Spun Trefoil Knot in S4

• The spun trefoil is a 2-knot in S4 produced by the Artin spin of the trefoil, resulting in a smoothly embedded, topologically knotted 2-sphere.
• Its construction extends to twist-spin and roll-spin variations that alter bridge trisections and mapping-torus complements with precise algebraic consequences.
• Applications of the spun trefoil include analyzing Gluck surgery, quandle invariants, and visualization models, providing deep insights into 4-dimensional topology.

The spun trefoil is the Artin spin $S(T)$ of the trefoil knot $T \subset S^3$: one removes a small ball so that the trefoil becomes a properly embedded knotted arc in a $3$-ball, then rotates that ball-and-arc through an $S^1$-family in $S^4$, producing a smoothly embedded $2$-sphere. It is the $0$-twist, $0$-roll member of a larger family of deform-spun $2$-knots, and it serves as a central example in the study of Gluck surgery, bridge trisections, broken Lefschetz fibrations, quandle invariants, and discrete models of knotted surfaces (Aranda et al., 2021, Naylor et al., 2020).

1. Definition and basic construction

For a classical knot $K \subset S^3$, the Artin spin is obtained by cutting $T \subset S^3$0 at a point so that it becomes a properly embedded arc $T \subset S^3$1 with endpoints at $T \subset S^3$2, and then forming

$T \subset S^3$3

When $T \subset S^3$4 is the trefoil, the resulting $T \subset S^3$5-knot is the spun trefoil (Aranda et al., 2021).

A concordance-based formulation is also standard. If $T \subset S^3$6, $T \subset S^3$7 is a small $T \subset S^3$8-ball meeting $T \subset S^3$9 in a trivial arc, $3$0, and $3$1, then the natural slice disk for $3$2 in $3$3 is

$3$4

In that language, the Artin spin is the double of $3$5 with the product concordance of $3$6: $3$7 This realization is useful because it extends directly to twist-spin and roll-spin constructions (Naylor et al., 2020).

For the trefoil, which is the torus knot $3$8, the spun knot is a smoothly embedded $3$9-sphere in $S^1$0 and is topologically knotted. Because the trefoil has bridge number $S^1$1, its spin admits a minimal bridge trisection with parameters $S^1$2, often abbreviated a $S^1$3-trisection (Aranda et al., 2021). This gives a precise low-complexity model of the spun trefoil inside the trisection framework.

2. Twist-spun and roll-spun relatives

The spun trefoil is only the first term in a broader family. Let $S^1$4 be a tubular neighborhood of $S^1$5, identified with $S^1$6, with meridian

$S^1$7

and longitude

$S^1$8

On a collar $S^1$9, the meridional and longitudinal Dehn twists are

$S^4$0

extended to ambient diffeomorphisms $S^4$1. The combined $S^4$2-twist $S^4$3-roll spin is then

$S^4$4

where $S^4$5 is the trace of $S^4$6 under an ambient isotopy from $S^4$7 to $S^4$8. The special cases are

$S^4$9

For the trefoil, $2$0 is therefore the untwisted, unrolled member of this family (Naylor et al., 2020).

In Zeeman’s twist-spinning notation, the $2$1-twist-spun knot $2$2 is produced by the gluing

$2$3

so $2$4 recovers the spun trefoil and $2$5 yields the trivial $2$6-knot (Fukuda et al., 2024). This distinction is sometimes blurred in expository usage, but the spun trefoil in the strict sense is the $2$7 case.

For torus knots there is a useful reduction. Litherland’s relation gives

$2$8

Specialized to the trefoil $2$9, this becomes

$0$0

Thus, within the torus-knot setting, roll parameters for the trefoil can be traded for twist parameters by shifting $0$1 by $0$2 (Naylor et al., 2020).

3. Complement, monodromy, and algebraic structure

The complement of a deform-spun knot admits a natural decomposition. For $0$3,

$0$4

The middle term is mapping-torus-like, and classical work describes deform-spun complements as mapping tori of $0$5 with boundary monodromy induced by meridional and longitudinal twists. Algebraically this produces an HNN extension of $0$6. For the trefoil,

$0$7

with the parameters $0$8 entering through the induced peripheral automorphisms (Naylor et al., 2020).

For spun and twist-spun fibered knots, the complement also has a bundle description. In the spun case, if $0$9 is the fiber surface of the original trefoil, then

$0$0

is a bundle over $0$1 with fiber

$0$2

and the monodromy extends the trefoil monodromy $0$3 by the identity on the extra factors. In Choi’s handle decomposition of the trefoil fiber surface, the trefoil monodromy is isotopic to a commuting product $0$4, with one $0$5-handle orbit of length $0$6, one $0$7-handle orbit of length $0$8, and one $0$9-handle orbit of length $2$0; this orbit data drives an explicit broken Lefschetz fibration of $2$1 with spun trefoil fiber (Choi, 2011).

Twist-spun complements admit additional fibered descriptions. Zeeman showed that for $2$2, the complement of $2$3 fibers over $2$4 with fiber the $2$5-fold cyclic branched cover of $2$6 branched along $2$7, with one open $2$8-ball removed. For the classical case $2$9, this applies to twist-spun trefoils in $K \subset S^3$0 (Fukuda et al., 2024). By contrast, visual treatments of the $K \subset S^3$1-twist-spun trefoil also emphasize the induced boundary action

$K \subset S^3$2

on the peripheral system, making the periodic meridional twisting explicit in a mapping-torus picture (Inoue, 2010).

The quandle-theoretic behavior of twist-spun trefoils exhibits a sharp transition. Inoue proved that the knot quandle of the $K \subset S^3$3-twist-spun trefoil is finite if and only if $K \subset S^3$4, with

$K \subset S^3$5

and identified these with quandles related to the $K \subset S^3$6-cell, $K \subset S^3$7-cell, and $K \subset S^3$8-cell respectively (Inoue, 2018). A later reformulation shows that for $K \subset S^3$9, the knot quandle of the $T \subset S^3$00-twist-spun trefoil is a central extension of a Schläfli quandle associated with the tessellation $T \subset S^3$01 (Inoue, 2021). That framework does not address the spun trefoil itself, since $T \subset S^3$02 lies outside the range of the Schläfli-quandle theorem.

4. Gluck surgery and the standard $T \subset S^3$03-sphere

If $T \subset S^3$04 is a smoothly embedded $T \subset S^3$05-sphere with tubular neighborhood $T \subset S^3$06, the Gluck twist is defined by regluing $T \subset S^3$07 via

$T \subset S^3$08

where $T \subset S^3$09 is rotation of $T \subset S^3$10 by angle $T \subset S^3$11 around a fixed axis. The resulting manifold is

$T \subset S^3$12

By Freedman, $T \subset S^3$13 is always homeomorphic to $T \subset S^3$14; the subtle question is whether it is diffeomorphic to the standard $T \subset S^3$15-sphere (Naylor et al., 2020).

For the spun trefoil and all its twist-roll relatives arising from an unknotting-number-one knot, Naylor and Schwartz proved a uniform standardness theorem: $T \subset S^3$16 for every integer pair $T \subset S^3$17 whenever $T \subset S^3$18. Since the trefoil satisfies $T \subset S^3$19, one obtains

$T \subset S^3$20

In particular,

$T \subset S^3$21

and likewise for every twist-spun, roll-spun, or mixed twist-roll spun trefoil (Naylor et al., 2020).

The proof passes through a stabilization argument. For $T \subset S^3$22, the $T \subset S^3$23-knot $T \subset S^3$24 is regularly homotopic to the unknot by exactly one finger move followed by one Whitney move. A theorem of Joseph–Klug–Ruppik–Schwartz then implies that a single stabilization yields an unknotted torus, Iwase identifies the Gluck twist with a multiplicity-one torus surgery on that stabilization, and Montesinos–Larson show that any multiplicity-one torus surgery on the unknotted torus in $T \subset S^3$25 gives back $T \subset S^3$26 (Naylor et al., 2020).

A common confusion is to conflate standardness of the Gluck-twisted ambient manifold with triviality of the $T \subset S^3$27-knot itself. Those are different assertions: $T \subset S^3$28 concerns the reglued $T \subset S^3$29-manifold, whereas the spun trefoil remains a knotted $T \subset S^3$30-sphere. This distinction is reinforced by trisection theory. For the spun trefoil case $T \subset S^3$31, the trisection diagrams obtained from Gay–Meier Gluck-surgery constructions were shown to be standard, in the sense of being slide-equivalent to a stabilization of the genus-$T \subset S^3$32 trisection of $T \subset S^3$33 (Isoshima et al., 2023).

An additional consequence of the Naylor–Schwartz theorem concerns Gompf’s twisted doubles. If $T \subset S^3$34, then for any unknotting-number-one knot $T \subset S^3$35, $T \subset S^3$36 is standard for all $T \subset S^3$37. Specialized to the trefoil, every $T \subset S^3$38 is diffeomorphic to $T \subset S^3$39 (Naylor et al., 2020).

5. Trisections, tri-planes, rainbows, and broken Lefschetz fibrations

Bridge trisections supply one of the most effective low-dimensional encodings of the spun trefoil. For a smooth closed surface $T \subset S^3$40, a bridge trisection decomposes $T \subset S^3$41 into three $T \subset S^3$42-balls $T \subset S^3$43 so that $T \subset S^3$44 is a trivial disk system and the pairwise intersections determine trivial tangles. The bridge number is $T \subset S^3$45, where $T \subset S^3$46 is the trisection sphere (Aranda et al., 2021). For the spun trefoil, every minimal bridge trisection is a $T \subset S^3$47-trisection, so its minimal bridge number is $T \subset S^3$48 (Aranda et al., 2021).

The Kirby–Thompson invariant gives a finer complexity measure. If $T \subset S^3$49 are efficient defining pairs in the pants complex of the punctured trisection sphere, then

$T \subset S^3$50

Aranda, Pongtanapaisan, Taylor, and Zhang proved the first sharp computation in this setting: $T \subset S^3$51 The lower bound comes from a universal estimate $T \subset S^3$52 for any $T \subset S^3$53-bridge trisection of a knotted connected surface, while the upper bound is realized by explicit length-$T \subset S^3$54 paths in the pants complex for each of the three sectors of the Meier–Zupan trisection (Aranda et al., 2021).

A related diagrammatic framework is the triplane or rainbow formalism. For the spun trefoil $T \subset S^3$55, explicit constructions give a $T \subset S^3$56-strand rainbow with common axis $T \subset S^3$57, crossingless middle tangle $T \subset S^3$58, and fully destabilizable pairwise unions $T \subset S^3$59. In this language,

$T \subset S^3$60

The same paper constructs a $T \subset S^3$61-strand rainbow for the $T \subset S^3$62-twist spun trefoil and verifies

$T \subset S^3$63

as well (Aranda et al., 5 Oct 2025).

Crossing-number questions are subtler. For the $T \subset S^3$64-twist spun trefoil $T \subset S^3$65, the tri-plane crossing number is exactly $T \subset S^3$66; this is the first exact computation of that invariant for a non-trivial knotted surface. The lower bound comes from the theorem that every $T \subset S^3$67-knot with a tri-plane diagram having at most five crossings is ribbon, together with Satoh’s result that the $T \subset S^3$68-twist spun trefoil has triple point number $T \subset S^3$69, hence is not ribbon (Gong et al., 2 Jun 2026). The same work exhibits a six-crossing tri-plane diagram for the spun trefoil and conjectures that the spun trefoil also has tri-plane crossing number $T \subset S^3$70 (Gong et al., 2 Jun 2026).

The spun trefoil also appears as an explicit fiber in a broken Lefschetz fibration $T \subset S^3$71. Choi constructed such a fibration with no cusps and no Lefschetz singularities. In the trefoil case, the singular image consists of round-handle loci winding $T \subset S^3$72, $T \subset S^3$73, $T \subset S^3$74, and $T \subset S^3$75 times in the base, corresponding respectively to the two $T \subset S^3$76-handle orbits, the single $T \subset S^3$77-handle orbit, and the $T \subset S^3$78 piece of the complement fiber (Choi, 2011). This gives a fully explicit handle-level realization of the spun trefoil as a regular fiber of a broken Lefschetz fibration on the standard $T \subset S^3$79-sphere.

6. Visual, cubical, and higher-dimensional extensions

The spun trefoil and its twist-spun relatives have long served as model examples for $T \subset S^3$80-dimensional visualization. In a motion-picture description defined by slicing with hyperplanes $T \subset S^3$81, the $T \subset S^3$82-twist-spun trefoil appears as a movie of links in $T \subset S^3$83. For the $T \subset S^3$84-twist-spun trefoil, a symmetric broken-surface diagram and motion picture make the $T \subset S^3$85-periodicity explicit: the movie decomposes into two congruent phases related by a $T \subset S^3$86 rotation about the spin axis. In that visualization the four triple points of the $T \subset S^3$87-twist-spun trefoil are organized into two symmetric pairs (Inoue, 2010).

A discrete counterpart is provided by cubical surface-knot theory. In the canonical cubulation $T \subset S^3$88 of $T \subset S^3$89, a cubical $T \subset S^3$90-knot is an embedded $T \subset S^3$91-sphere in the $T \subset S^3$92-skeleton $T \subset S^3$93, and its area is the number of unit squares. For a reduced cubical spin $T \subset S^3$94,

$T \subset S^3$95

Using this formula together with a lower-bound argument based on $T \subset S^3$96-level combinatorics and crossing steps, the spun trefoil was shown to have cubical-spin minimal area

$T \subset S^3$97

Moreover, there exists a weakly minimal cubical $T \subset S^3$98-knot with area $T \subset S^3$99 isotopic to the spun trefoil, and the unrestricted global invariant satisfies

$3$00

The equality $3$01 for all cubical representatives was not proved (Baray et al., 12 Jul 2025).

Higher-dimensional iteration leads to a further extension. If $3$02 is first formed in $3$03 and then twist-spun again, the resulting $3$04-knot $3$05 is trivial whenever $3$06. The proof uses the fibered structure of twist-spun complements together with cyclic branched-cover arguments and Pao’s theorem on branched twist spins (Fukuda et al., 2024). This suggests that the spun trefoil is best understood ոչ as an isolated object but as the base case of an extensive hierarchy of spinning operations across dimensions.

In that sense, the spun trefoil occupies a distinctive position. It is simultaneously a concrete $3$07-sphere in $3$08, the $3$09-parameter point of the twist-roll deformation space, a source of exact complexity computations such as $3$10 and cubical area $3$11, and a stable test object for cut-and-paste operations whose ambient outcome remains the standard $3$12-sphere (Aranda et al., 2021, Baray et al., 12 Jul 2025).