Two-bridge knots admit no purely cosmetic surgeries

In Dae Jong; Kazuhiro Ichihara; Thomas W. Mattman; Toshio Saito

REVIEW

Two-bridge knots admit no purely cosmetic surgeries

Kazuhiro Ichihara, In Dae Jong, Thomas W. Mattman, Toshio Saito

Geometric Topology · Submitted 5 Sep 2019 · 28 references

Pith's verdict: A solid proof of the Cosmetic Surgery Conjecture for two-bridge and alternating fibered knots, with a load-bearing hand enumeration that needs independent verification.

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CitationIn Dae Jong, Kazuhiro Ichihara, Thomas W. Mattman, Toshio Saito. "Two-bridge knots admit no purely cosmetic surgeries." arXiv:1909.02340 (2019). https://pith.science/paper/4ZXJLJIE

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Editor's note

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This paper proves that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, confirming the Cosmetic Surgery Conjecture for two infinite families. The main novelty is the combination of Hanselman's recent Heegaard Floer constraints with older classification and invariant techniques, and the new computation of the Jones coefficient j4 for the family C[4n,-2n,-2n,4n].

The structure is clean: Hanselman's lemma restricts candidates to genus 2, signature 0, slopes ±1,±2, and a specific Alexander polynomial. For two-bridge knots, the genus-two classification from the authors' earlier work reduces to K = C[2x,2y,-2(x+y),2x]; signature computations (Proposition 2.1) eliminate all but y<0, x+y>0; the SL(2,C) Casson invariant (Proposition 2.2) then forces the amphichiral case x=-2y; and finite type invariants (Proposition 2.3) rule that out. The alternating fibered case follows from a short list of diagrams after applying Stoimenow's classification and Murasugi's monic Alexander criterion.

The proof is convincing in its main line. The soft spots are computational. Proposition 2.2's boundary-slope enumeration is the load-bearing step: 41 cases (across four subfamilies) are listed by hand, with elimination rules stated but not proved, and the final formula S- - S+ = 2(x+2y)(4x^2 - 6x + 5) is asserted after the case lists. There is also a sign slip: for x+2y<0, the difference is negative, not "positive" as written. This does not affect the nonvanishing conclusion, but it signals that the arithmetic deserves independent checking. I'd like to see either a short justification of the enumeration or a computer-assisted verification of the final sums. The omitted proof of Lemma 2.5 is a non-issue; the Mathematica verification of Lemma 2.7, however, should be supplemented with explicit code or an expanded derivation, since the current text is not reproducible.

The dependence on Hanselman's theorem is real: if that result fails, both theorems fail. But the result is now published (JDG), so this is a reasonable foundation. The citation pattern is fine; the authors use their previous papers as tools, not as an undeclared assumption of the target result.

Bottom line: this is a solid, useful paper. It deserves a serious referee, but I'd request revisions that fill the computational gaps and correct the sign statement. If those checks pass, the theorem is established.

Referee report

Major objections (3)

[§2.2, Proposition 2.2] The proof of Proposition 2.2 rests on an unverified enumeration of 41 boundary-slope continued fractions. The substitution rules for the six-tuples, the elimination criteria (e.g., 'no adjacent 1s', 'each term at least two in absolute value'), and the resulting values of n±, N_i, W_i are asserted without derivation or an explicit algorithm. A reader cannot easily check that the list is complete or that the arithmetic in the final S+ and S- expressions is correct. Since this proposition is the step that forces x=-2y, a single error would invalidate Theorem 1.1. The authors should either provide a complete derivation or include a computer-verifiable script that generates the list and the sums.
[§2.3, Lemma 2.7] The proof of Lemma 2.7 derives a closed expression for V_K(t) but then states 'Using Mathematica, we verify that j4(K)=-12n^4' without providing the code or the resulting series expansion. The coefficient extraction is a finite computation that should be reproducible. Please include the series expansion or the Mathematica code.
[§2.3, Lemma 2.5] The proof of Lemma 2.5 is omitted with the comment that it can be 'calculated easily by hand'. Since this lemma provides the Conway polynomial coefficients a2, a4, a6 used in the finite type invariant obstructions, it is load-bearing. Please include the computation or a reference with the explicit formula.

Minor objections (3)

[§2.2, proof of Proposition 2.2] The sentence 'If we are not in the amphichiral case, that is, if x≠-2y, this difference is positive as 4x^2−6x+5 has imaginary roots' is not always correct; when x+2y<0, S- − S+ is negative. The conclusion that the difference is nonzero remains valid, so the argument is unaffected, but the wording should be changed to 'non-zero' or 'positive up to sign'.
[Throughout] There are several typos and formatting artifacts, including 'the the' in the proof of Proposition 2.1 and a line break inside 'PURELY' in the title. These should be corrected in revision.
[§3] The sentence 'alternating fibered knots of genus 2 are exactly the fibered knots contained in 6_3, 7_7, 8_12, 9_41, 10_58 or 12_1202, or 3_1#3_1^*, 4_1#4_1' is confusing because the next sentence states that only five of these are fibered. Please rephrase to clearly separate the classification of genus-two zero-signature alternating knots from the subsequent fiberedness filter.

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The paper

The paper claims

Two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries: no two distinct Dehn surgeries on such a knot can produce 3-manifolds that are homeomorphic as oriented manifolds.

The reading

The paper proves that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries: no two distinct Dehn surgeries on such a knot yield 3-manifolds that are homeomorphic as oriented manifolds. This confirms the Cosmetic Surgery Conjecture for these two infinite families. The proof funnels all possible candidates through a structural restriction on alternating knots, then eliminates them with a signature formula, the $SL(2,\mathbb{C})$ Casson invariant, and finite-type invariants. For the two-bridge case, the surviving amphichiral family is ruled out by a Jones-polynomial coefficient.

What carries the argument

The load-bearing mechanism is a structural restriction on alternating knots with purely cosmetic surgeries (Lemma 1.3, taken from a recent preprint): such a knot must have genus $2$, signature $0$, slopes $\pm1$ or $\pm2$, and Alexander polynomial $n t^2-4n t+(6n+1)-4n t^{-1}+n t^{-2}$. Around this, the paper deploys three tools: the signature formula $\sigma(L)=o(D)-y(D)-1$ for reduced alternating diagrams; a difference formula for the $SL(2,\mathbb{C})$ Casson invariant under $\pm p$ surgeries on two-bridge knots, which reduces the slope check to comparing weighted sums of boundary slopes; and finite-type invariants, namely the Conway coefficient $a_2$ and the Jones coefficient $j_4$, which rule out the last amphichiral family.

Extended reading notes

Core claim

The central discovery is Theorem 1.1 together with Theorem 1.2: if $K$ is a two-bridge knot or an alternating fibered knot in $S^3$, then $K(r)\not\cong K(r')$ for every pair of distinct slopes $r\ne r'$, where $\cong$ denotes orientation-preserving homeomorphism. Relying on a structural lemma (Lemma 1.3) that restricts alternating knots with purely cosmetic surgeries to genus $2$, signature $0$, slopes $\pm1$ or $\pm2$, the proof for two-bridge knots isolates the Conway form $[2x,2y,-2(x+y),2x]$, uses a signature formula to force $y<0$ and $x+y>0$, uses the $SL(2,\mathbb{C})$ Casson invariant to force $x=-2y$, then rules out the remaining amphichiral family $[4n,-2n,-2n,4n]$ by computing the Conway polynomial and a fourth-order Jones coefficient. For alternating fibered knots, a classification of genus-two alternating knots with zero signature reduces to a short list, and the fibered condition leaves only four knots whose Alexander polynomials violate the structural lemma.

Load-bearing premise

The argument stands on a recent structural lemma (Lemma 1.3 in the paper) asserting that any alternating knot with purely cosmetic surgeries must have genus 2, signature 0, slopes $\pm1$ or $\pm2$, and a specific Alexander polynomial; if that lemma is false, the paper's conclusions do not follow.

Editorial extensions

If this is right

The Cosmetic Surgery Conjecture holds for every two-bridge knot.
The Cosmetic Surgery Conjecture holds for every alternating fibered knot.
Any two-bridge counterexample to the conjecture would have to lie in the family $C[4n,-2n,-2n,4n]$, and the finite-type computation rules that family out.
Alternating pretzel knots also admit no purely cosmetic surgeries, as demonstrated in Remark 1.1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the structural lemma is extended from alternating knots to all genus-two knots, the same strategy—signature, $SL(2,\mathbb{C})$ Casson difference, finite-type coefficients—could potentially settle the genus-two case, since the boundary-slope computation is not inherently alternating.
The difference $S_- - S_+ = 2(x+2y)(4x^2-6x+5)$ suggests the $SL(2,\mathbb{C})$ Casson invariant detects the chirality of the surgery pair; one could test whether this difference is always sign-definite for other two-bridge families.
For the amphichiral family $C[4n,-2n,-2n,4n]$ the Conway polynomial is $1+4n^4 z^4$, so the classical second-coefficient obstruction vanishes; a natural extension is to study higher Conway or Jones coefficients for neighboring amphichiral two-bridge knots.

Original abstract

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for 3-manifolds, we deploy the $SL(2,\mathbb{C})$ Casson invariant.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for 3-manifolds, we deploy the $SL(2,\mathbb{C})$ Casson invariant.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds.	result 0.98	0.94	0.97	asserted 1.00
s1	Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for 3-manifolds, we deploy the $SL(2,\mathbb{C})$ Casson invariant.	method 0.99	0.78	0.04	asserted 0.95

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

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full text
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Figures

**Figure 1.** Diagram of knots [2x, 2y − 1, 1, 2x + 2y − 2, 1, 2x − 1] Then, by direct calculation, we see that o(D) = 4x+4y−3 and y(D) = 4x+4y−2. It follows that σ(K) = −2. In the same way, when y < 0 and (x + y) < 0 (case (iii)), since [2x, 2y, −2(x + y), 2x] = [2x−1, 1, −2y−2, 1, −2x−2y−1, 2x], the knot K has a reduced alternating diagram D for which o(D) = 2x + 3 and y(D) = 2x. It follows that σ(K) = 2. That is, in both cases, the signature, σ(K), is non-zero. From Lemma 1.3, this implies that if K admits purely cosmetic surgeries we must be in the remaining case, case (ii), where y < 0 and (x

**Figure 2.** Apply the skein relation at the positive crossing marked by ∗

**Figure 3.** The link Ln is the connected sum of the knot C[4n, −2n] and T(2, −4n). For T(2, −4n), applying the skein relation (4) 2n times, we have VT(2,−4n)(t) = t 2VT(2,−4n+2)(t) + t(t 1 2 − t − 1 2 ) = · · · = t 4nV◦◦ + (1 + t 2 + · · · + t 4n−2 )t(t 1 2 − t − 1 2 ) = t 4n (−t 1 2 − t − 1 2 ) + 1 − t 4n 1 − t 2 t(t 1 2 − t − 1 2 ) = −t 1 2 1 + t

Figures reproduced from arXiv:1909.02340 by the authors.

References

Works cited (28)

[1]

H. U. Boden and C. L. Curtis, The SL(2, C) Casson invariant for Dehn surgeries on two-bridge knots, Algebr. Geom. Topol. 12 (2012), no. 4, 2095–2126

View on Pith 2012
[2]

S. Boyer and D. Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990), 181–220

View on Pith 1990
[3]

T. D. Cochran and R. E. Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology 3-spheres and property P , Topology 27 (1988), no. 4, 495–512

View on Pith 1988
[4]

R. Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959), 258–275

View on Pith 1959
[5]

C. L. Curtis, An intersection theory count of the SL 2(C)-representations of the fundamental group of a 3-manifold, Topology 40 (2001), no. 4, 773–787

View on Pith 2001
[6]

C. L. Curtis, Erratum to: “An intersection theory count of the SL 2(C)-representations of the fundamental group of a 3-manifold” [Topology 40 (2001), no. 4, 773–787], Topology 42 (2003), no. 4, 929

View on Pith 2001
[7]

P. R. Cromwell, Homogeneous links, J. London Math. Soc. (2) 39 (1989), no. 3, 535–552

View on Pith 1989
[8]

J. Hanselman, Heegaard Floer homology and cosmetic surgeries in S3, preprint, arXiv:1906.06773

View on Pith Pith review arXiv 1906
[9]

A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246

View on Pith 1985
[10]

K. Ichihara and T. Saito, Cosmetic surgery and SL(2, C) Casson invariant for two-bridge knots, Hiroshima Math. J. 48 (2018), no. 1, 21–37

View on Pith 2018
[11]

K. Ichihara, Z. Wu, A note on Jones polynomial and cosmetic surgery, Comm. Anal. Geom. to appear

View on Pith
[12]

T. Ito, On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in S3, Comm. Anal. Geom. to appear

View on Pith
[13]

I. D. Jong, Alexander polynomials of alternating knots of genus two, Osaka J. Math. 46 (2009), no. 2, 353–371

View on Pith 2009
[14]

Problems in low-dimensional topology, Edited by Rob Kirby. AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35–473, Amer. Math. Soc., Providence, RI, 1997

View on Pith 1993
[15]

K. Kishimoto, T. Shibuya, and T. Tsukamoto, in preparation

View on Pith
[16]

E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554–586

View on Pith 2005
[17]

T. W. Mattman, G. Maybrun and K. Robinson, 2-bridge knot boundary slopes: diameter and genus, Osaka J. Math. 45 (2008), no. 2, 471–489

View on Pith 2008
[18]

Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019)

View on Pith 2019
[19]

W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topol- ogy 23 (1984), no. 1, 37–44

View on Pith 1984
[20]

K. Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248

View on Pith 1958
[21]

K. Murasugi, On a certain subgroup of the group of an alternating link, Amer. J. Math. 85 (1963), 544–550

View on Pith 1963
[22]

L. Neuwirth, The algebraic determination of the genus of knots, Amer. J. Math. 82 (1960), 791–798

View on Pith 1960
[23]

J. H. Przytycki, Positive knots have negative signature, Bull. Polish Acad. Sci. Math. 37 (1989), no. 7-12, 559–562 (1990)

View on Pith 1989
[24]

A. Stoimenow, Knots of genus one or on the number of alternating knots of given genus, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2141–2156

View on Pith 2001
[25]

A. Stoimenow, Knots of (canonical) genus two, Fund. Math. 200 (2008), no. 1, 1–67. TWO-BRIDGE KNOTS ADMIT NO PURELY COSMETIC SURGERIES 13

View on Pith 2008
[26]

A. Stoimenow, Positive knots, closed braids and the Jones polynomial, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 2, 237–285

View on Pith 2003
[27]

P. Traczyk, Nontrivial negative links have positive signature, Manuscripta Math. 61 (1988), no. 3, 279–284

View on Pith 1988
[28]

P. Traczyk, A combinatorial formula for the signature of alternating diagrams, Fund. Math. 184 (2004), 311–316. Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, JAPAN E-mail address: ichihara.kazuhiro@nihon-u.ac.jp Department of Mathematics, Kindai University, 3-4-1 Kowakae,...

View on Pith 2004

How the literature uses it

1 arXiv paper cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as support 1 paper

Indeed, if we have Case (1) of Corollary [ref], then $P$ is a two-bridge knot, and PCSC follows from [ref] .

Purely cosmetic surgeries and pretzel knots · More than three strands · 0.96

What it cites · 3 sentences without a confident label

Uses the method or tool of 1 paper

Our argument, based on a recent result by Hanselman [ref] , uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for $3$-manifolds we deploy the $SL(2,ℂ)$ Casson invariant.

Heegaard Floer homology and cosmetic surgeries in $S^3$ · Introduction · 0.81

Cites as support 1 paper

The latter two assertions follow from [ref] directly.

Heegaard Floer homology and cosmetic surgeries in $S^3$ · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

H. U. Boden and C. L. Curtis, The SL(2, C) Casson invariant for Dehn surgeries on two-bridge knots, Algebr. Geom. Topol. 12 (2012), no. 4, 2095–2126

View on Pith 2012

[2] [2]

S. Boyer and D. Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990), 181–220

View on Pith 1990

[3] [3]

T. D. Cochran and R. E. Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology 3-spheres and property P , Topology 27 (1988), no. 4, 495–512

View on Pith 1988

[4] [4]

R. Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959), 258–275

View on Pith 1959

[5] [5]

C. L. Curtis, An intersection theory count of the SL 2(C)-representations of the fundamental group of a 3-manifold, Topology 40 (2001), no. 4, 773–787

View on Pith 2001

[6] [6]

C. L. Curtis, Erratum to: “An intersection theory count of the SL 2(C)-representations of the fundamental group of a 3-manifold” [Topology 40 (2001), no. 4, 773–787], Topology 42 (2003), no. 4, 929

View on Pith 2001

[7] [7]

P. R. Cromwell, Homogeneous links, J. London Math. Soc. (2) 39 (1989), no. 3, 535–552

View on Pith 1989

[8] [8]

J. Hanselman, Heegaard Floer homology and cosmetic surgeries in S3, preprint, arXiv:1906.06773

View on Pith Pith review arXiv 1906

[9] [9]

A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246

View on Pith 1985

[10] [10]

K. Ichihara and T. Saito, Cosmetic surgery and SL(2, C) Casson invariant for two-bridge knots, Hiroshima Math. J. 48 (2018), no. 1, 21–37

View on Pith 2018

[11] [11]

K. Ichihara, Z. Wu, A note on Jones polynomial and cosmetic surgery, Comm. Anal. Geom. to appear

View on Pith

[12] [12]

T. Ito, On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in S3, Comm. Anal. Geom. to appear

View on Pith

[13] [13]

I. D. Jong, Alexander polynomials of alternating knots of genus two, Osaka J. Math. 46 (2009), no. 2, 353–371

View on Pith 2009

[14] [14]

Problems in low-dimensional topology, Edited by Rob Kirby. AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35–473, Amer. Math. Soc., Providence, RI, 1997

View on Pith 1993

[15] [15]

K. Kishimoto, T. Shibuya, and T. Tsukamoto, in preparation

View on Pith

[16] [16]

E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554–586

View on Pith 2005

[17] [17]

T. W. Mattman, G. Maybrun and K. Robinson, 2-bridge knot boundary slopes: diameter and genus, Osaka J. Math. 45 (2008), no. 2, 471–489

View on Pith 2008

[18] [18]

Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019)

View on Pith 2019

[19] [19]

W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topol- ogy 23 (1984), no. 1, 37–44

View on Pith 1984

[20] [20]

K. Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248

View on Pith 1958

[21] [21]

K. Murasugi, On a certain subgroup of the group of an alternating link, Amer. J. Math. 85 (1963), 544–550

View on Pith 1963

[22] [22]

L. Neuwirth, The algebraic determination of the genus of knots, Amer. J. Math. 82 (1960), 791–798

View on Pith 1960

[23] [23]

J. H. Przytycki, Positive knots have negative signature, Bull. Polish Acad. Sci. Math. 37 (1989), no. 7-12, 559–562 (1990)

View on Pith 1989

[24] [24]

A. Stoimenow, Knots of genus one or on the number of alternating knots of given genus, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2141–2156

View on Pith 2001

[25] [25]

A. Stoimenow, Knots of (canonical) genus two, Fund. Math. 200 (2008), no. 1, 1–67. TWO-BRIDGE KNOTS ADMIT NO PURELY COSMETIC SURGERIES 13

View on Pith 2008

[26] [26]

A. Stoimenow, Positive knots, closed braids and the Jones polynomial, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 2, 237–285

View on Pith 2003

[27] [27]

P. Traczyk, Nontrivial negative links have positive signature, Manuscripta Math. 61 (1988), no. 3, 279–284

View on Pith 1988

[28] [28]

P. Traczyk, A combinatorial formula for the signature of alternating diagrams, Fund. Math. 184 (2004), 311–316. Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, JAPAN E-mail address: ichihara.kazuhiro@nihon-u.ac.jp Department of Mathematics, Kindai University, 3-4-1 Kowakae,...

View on Pith 2004