and uniqueness up to perturbation

A trisection of $S \subset X$49 is a decomposition $S \subset X$50 with $S \subset X$51, pairwise intersections $S \subset X$52-dimensional handlebodies of genus $S \subset X$53, and triple intersection a closed surface $S \subset X$54. A surface $S \subset X$55 is in $S \subset X$56-bridge position with respect to a trisection $S \subset X$57 if $S \subset X$58 is a union of $S \subset X$59 boundary-parallel disks and $S \subset X$60 is a trivial $S \subset X$61-strand tangle, with

$S \subset X$62

(Hughes et al., 2018).

The paper proves the Meier–Zupan conjecture in full generality. A perturbation is the standard local move that increases $S \subset X$63 by $S \subset X$64 by compressing along a suitable disk $S \subset X$65 in one sector $S \subset X$66, and simultaneously increases $S \subset X$67 by $S \subset X$68; a deperturbation is the inverse move. Theorem 4.3 states that if $S \subset X$69 and $S \subset X$70 are surfaces in bridge position with respect to a trisection $S \subset X$71 of $S \subset X$72 and $S \subset X$73 is isotopic to $S \subset X$74, then $S \subset X$75 can be taken to $S \subset X$76 by a sequence of perturbations and deperturbations, followed by a $S \subset X$77-regular isotopy (Hughes et al., 2018).

The proof uses an explicit dictionary between banded unlink diagrams and bridge trisections relative to a Heegaard splitting of $S \subset X$78. Any banded unlink can be put into bridge position with respect to a chosen splitting $S \subset X$79, with $S \subset X$80 in a core of $S \subset X$81 and $S \subset X$82 in a core of $S \subset X$83, and conversely a bridged surface determines a banded unlink in $S \subset X$84. The completeness theorem for banded unlink diagrams is then translated into sequences of perturbations, deperturbations, and $S \subset X$85-regular isotopies (Hughes et al., 2018).

This establishes banded unlink diagrams as a bridge between Kirby calculus and trisection theory. A plausible implication is that the diagrammatic calculus is not merely a presentation tool for surfaces, but a transport mechanism between different decompositional languages for $S \subset X$86-manifolds. The paper itself states the qualitative uniqueness theorem, while quantitative bounds on the number of perturbations or deperturbations required are not addressed (Hughes et al., 2018).

5. Unit surfaces in $S \subset X$87 and the Gluck twist

Write $S \subset X$88 as $S \subset X$89 and its standard complex line $S \subset X$90 as $S \subset X$91. A unit surface is an embedded $S \subset X$92-sphere $S \subset X$93 representing the generator $S \subset X$94 and intersecting $S \subset X$95 transversely in a single point; equivalently $S \subset X$96 and $S \subset X$97. For a surface $S \subset X$98, the associated unit surface is $S \subset X$99, obtained by placing $M_{3/2}$00 in a ball in $M_{3/2}$01 and tubing $M_{3/2}$02 once to $M_{3/2}$03 (Hughes et al., 2018).

The banded unlink calculus yields several explicit standardization results. If $M_{3/2}$04 is a ribbon surface of genus $M_{3/2}$05, then $M_{3/2}$06 is isotopic to $M_{3/2}$07, where $M_{3/2}$08 is an unknotted torus in $M_{3/2}$09. More generally, if $M_{3/2}$10 and $M_{3/2}$11 are $M_{3/2}$12-concordant, then $M_{3/2}$13 is isotopic to $M_{3/2}$14; in particular, if $M_{3/2}$15 is $M_{3/2}$16-concordant to the unknot, then $M_{3/2}$17 (Hughes et al., 2018).

For twist-spun and deform-spun knots, the paper proves that if $M_{3/2}$18 is a $M_{3/2}$19-knot and $M_{3/2}$20 is its $M_{3/2}$21-twist spin, then $M_{3/2}$22 for all $M_{3/2}$23. More generally, if $M_{3/2}$24 is any deform-spun $M_{3/2}$25-knot, then $M_{3/2}$26 for all $M_{3/2}$27. A further corollary states that if $M_{3/2}$28 admits an integral lens space surgery, then $M_{3/2}$29 for all $M_{3/2}$30 (Hughes et al., 2018).

The calculus also trivializes certain band-sums and satellite constructions. If $M_{3/2}$31 is a band-sum of disjoint surfaces $M_{3/2}$32, then $M_{3/2}$33. For a satellite $M_{3/2}$34 of companion $M_{3/2}$35 with pattern $M_{3/2}$36 representing $M_{3/2}$37 in homology, the paper proves: if $M_{3/2}$38, then $M_{3/2}$39; if $M_{3/2}$40, then $M_{3/2}$41; if $M_{3/2}$42, then $M_{3/2}$43, where $M_{3/2}$44 is the satellite with pattern the unknotted sphere $M_{3/2}$45 representing $M_{3/2}$46 (Hughes et al., 2018).

These isotopies have consequences for the Gluck twist. Melvin showed that the Gluck twist on $M_{3/2}$47 along a sphere $M_{3/2}$48 yields $M_{3/2}$49 if and only if there is a diffeomorphism of pairs $M_{3/2}$50. The unit-surface isotopies therefore imply standardness of the Gluck twist in several families, including ribbon surfaces, surfaces $M_{3/2}$51-concordant to a band-sum of twist-spins, and the satellite families with $M_{3/2}$52 (Hughes et al., 2018).

A frequent overstatement is that the calculus proves all unit spheres in $M_{3/2}$53 are smoothly isotopic to $M_{3/2}$54. The paper explicitly rejects that conclusion: an earlier version claimed it, but a gap was found and acknowledged. The present version proves many new isotopies in $M_{3/2}$55, but the full statement that all unit spheres are standard remains open (Hughes et al., 2018).

6. Subsequent developments and related variants

Later work has treated banded unlink diagrams as a unifying combinatorial model rather than an isolated isotopy calculus. One major extension is to the fundamental quandle of a surface link in an arbitrary $M_{3/2}$56-manifold. Building on the Hughes–Kim–Miller framework, a 2026 paper gives a Wirtinger-type presentation $M_{3/2}$57 from a banded unlink diagram $M_{3/2}$58, proves that the operator group of $M_{3/2}$59 is the knot group $M_{3/2}$60, and establishes an isomorphism of augmented quandles $M_{3/2}$61 for any surface link $M_{3/2}$62 and any banded unlink diagram $M_{3/2}$63 of $M_{3/2}$64. The same paper derives the bridge-number bound

$M_{3/2}$65

and uses it to obtain existence theorems for infinitely many pairwise non-local surface knots with prescribed bridge number in $M_{3/2}$66 (Zhou, 14 May 2026).

A second extension addresses immersed rather than embedded surfaces. Singular banded unlink diagrams for self-transverse immersed surfaces in smooth, orientable, closed $M_{3/2}$67-manifolds are developed by replacing the unlink at the level $M_{3/2}$68 with a marked singular link together with bands. Every self-transverse immersed surface admits such a diagram, and equivalence is generated by singular band moves; the same paper adds finger, Whitney, and cusp moves to describe regular homotopy and homotopy, and proves that immersed bridge trisections exist and are unique up to simple perturbation moves (Hughes et al., 2021). In the special case of immersed surface-links in $M_{3/2}$69, a related 2025 work uses singular banded unlink or singular marked graph diagrams, a twelve-move calculus, and biquandle colorings, while also proving that Yoshikawa’s oriented fifth move $M_{3/2}$70 is independent of the other nine moves and planar isotopies (Jablonowski, 19 May 2025).

There are also adjacent uses of band diagrams outside the Hughes–Kim–Miller $M_{3/2}$71-manifold context. A 2025 note studies diagrams of links and bands on almost special spines and flow-spines of $M_{3/2}$72-manifolds, extending the Reidemeister theorem to bands, cylinders, and Möbius strips with complete local move sets on spines and flow-spines (Petronio, 17 Jun 2025). Earlier, in classical link theory, “banded unlink diagram” denoted an unlink with bands attached whose boundary is a link; in that setting every link bounds an unknotted flat banded surface, and band index and flat band index quantify minimal band presentations (Kim et al., 2011).

These developments indicate that banded unlink diagrams now serve several related but non-identical purposes: isotopy classification of embedded surfaces, singular calculus for immersed surfaces, algebraic presentation of quandles and knot groups, and diagrammatics of bands in $M_{3/2}$73-manifolds. What remains specific to the Hughes–Kim–Miller theory is the complete calculus for embedded surfaces in arbitrary Kirby-presented $M_{3/2}$74-manifolds and its tight interface with bridge trisections and unit-surface problems (Hughes et al., 2018).

The principal open directions recorded in the cited work are correspondingly structural. The full “all unit spheres are standard” statement in $M_{3/2}$75 remains open (Hughes et al., 2018). Quantitative bounds for perturbation and deperturbation in bridge trisection uniqueness are not addressed (Hughes et al., 2018). For quandle presentations, practical complexity grows quickly with the number of crossings, bands, and handle components, and normal forms or confluent rewriting systems under HKM moves remain open (Zhou, 14 May 2026). This suggests that the mature theory is complete at the level of existence and move-generation, while effective simplification and classification remain active problems.