The effect of link Dehn surgery on the Thurston norm

Maggie Miller

arxiv: 1906.08458 · v2 · pith:M4QZL6M4new · submitted 2019-06-20 · 🧮 math.GT

The effect of link Dehn surgery on the Thurston norm

Maggie Miller This is my paper

Pith reviewed 2026-05-25 19:28 UTC · model grok-4.3

classification 🧮 math.GT

keywords Thurston normDehn surgerylink complementsnorm-minimizing surfacesProperty R conjecturerational homology spheres3-manifold topology

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The pith

For two-component links with nonzero linking numbers, norm-minimizing surfaces in the complement remain norm-minimizing after Dehn surgery and capping except along finitely many homology rays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how Dehn surgery on a multi-component link affects whether a surface that minimizes the Thurston norm in the link complement continues to minimize the norm after the surgery is performed and the surface is capped with disks. For links with exactly two components, the capped surface stays norm-minimizing whenever its homology class avoids a finite collection of rays in the relative second homology of the complement. The setting requires that the components have nonzero pairwise linking numbers and that the complement has nondegenerate Thurston norm; when the ambient manifold is an integer homology sphere the result limits the surgeries that can produce S1 times S2. The argument adapts techniques from Gabai's proof of the Property R conjecture.

Core claim

Let L be an n-component link (n>1) with pairwise nonzero linking numbers in a rational homology 3-sphere Y. Assume the link complement X has nondegenerate Thurston norm. A properly embedded Thurston norm-minimizing surface S in X remains norm-minimizing after Dehn filling every boundary torus of X along the slope given by ∂S and then capping the resulting closed surface with disks. When n=2 this holds for every class [S] lying outside a finite set of rays in H2(X,∂X;R). When Y is an integer homology sphere the statement supplies an upper bound on the number of surgeries on L that can produce S1×S2.

What carries the argument

Dehn filling of the link complement along the boundary slopes of a norm-minimizing surface, followed by disk capping, measured in the Thurston norm.

If this is right

For n=2 the capped-off surface is norm-minimizing when [S] lies outside a finite set of rays in H2(X,∂X;R).
When Y is an integer homology sphere this gives an upper bound on the number of surgeries on L which may yield S1×S2.
The result applies only under the standing assumptions of nonzero pairwise linking numbers and nondegenerate Thurston norm on the complement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The finite exceptional rays could be identified explicitly for particular links such as the Hopf link, yielding a complete list of surgeries to check.
The same capping argument may constrain which surgeries produce other reducible or toroidal manifolds beyond S1×S2.
For links with more than two components the paper leaves open whether an analogous finite-exception statement holds or whether infinitely many rays can appear.
The nondegeneracy assumption on the Thurston norm may itself be checkable via existing algorithms for many explicit links.

Load-bearing premise

The link complement has nondegenerate Thurston norm.

What would settle it

A concrete two-component link whose complement has nondegenerate Thurston norm, together with a homology class outside the finite exceptional rays, for which the capped surface after surgery fails to minimize the Thurston norm of its class in the filled manifold.

Figures

Figures reproduced from arXiv: 1906.08458 by Maggie Miller.

**Figure 1.** Figure 1: First: A 2-component link L = L1 t L2 in S 3 . Second: A norm-minimizing surface S1 in the homology class of a punctured Seifert surface for L1. Third: A normminimizing surface in the homology class of a punctured surface for L2. Fourth: The unit ball of the Thurston norm on S 3 \ν(L), in which we indicate [±S1] and [±S2]. The surfaces Sc1 and Sc2 are not norm-minimizing. L1 L2 1 4 0 2[S1] + [S2] [S1] [S… view at source ↗

**Figure 2.** Figure 2: First: A 2-component link L = L1 t L2 in S 3 . Second: Let Si be a surface in the homology class of a punctured Seifert surface for Li . We indicate the boundary slopes of 2[S1] + [S2]. Third: Surgery on Li according to these slopes yields S 1 × S 2 . Fourth: The unit ball of the Thurston norm on S 3\ν(L), in which we highlight 2[S1]+[S2]. A norm-minimizing surface S in this homology class is genus1 with … view at source ↗

**Figure 3.** Figure 3: Left: A surgery diagram for the Brieskorn sphere Y := Σ(2, 5, 7). Zero-framed surgery on the knot L ⊂ Y yields S 1 × S 2 , but L does not bound a disk in Y . Right: A norm-minimizing surface in Y \ ν(L). classes. (We remind the reader that a primitive class α is one which is integral and not equal to cβ for any integral class β and integer c > 1.) If cSc is not norm-minimizing, then neither is Sb. Thus, E… view at source ↗

**Figure 4.** Figure 4: Left: A norm-minimizing surface S near a torus boundary-component P of 3-manifold X. The components of ∂S on P need not have parallel orientations. Right: We increase the genus of S (while preserving Euler characteristic) to find a properly norm-minimizing surface S 0 homologous to S. When M has boundary a collection of tori, any norm-minimizing surface S can be transformed into a properly norm-minimizing … view at source ↗

**Figure 5.** Figure 5: Left: S1 \ S2 includes a disk C. Right: We surger S2 along C to find another properly norm-minimizing surface S 0 2 . We replace S2 := S 0 2 and repeat until every component of S1 \ S2 not meeting ∂S1 has negative Euler characteristic. Then aS1 + bS2 cannot have any sphere or torus components. Therefore, aS1 +bS2 has no closed sphere or torus components, so aS1 +bS2 is proeprly norm-minmizing. There is a… view at source ↗

**Figure 6.** Figure 6: Left: a Seifert surface S in S 3 \ ν(K). Right: The complementary sutured manifold (M, γ). In the picture we draw a genus-4 surface ∂M in R 3 . The manifold M is the unbounded region in R 3 , compactified by R 3 ⊂ S 3 . The elements of A(γ) are drawn as thin black annuli, with orientations of the sutures indicated by arrows. Note that the normal vector to R+(γ) points out of M, while the normal vector to … view at source ↗

**Figure 7.** Figure 7: Left: a product disk D in a sutured manifold (M, γ). Middle: part of a taut foliation F on (M, γ). Right: Up to perturbation of D, F|D is a product foliation. 3.2. Sutured manifold and foliation operations. The most basic operations one can perform on a sutured manifold (M, γ) (and the only ones needed in this paper) are product-disk decomposition and product-annulus decomposition. Definition 3.11 ([G1]).… view at source ↗

**Figure 8.** Figure 8: Left: a sutured manifold (M, γ). In the picture we draw a genus-4 surface ∂M in R 3 . The manifold M is the unbounded region in R 3 , compactified by R 3 ⊂ S 3 . The elements of A(γ) are drawn as thin black annuli, with orientations of the sutures indicated by arrows. The sutured manifold (M, γ) is the complementary manifold to a Seifert surface S in S 3 , as shown in [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗

**Figure 9.** Figure 9: Top Middle: Two distinct elements A1, A2 of A(γ) in a sutured manifold (M, γ). We indicate a product disk D connecting A1 and A2. We draw a disk E with boundary the core of A1 to indicate the core of a 2-handle E × I that we might attach to M. Top Left: The sutured manifold (M1, γ1), where M1 = M ∪ (E × I) and γ1 = γ \ A1. Top Right: The sutured manifold (M2, γ2), which is obtained from (M, γ) by produc… view at source ↗

**Figure 10.** Figure 10: Left: An arc of intersection between two surfaces R and T. Middle: The cut-and-paste surface S = R + T. Right: A product disk in the complementary sutured manifold to S. (M, γ) implies that (M0 , γ0 ) is taut, so Sb is norm-minimizing. This concludes the proof of Theorem 2.11. Now we prove Theorem 2.16. It is again sufficient to prove the claim when S is properly norm-minimizing. Since [S] is not a corn… view at source ↗

**Figure 11.** Figure 11: Left: V = D × I, where D is a product disk in a sutured manifold (M, γ). We draw the intersection of V with a taut foliation F. Right: We perform a suspension change operation (Operation 3.21) on V to obtain a new taut foliation F 0 . We draw the intersection of F 0 with the reglued V . Note that F 0 and F do not agree on the elements of A(γ) met by D. V = I × I × I via    V 3 ((x, t), s) ∼ ((x, t), … view at source ↗

**Figure 12.** Figure 12: Left: F × I, where F is a connected positivegenus surface. Let C be a boundary component of F. The curve α in F is non-separating. We highlight α × I. Right: In H × I (where H = F \ ν(α)), we can find product disks connecting C × I to the two new vertical (transverse to the product foliation G) annuli C1 × I, C2 × I in (∂H) × I. By performing the suspension change operation on G at these product disks, w… view at source ↗

**Figure 13.** Figure 13: Left: a face Ci of G, where G is a fat-vertex graph describing product disks in (M, γ). Right: we use Ci to construct a product annulus Ai in (M, γ). product annulus Ai corresponds to a face Ci of G, we will always mean as in this construction. Remark 4.3. Say G has components G1, . . . , Gc. Let (M0 , γ0 ) be the sutured manifold obtained by decomposing (M, γ) by product annulus Ai corresponding to a f… view at source ↗

**Figure 14.** Figure 14: Left: paths in ν(G) connecting each ∂ν(vk) to a point q in ∂ν(G). The face containing q corresponds to a product annulus A in (M, γ). Right: The paths yields product disks connecting corresponding elements of A(γ) to an element of A(γ 0 ) \ A(γ), where (M, γ) Ai −→ (M0 , γ0 ). If Sb is not norm-minimizing, then either g(S) = 1 or g([S]) is minimal among all classes in the interior of the same face of the … view at source ↗

**Figure 15.** Figure 15: We have not yet specified our choice of foliation on the [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗

**Figure 15.** Figure 15: Left: the boundary of Vi in (Mc0 , γb0). We highlight where we would like to glue a disk when filling ∂Vi . Right: The foliation G on the two annuli in A(γ 0 ). Each annulus is subdivided into three annuli on which G induces some suspension foliation. We can fill ∂Vi by a solid torus and extend G if the concatenations µ − 1 fµ+ 1 and ¯µ − 2 g¯µ¯ + 2 are conjugate as automorphisms of I. Since ∂−A is non-s… view at source ↗

**Figure 16.** Figure 16: First: On Pi , an intersection of ∂R and ∂T, where R and T are properly norm-minimizing surfaces. Second: One intersection of ∂R and ∂T gives rise to ab intersections of ∂aR and ∂bT. Third: On Pi , ∂(aR + bT). Fourth: We indicate product disks in the complementary sutured manifold to aR + bT arising from the cut-and-paste construction. Specifically, we draw the intersection of these disks with Pi . Fift… view at source ↗

read the original abstract

Let $L$ be an $n$-component link ($n>1$) with pairwise nonzero linking numbers in a rational homology $3$-sphere $Y$. Assume the link complement $X:=Y\setminus\nu(L)$ has nondegenerate Thurston norm. In this paper, we study when a Thurston norm-minimizing surface $S$ properly embedded in $X$ remains norm-minimizing after Dehn filling all boundary components of $X$ according to $\partial S$ and capping off $\partial S$ by disks. In particular, for $n=2$ the capped-off surface is norm-minimizing when $[S]$ lies outside of a finite set of rays in $H_2(X,\partial X;\mathbb{R})$. When $Y$ is an integer homology sphere this gives an upper bound on the number of surgeries on $L$ which may yield $S^1\times S^2$. The main techniques come from Gabai's proof of the Property R conjecture and related work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Gabai's foliation methods to give a finite-ray exception for norm-minimizing surfaces after surgery on two-component links, yielding a surgery bound when the ambient space is an integer homology sphere.

read the letter

The main new content is the statement that for two-component links with nonzero linking numbers, once the complement has nondegenerate Thurston norm, a norm-minimizing surface stays norm-minimizing after the relevant Dehn fillings and capping except along a finite set of rays in homology. When the ambient Y is an integer homology sphere this produces an explicit upper bound on the number of surgeries that can produce S1 x S2. The argument recycles Gabai's techniques from the Property R work and applies them to the multi-component setting with the stated linking-number hypothesis. That is a direct and useful extension; the finite-ray claim and the resulting surgery count are not in the earlier Gabai papers referenced in the abstract. The paper states the nondegeneracy assumption up front and treats the results as conditional on it, which keeps the claims honest. The stress-test note indicates no internal inconsistency once the assumption is granted, and the techniques are the standard ones for detecting norm-minimizing surfaces via foliations. The main soft spot is that nondegeneracy is an assumption rather than a conclusion, so the result applies only where that holds; the abstract gives no indication that the paper removes or weakens this hypothesis in general. Without the body of the manuscript it is impossible to check every step of the derivation for post-hoc choices, but nothing in the provided material suggests circularity or invented entities. This is a paper for people already working on the Thurston norm, Dehn surgery on links, and foliation methods in three-manifolds. It is a modest, technically grounded increment rather than a broad advance, but the new statement is cleanly stated and the logic appears to hold under the given hypotheses. I would send it to a serious referee.

Referee Report

0 major / 2 minor

Summary. The paper studies the effect of Dehn surgery on an n-component link L (n>1) with pairwise nonzero linking numbers in a rational homology 3-sphere Y, assuming the link complement X has nondegenerate Thurston norm. It shows that a Thurston norm-minimizing properly embedded surface S in X remains norm-minimizing after Dehn filling along ∂S and capping off the boundary components by disks, with the n=2 case holding outside a finite set of rays in H2(X,∂X;R). When Y is an integer homology sphere this yields an upper bound on the number of surgeries on L producing S1×S2. The arguments rely on Gabai's foliation techniques.

Significance. If the results hold under the stated nondegeneracy assumption, the work extends Gabai's methods from the Property R conjecture to links, producing a concrete upper bound on surgeries yielding S1×S2. This is a useful contribution to Dehn surgery problems in 3-manifold topology, with the finite-ray exception arising directly from the finitely many faces of the Thurston norm unit ball.

minor comments (2)

[Introduction] The abstract and introduction state the nondegeneracy assumption clearly, but a brief remark in §1 on classes of links where this assumption is known to hold (or is expected to fail) would help readers assess applicability.
[§2] Notation for the capped-off surface after surgery could be introduced once in §2 and used consistently thereafter to avoid minor ambiguity in the n=2 case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the work is viewed as a useful contribution extending Gabai's techniques to links.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states its central results as conditional on the explicit nondegeneracy assumption for the Thurston norm of the link complement X. All load-bearing steps invoke external techniques from Gabai's foliation methods and Property R work rather than any internal fit, self-definition, or self-citation chain. No equation or claim reduces by construction to a renamed input or prior result by the same author.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on two domain assumptions stated in the abstract and on background results from Gabai; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The link complement X has nondegenerate Thurston norm.
Explicitly assumed for the main theorems to apply.
domain assumption Pairwise nonzero linking numbers for the n components of L.
Stated as a hypothesis on the link L in Y.

pith-pipeline@v0.9.0 · 5698 in / 1303 out tokens · 31458 ms · 2026-05-25T19:28:56.145245+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Assume the link complement X := Y ∖ ν(L) has nondegenerate Thurston norm... Let S be a norm-minimizing surface... after Dehn filling... capping off ∂S by disks.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The main techniques come from Gabai’s proof of the Property R conjecture and related work... taut foliation F on X achieving S as a leaf... product-disk decomposition.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Introduction to Machine Protection

Kenneth L. Baker and Scott A. Taylor, Dehn filling and the Thurston norm , arXiv:1608.02433 [math.GT], Aug. 2016. To appear in J. Diff. Geom

work page internal anchor Pith review Pith/arXiv arXiv 2016
[2]

Monogr., Clarendon Press, Oxford, 2014, MR2327361, Zbl 1118.57002

Danny Calegari, Foliations and the Geometry of 3-Manifolds , Oxford Math. Monogr., Clarendon Press, Oxford, 2014, MR2327361, Zbl 1118.57002

work page arXiv 2014
[3]

Arnaud Denjoy, Sur les courbes d\' e finies par les \' e quations diff\' e rentielles \` a la surface du tore , J. Math. Pure. Appl. 11 (1932) 333--376, MR0101369, Zbl 0006.30501

work page arXiv 1932
[4]

David Gabai, Foliations and genera of links , Topology 23 :4 (1984) 381--394, MR0780731, Zbl 0567.57021

work page arXiv 1984
[5]

David Gabai, Foliations and the topology of 3 -manifolds , J. Diff. Geom. 18 :3 (1983) 445-503, MR0723813, Zbl 0533.57013

work page arXiv 1983
[6]

David Gabai, Foliations and the topology of 3 -manifolds. II , J. Diff. Geom. 26 :3 (1987) 461--478, MR0910017, Zbl 0627.57012

work page arXiv 1987
[7]

David Gabai, Foliations and the topology of 3 -manifolds. III , J. Diff. Geom. 26 :3 (1987) 479--536, MR0910018, Zbl 0639.57008

work page arXiv 1987
[8]

David Gabai, Taut foliations of 3 -manifolds and suspensions of S^1 , Ann. Inst. Fourier 42 :1--2 (1992)193--208, MR1162560, Zbl 0736.57010

work page arXiv 1992
[9]

Blaine Lawson, Jr., Foliations , Bull

H. Blaine Lawson, Jr., Foliations , Bull. Amer. Math. Soc. 80 :3 (1974) 369--418, MR0343289, Zbl 0293.57014

work page arXiv 1974
[10]

McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology , Ann

Curtis T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology , Ann. Sci. \' E c. Norm. Sup\' e r. 35 :2 (2002) 153--171, MR1914929, Zbl 1009.57021

work page arXiv 2002
[11]

Novikov, Topology of foliations , Trans

Sergei P. Novikov, Topology of foliations , Trans. Moscow Math. Soc. 14 (1963) 268--305, MR0200938, Zbl 0247.57006

work page arXiv 1963
[12]

Robert Roussarie, Plongements dans les vari\' e t\' e s feuillet\' e es et classification de feuilletages sans holonomie , Publ. Math. Inst. Hautes. \' E tudes Sci. 43 (1974) 101--141, MR0358809, Zbl 0356.57017

work page arXiv 1974
[13]

Zlil Sela, Dehn fillings that reduce Thurston norm , Israel J. Math. 69 :3 (1990) 371--378, MR1049294, Zbl 0714.57009

work page arXiv 1990
[14]

Thurston, A norm for the homology of 3 -manifolds , Mem

William P. Thurston, A norm for the homology of 3 -manifolds , Mem. Amer. Math. Soc., No. 339, Amer. Math. Soc., Providence, RI (1986) 99--130, MR0823443, Zbl 0585.57006

work page arXiv 1986

[1] [1]

Introduction to Machine Protection

Kenneth L. Baker and Scott A. Taylor, Dehn filling and the Thurston norm , arXiv:1608.02433 [math.GT], Aug. 2016. To appear in J. Diff. Geom

work page internal anchor Pith review Pith/arXiv arXiv 2016

[2] [2]

Monogr., Clarendon Press, Oxford, 2014, MR2327361, Zbl 1118.57002

Danny Calegari, Foliations and the Geometry of 3-Manifolds , Oxford Math. Monogr., Clarendon Press, Oxford, 2014, MR2327361, Zbl 1118.57002

work page arXiv 2014

[3] [3]

Arnaud Denjoy, Sur les courbes d\' e finies par les \' e quations diff\' e rentielles \` a la surface du tore , J. Math. Pure. Appl. 11 (1932) 333--376, MR0101369, Zbl 0006.30501

work page arXiv 1932

[4] [4]

David Gabai, Foliations and genera of links , Topology 23 :4 (1984) 381--394, MR0780731, Zbl 0567.57021

work page arXiv 1984

[5] [5]

David Gabai, Foliations and the topology of 3 -manifolds , J. Diff. Geom. 18 :3 (1983) 445-503, MR0723813, Zbl 0533.57013

work page arXiv 1983

[6] [6]

David Gabai, Foliations and the topology of 3 -manifolds. II , J. Diff. Geom. 26 :3 (1987) 461--478, MR0910017, Zbl 0627.57012

work page arXiv 1987

[7] [7]

David Gabai, Foliations and the topology of 3 -manifolds. III , J. Diff. Geom. 26 :3 (1987) 479--536, MR0910018, Zbl 0639.57008

work page arXiv 1987

[8] [8]

David Gabai, Taut foliations of 3 -manifolds and suspensions of S^1 , Ann. Inst. Fourier 42 :1--2 (1992)193--208, MR1162560, Zbl 0736.57010

work page arXiv 1992

[9] [9]

Blaine Lawson, Jr., Foliations , Bull

H. Blaine Lawson, Jr., Foliations , Bull. Amer. Math. Soc. 80 :3 (1974) 369--418, MR0343289, Zbl 0293.57014

work page arXiv 1974

[10] [10]

McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology , Ann

Curtis T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology , Ann. Sci. \' E c. Norm. Sup\' e r. 35 :2 (2002) 153--171, MR1914929, Zbl 1009.57021

work page arXiv 2002

[11] [11]

Novikov, Topology of foliations , Trans

Sergei P. Novikov, Topology of foliations , Trans. Moscow Math. Soc. 14 (1963) 268--305, MR0200938, Zbl 0247.57006

work page arXiv 1963

[12] [12]

Robert Roussarie, Plongements dans les vari\' e t\' e s feuillet\' e es et classification de feuilletages sans holonomie , Publ. Math. Inst. Hautes. \' E tudes Sci. 43 (1974) 101--141, MR0358809, Zbl 0356.57017

work page arXiv 1974

[13] [13]

Zlil Sela, Dehn fillings that reduce Thurston norm , Israel J. Math. 69 :3 (1990) 371--378, MR1049294, Zbl 0714.57009

work page arXiv 1990

[14] [14]

Thurston, A norm for the homology of 3 -manifolds , Mem

William P. Thurston, A norm for the homology of 3 -manifolds , Mem. Amer. Math. Soc., No. 339, Amer. Math. Soc., Providence, RI (1986) 99--130, MR0823443, Zbl 0585.57006

work page arXiv 1986