aa r X i v : . [ m a t h . G T ] F e b A QUANTUM OBSTRUCTION TO PURELY COSMETIC SURGERIES
RENAUD DETCHERRY
Abstract.
We present new obstructions for a knot K in S to admit purely cosmeticsurgeries, which arise from the study of Witten-Reshetikhin-Turaev invariants at fixed level,and can be framed in terms of the colored Jones polynomials of K. In particular, we strengthen a recent result of Hanselman [9], showing that if K has purelycosmetic surgeries then the slopes of the surgery are of the form ± k , except if J K ( e iπ ) = 1 , where J K is the Jones polynomial of K. For any odd prime r, we also give an obstructionfor K to have a ± k surgery slope with r ∤ k that involves the values of the first r − coloredJones polynomials of K at an r -th root of unity. We verify the purely cosmetic surgeryconjecture for all knots with at most 17 crossings. Introduction
Given an oriented 3-manifold M with ∂M = T , a pair of slopes r = r ′ on the boundaryis said to form a cosmetic surgery pair if the Dehn-fillings M ( r ) and M ( r ′ ) are homeomor-phic, and a purely cosmetic surgery pair if M ( r ) and M ( r ′ ) are homeomorphic as oriented3-manifolds. We will write M ( r ) ≃ M ( r ′ ) when the two manifolds M ( r ) and M ( r ′ ) arepositively homeomorphic.In the case where M is the complement E K of a non-trivial knot K in S , the cosmeticsurgery conjecture, first formulated by Gordon in [8], asserts that: Conjecture 1.1. (Cosmetic Surgery Conjecture) If K is a non-trivial knot in S and r = r ′ are two slopes, then E K ( r ) ≇ E K ( r ′ ) . In other words, no non-trivial knot in S has a purely cosmetic surgery pair. In the moregeneral setting of a 3-manifold with torus boundary M, the cosmetic surgery conjecture saysthat M ( r ) ≃ M ( r ′ ) if and only if there is a positive self-homeomorphism of M which sendsthe slope r to r ′ . Cosmetic surgery pairs which are not purely cosmetic are called chirally cosmetic. Chi-rally cosmetic pairs do exist: indeed, for any amphichiral knot, any pair of opposite slopeswill form a chirally cosmetic pair. Moreover, some chirally cosmetic pairs on the right-handtrefoil knot, and more generally, (2 , n )-torus knots, were found by Mathieu [13].The first result on Conjecture 1.1 was given by Boyer and Lines [3]. Using surgery for-mulas for the Casson-Walker and Casson-Gordon invariants, they proved the conjecture forany knot K such that ∆ ′′ K (1) = 0 , where ∆ K is the Alexander polynomial of K. The next
Date : February 24, 2021. This research has been funded by the project “ITIQ-3D” of the R´egion Bourgogne Franche–Comt´e andwas carried out at the Institut de Math´ematiques de Bourgogne. The IMB receives support from the EIPHIGraduate School (contract ANR-17-EURE-0002) finite type invariant of knots also gives an obstruction: Ichihara and Wu showed [10] that if K admits purely cosmetic surgeries then J ′′′ K (1) = 0 where J K is the Jones polynomial.Later on, much of the progress on Conjecture 1.1 has been brought by studying Heegaard-Floer homology of knots. First, it was proved than genus one knots do not have purelycosmetic surgeries [22]. Some conditions on the set of possible slopes in a purely cosmeticpair were established: it was first shown that the slopes in a purely cosmetic pair for a knot K ⊂ S have opposite signs [15] [24], then Ni and Wu showed that the slopes must actuallybe opposite and furthermore that the Ozv´ath-Szab´o-Rasmussen τ invariant of K must van-ish [14]. Also, the Heegaard-Floer homology \ HF K ( K ) must satisfy some other constraints[14][6].Then, the work of Futer, Purcell and Schleimer [5] showed for any 3-manifold with torusboundary, it can be algorithmically checked whether it admits a purely cosmetic pair. Theirresult came out of a different direction, using hyperbolic geometry to bound the length ofslopes in a purely cosmetic pair. With their method they proved the conjecture for knotswith less than 15 crossings.Using a new method to express the Heegaard-Floer homology of surgeries, Hanselman putfurther restrictions on the surgery slopes [9]. Similarly to Futer, Purcell and Schleimer’sresult, those conditions restrict the set of possible slopes to a finite set. Hanselman’s boundsseem to be more powerful in practice, with the downside that so far they apply only to thecase of knots in S . Indeed, Hanselman was able to use his results to show that no knot with prime summandswith less than 16 crossings has a purely cosmetic surgery pair.Finally, let us mention that, analyzing JSJ decompositions, Tao has proved that compositeknots [18] and cable knots [17] do not have purely cosmetic surgeries. It reduces the conjec-ture to the case of prime knots.We will now describe some of Hanselman’s main theorem from [9] in more detail below, asour results will build up on his work.For K a knot in S , let g ( K ) be its Seifert genus. Let \ HF K ( K ) be its Heegaard-Floerknot homology, which is bigraded with Alexander grading A and Maslov grading µ. Wedefine the δ -grading by δ = A − µ, and let the Heegaard-Floer thickness of K be th ( K ) = max { δ ( x ) | x = 0 ∈ \ HF K ( K ) } − min { δ ( x ) | x = 0 ∈ \ HF K ( K ) } Said differently, th ( K ) + 1 is the number of diagonals on which the Heegaard-Floer homologyof K is supported. Theorem 1.2. [9]
Let K be a knot in S and r = r ′ be slopes such that E K ( r ) ≃ E K ( r ′ ) then: - The pairs of slopes { r, r ′ } is either {± } or of the form {± k } for some non-negativeinteger k. - If { r, r ′ } = {± } then g ( K ) = 2 . - If { r, r ′ } = {± k } then k th ( K ) + 2 g ( K )2 g ( K )( g ( K ) − K ⊂ S for which Heegaard-Floer homology does not distinguish between E K ( r ) and E K ( − r ) for r = 1 or 2 . QUANTUM OBSTRUCTION TO COSMETIC SURGERIES 3
Invariants that have nice surgery expressions are natural candidates for applications to thecosmetic surgery conjecture. Therefore, the author tried to investigate what information theReshetikhin-Turaev invariants have to offer about cosmetic surgeries. As Reshetikhin-Turaevinvariants are part of TQFTs, they admit some natural surgery expressions. The case ofthe so-called SO -Reshetikhin-Turaev invariants at level 5 seems to give the most interestingcondition. We find: Theorem 1.3.
Let K ⊂ S be a knot, let J K be its Jones polynomial, and assume that K has a purely cosmetic surgery pair {± r } . Then r is of the form ± k for some k ∈ Z , unless J K ( e iπ ) = 1 . At this point, the role of the 5-th root of unity, and multiples of 5 in the denominatorof cosmetic surgery slopes may seem mysterious. This is mostly a matter of exposition: wenote that it is still possible to get similar conditions out of Reshetikhin-Turaev TQFTs atdifferent levels than 5 . From the SO -TQFT at odd level r, we get similar obstructions; withthe drawback that they involve several colored Jones polynomials of K. Here, let us note J K,n the n -th normalized colored Jones polynomial of K, with the convention that J K, = 1 and J K, is the Jones polynomial. Theorem 1.4.
Let r > be an odd prime, ζ r be a primitive r -th root of unity and K be aknot. Let also [ n ] = ζ nr − ζ − nr ζ r − ζ − r . There exists a finite set F r of nonzero vectors in C r − , withcardinal | F r | r +12 such that if K has a purely cosmetic surgery pair, then either the slopesare of the form {± rk } with k ∈ Z , or the vector v = − [2] J K, ( ζ r )[3] J K, ( ζ r ) ... ( − r − [ r − ] J K, r − ( ζ r ) is orthogonal to an element of F r . Let us now restrict to the simpler condition we get from the case r = 5 and focus on thecorollaries of Theorem 1.3.Theorem 1.3 combined with Hanselman’s results implies that if a knot K has a purely cos-metic surgery pair and J K ( e iπr ) , it must have a rather large crossing number: Corollary 1.5. If K is a non-trivial knot with at most crossings and such that J K ( e iπ ) =1 , then K has no purely cosmetic surgery pair.Proof. By Theorem 1.3, if K has a purely cosmetic surgery pair then the slopes must be ofthe form ± k , so the denominator is at least 5 . By Hanselman’s inequality and the fact that g ( K ) > th ( K ) > . However, Lowrance proved that th ( K ) g T ( K ) , where g T ( K ) is the Turaev genus of K [12].But for any knot K, the Turaev genus g T ( K ) is bounded above by c ( K ) / c ( K ) is thecrossing number of K. (see for example [1, Proposition 2.4]).Thus a knot with c ( K )
31 crossings has thickness at most 15 , and can not have a purelycosmetic surgery pair if J K ( e iπ ) = 1 . (cid:3) RENAUD DETCHERRY
Let us note that according to Hanselman’s computations [9] a prime knot with at most 16crossings has thickness at most 2 , so the inequality th ( K ) > c ( K )2 we used seems to be farfrom optimal.The litterature has been in particular interested in the special case of alternating knots. Inthat case, hypothetical alternating counterexamples of the conjecture have been shown to havea very special form. Indeed, Hanselman showed also that in [9, Theorem 3] that an alternatingknot K (or, more generally, a thin knot , which has th ( K ) = 0) with a purely cosmetic surgerymust have signature 0 and Alexander polynomial ∆ K ( t ) = nt − nt + (6 n + 1) − nt − + nt − for some n ∈ Z . As a corollary of our main result, we can put an extra condition on the Jonespolynomial of K : Corollary 1.6. If K is an alternating (or thin) knot with a purely cosmetic surgery, then J K ( e iπ ) = 1 . Proof.
It is also part of [9, Theorem 3] that the only possible purely cosmetic surgery pair ofthin knots are the pairs {± } or {± } . By Theorem 1.3, K must satisfy J K ( e iπ ) = 1 . (cid:3) Finally, let us discuss some numerical estimates of the strength of this obstruction. Look-ing at knot tables on KnotInfo, we found that the only knots with less than 12 crossingswith J K ( e iπ ) = 1 are the knots 8 , , a , a , a , a , and 12 n . We notehowever that none of those knots also satisfy ∆ ′′ K (1) = 0 , hence the combination of those twoobstructions suffices to exclude the existence of knots with less than 12 crossings.We note that knots with J K ( e iπ ) = 1 seem to become fairly rare when the number ofcrossings increases. Using a census of knots with 17 crossings and their Jones polynomialsgenerated with Regina, we found that among all 8,053,363 hyperbolic knots with 17 crossings,there are only 45 that have J K ( e iπ ) = 1 . Following Regina’s numerotation, those are theknots: 17 ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , ah , nh , nh , nh , nh , nh , nh , nh , nh , nh , nh , nh Seven of these knots also satisfy ∆ ′′ K (1) = 0 , but none of the latter also satisfy J ′′′ K (1) = 0 . We get the following corollary:
Corollary 1.7.
No non-trivial knot with at most crossings admits purely cosmetic surg-eries.Proof. Hanselman [9] treated the case of knots with at most 16 crossings, and by [18], aknot with a purely cosmetic surgery pair must be prime. Moreover, none of the 29 primesatellite knots with 17 crossings or the torus knot T , satisfy J K ( e iπ ) = 1 , and by the abovediscussion, no hyperbolic knot with 17 crossings satisfies J K ( e iπ ) = ∆ ′′ K (1) = J ′′′ K (1) = 0 . (cid:3) QUANTUM OBSTRUCTION TO COSMETIC SURGERIES 5
We note that Sikora and Tuzun [20] have done extensive computations of Jones polynomialsof knots with at most 22 crossings, in order to verify the Jones unknot detection conjectureup to that crossing number [20]. We hope that a similar search may find all knots with J K ( ζ ) = 1 with at most ≃
22 crossings, and be used to prove the cosmetic surgery conjectureup to crossing number ≃ . At the very least it feels doable to use this obstruction to verifythe conjecture for the whole census of knots with at most 19 crossings on Regina.The paper is organized as follows: in Section 2, we review the basics of the SO(3)-Witten-Reshetikhin-Turaev TQFTs that we will use. In Section 3, we derive the formula of the SO(3)-Reshetikhin-Turaev TQFTs of Dehn-surgeries of a knot from TQFT axioms. In Section 4,we prove Theorem 1.4 using the surgery formula and deduce Theorem 1.3.
Acknowledgements:
We would like to thank Fran¸cois Costantino, Andras Stipsciz andEffie Kalfagianni for their interest and helpful conservations. We also thank Cl´ement Mariafor helping us generate the census of Jones polynomials with Regina.2.
Preliminaries -Reshetikhin-Turaev TQFTs. In this section, we review some well-known prop-erties of Witten-Reshetikhin-Turaev TQFTs needed in this paper. Although they were ini-tially defined by Reshetikhin and Turaev in [16] to realize Witten’s [23] interpretation of theJones polynomial as a quantum field theory based on the Chern-Simons invariant, it is reallythe so-called SO -TQFTs in the framework of Blanchet, Habegger, Masbaum and Vogel [2]that we will present here. For every odd integer r > , let A r be a primitive 2 r -th root ofunity, and u r be a square root of A r , thus a 4 r -th root of unity. Let also U r be the multi-plicative group generated by u r , that is the set of 4 r -th root of unity. Finally, for any integer n ∈ Z the quantum integer [ n ] is given by[ n ] = A nr − A − nr A r − A − r The SO -Reshetikhin-Turaev TQFT RT r is a monoidal functor (with anomaly) from thecategory of dimension 2 + 1 cobordisms to the category of complex vector spaces. Concretelyspeaking, we have the following: Theorem 2.1. [2]
For any odd integer r > , we have: - For any closed compact oriented -manifold M, we have RT r ( M ) ∈ C / U r , a topologicalinvariant. Similarly, if M contains a framed link L then we can define a topologicalinvariant RT r ( M, L ) ∈ C / U r . - For every closed compact oriented surface Σ , RT r (Σ) is a finite dimensional vectorspace with a natural Hermitian form h· , ·i . - For any closed oriented -manifold M, containing a link L ⊂ M, and with a fixedhomeomorphism ∂M ≃ Σ , RT r ( M, L ) is a vector in RT r (Σ) , and RT r (Σ) is spannedby such vectors. - The mapping class group
Mod(Σ) , acting on such cobordisms with boundary Σ , givesrise to a projective representation ρ r : Mod(Σ) PAut( RT r (Σ)) called the SO -quantum representation. RENAUD DETCHERRY - For any two manifolds (possibly with links) M , M , with ∂M ≃ ∂M ≃ Σ , the closedmanifold M = M ` Σ M has invariant RT r ( M ) = h RT r ( M ) , RT r ( M ) i . Note that as the invariant RT r ( M ) is only defined up to a power of u r , the last equalityis also up to a power of u r . Moreover, the projective anomaly of the quantum representation ρ r is also always a power of u r . However, it is possible to lift this invariant to an invariant valued in C . To do so, one needsto equip M with an additional structure, either a so-called p -structure as in [2], or following[21] a so-called extended structure . This means we equip 3-manifolds with a weight , whichis an integer, and surfaces Σ with a Lagrangian l ⊂ H (Σ) . Then there is an invariant ofextended 3-manifold Z r ( M, n ) that lifts RT r to C , such that Z r ( M, n ) = u n Z r ( M,
0) where u is some 4 r -th root of unity. Moreover, when gluing extended manifolds M and M withboundary an extended surface (Σ , l ) , the gluing formula becomes: Z r ( M,
0) = u − µ ( λ M (Σ) ,l,λ M (Σ)) h Z r ( M , , Z r ( M , i where µ is the Maslov index, and λ Σ ( M i ) is the kernel of H (Σ , Z ) → H ( M i , Z ) . Morally,the weight of M encodes the signature of a 4-manifold bounding M. We refer to [21] and [19]for a precise definition of the Maslov index and the category of extended 2 + 1-cobordisms.In Section 3, we will compute the RT r invariants of Dehn-surgeries first, then use that thecorrection given by the Maslov index depends only on homological information to lift to the Z r invariant, by normalizing the invariant correctly for the 3-sphere.Below we recall a skein theoretic definition of the invariant RT r ( M ) . The invariant RT r of a 3-manifold M with a surgery presentation M = S ( L ) can be com-puted explicitly as a sum of colored Jones polynomials L evaluated at the 2 r -th root of unity A r = u r . We summarize its definition below.Let us first recall the definition of the Kauffman bracket h L i of a framed link in S . It is aninvariant of framed links completely determined by the normalization h∅i = 1 , and the twoKauffman relations: h L ∪ U i = ( − A − A − ) h L i , where L ∪ U denotes the disjoint union ofa framed link and a 0-framed unknot, and h L + i = A h L i + A − h L ∞ i , where L + , L and L ∞ are 3 framed links that are the same expect in the small ball where they look like a positivecrossing, and its positive and negative resolutions respectively.Furthermore, the colored Kauffman bracket h L, c i of a link L whose components are col-ored by elements of C [ z ] is defined in the following way: if all components L i ’s are coloredby monomials z d i then just replace the component L i by d i parallel copy and compute theKauffman bracket. Otherwise, expand by multilinearity.Now, suppose that M has a surgery presentation M = S ( L ) , when L is a framed link in S . Let us write r = 2 m + 1 , and let ω = m P i =1 ( − i − [ i ] e i , when e i ( z ) is the i -th Chebishevpolynomial, defined by e ( z ) = 1 , e ( z ) = z and e i +1 ( z ) = ze i ( z ) − e i +1 ( z ) . In the remaining of the paper, we will often talk about the Kauffman bracket of a link coloredwith integers i > . By this we will simply mean that the components have been colored withthe respecting e i ( z ) . Thus we will write h K, i i for h K, e i i . Finally, we introduce η r = ( A r − A − r ) √− r . QUANTUM OBSTRUCTION TO COSMETIC SURGERIES 7
Theorem 2.2. [2] If M is obtained by surgery on a framed link L ⊂ S , with n ( L ) componentsthen RT r ( M ) = η n ( L ) r h L, ω ⊗ n ( L ) i is a topological invariant, up to a r -root of unity. If furthermore, M contains a colored link ( K, c ) then RT r ( M, K, c ) = η n ( L ) r h L ∪ K, ω ⊗ n ( L ) , c i . is a topological invariant, up to a r -root of unity. In this paper, we will only compute Reshetikhin-Turaev invariants of manifolds that areDehn surgery of knots. We will not make use of the above formula directly, but rather appealto the TQFT properties of RT r , that is, use the last point of Theorem 2.1 in the special casewhere Σ is a torus, M is a knot complement in S and M is a solid torus. Therefore wewill now focus on the TQFT space and quantum representation of the torus.2.2. Quantum representations of the torus.
We first describe a basis of the RT r TQFT-space of the torus T . If 1 i r − , let e i be the vector in RT r ( T ) corresponding to thesolid torus D × S with boundary T and meridian S × { } , containing the framed knot { [0 , ε ] } × S colored by e i ( z ) . In particular, e corresponds to the solid torus with the emptylink inside. Proposition 2.3. [2, Corollary 4.10]
Let r = 2 m + 1 > be an odd integer. The vector space RT r ( T ) admits e , . . . , e m as an orthonormal basis. With this in mind, we will now describe the quantum representations of the torus, in theabove basis.Let us recall that the mapping class group of the torus is isomorphic to SL ( Z ) , and generatedby the two matrices T = (cid:18) (cid:19) and S = (cid:18) −
11 0 (cid:19) . As mapping classes, T corresponds tothe Dehn-twist along the meridian S × { } of T = S × S , and S to the map S ( u, v ) =( − v, u ) of order 4 . A presentation of SL ( Z ) is then given bySL ( Z ) = h S, T | S = 1 , S = ( ST ) i . Proposition 2.4. [7]
We have, in the basis e i , that ρ r ( T ) e i = ( − A r ) i − e i and ρ r ( S ) e i = η r X j r − ( − i + j [ ij ] e j The following lemma shows that the above definition indeed yields a projective represen-tation of SL ( Z ) : Lemma 2.5.
The matrices ρ r ( S ) and ρ r ( T ) are unitary for h , i , and we have ρ r ( T ) r = Id,ρ r ( S ) = Id and ( ρ r ( ST )) = κ r Id where κ r = − ie − iπr if r = 1 mod − e − iπr if r = 3 mod ie − iπr if r = 5 mod e − iπr if r = 7 mod RENAUD DETCHERRY
The lemma is a very standard computation in the field and the proof is therefore leftas exercise for the reader. The last identity is shown by proving the equivalent identity ρ r ( T ST S ) = κ r ρ r ( ST − ) , and involves computing the Gauss sum G ( r ) = P k ∈ Z /r Z ( − A r ) k . The above formula already shows that the representation has non-trivial kernel. Actuallymore is known:
Theorem 2.6. [4] If r > is prime, the representation ρ r : SL ( Z ) PGL r − ( C ) factorsthrough SL ( Z /r Z )The above theorem was proved by Freedman and Krushkal, who recognized from the aboveformulas that when r is prime, the quantum representation of the torus is isomorphic to aso-called metaplectic representation over the finite field Z /r Z . The quantum representationsof the torus had already been known to have finite image by work of Gilmer [7].3.
Surgery formulas for RT r and Z r In this section, we will present formulas that express the RT r and Z r invariant of a Dehn-surgery on a knot in terms of its colored Jones polynomials. We will first describe the surgeryformulas for RT r for arbitrary slopes, then for the anomaly corrected invariant Z r we willspecialize to slopes of the form k , k ∈ Z or ± . We have the following:
Proposition 3.1.
Let K be a framed knot in S , and r = 2 m + 1 > be an odd integer.Then, in the basis e , . . . , e m , we have RT r ( E K ) = η r h K, ih K, i ... h K, m i Moreover, if φ is any element of Mod( ∂E K ) which sends the meridian of K to the curve ofslope s, then RT r ( E K ( s )) = h RT r ( E K ) , ρ r ( φ ) e i . Proof.
The first identity expresses the fact that pairing E K with the basis vectors e i , wesimply get the manifold S with the link K colored by i in it. By Theorem 2.2, η r is the RT r invariant of S , and adding a knot K colored by i multiplies the invariant by h K, i i . As for the second identity, recall that e is the vector corresponding to the solid torus (withmeridian the meridian of K. Thus the second identity follows from the TQFT axioms for RT r and the definition of the quantum representation. (cid:3) Remark 3.2.
One may want to compare the vector RT r ( E K ) with the one in Theorem 1.4.To do this, we should say that while the colored Jones polynomial are invariants of knots, thecolored Kauffman bracket are invariants of framed knots only. However when talking aboutthe cosmetic surgery problem for a knot K , a framing on K is somewhat implicit, if we wantto be able to talk about slopes on the knot.Taking the convention that we chose as framing for K the longitude with zero windingnumber, we have for any A ∈ C , h K, i i ( A ) = ( − i − [ i ] J K,i ( A )where [ i ] = A i − A − i A − A − . QUANTUM OBSTRUCTION TO COSMETIC SURGERIES 9
Remark 3.3.
The map φ ∈ Mod( T ) which sends the meridian to the curve of slope s is notunique. However, any two such maps differ by multiplication on the right by T k , with k ∈ Z . As ρ r ( T ) e = e , the pairing h RT r ( E K ) , ρ r ( φ ) e i does not depend on the choice of φ. Proposition 3.4.
Let K be a knot in S then we have Z r ( E K ( 1 k ) ,
0) = h RT r ( E K ) , ρ r ( ST − k S ) e i and Z r ( E K (2) ,
0) = u d r r h RT r ( E K ) , ρ r ( T S ) e i Z r ( E K ( − ,
0) = κ r u d r r h RT r ( E K ) , ρ r ( T − S ) e i where κ r is the constant appearing in Lemma 2.5, u r = e iπ r is a r -root of unity, and d r isan integer which does not depend on K. Remark 3.5.
In the above theorem the vector RT r ( E K ) has to be understood to be exactlythe vector v featured in Theorem 1.4 (not up to a 4 r -th root of unity), and the matrices ρ r ( ST − k S ) , ρ r ( T S ) , ρ r ( T − S ) are computed using those exact words in the generators S and T. We note that it would be possible to actually compute the integer d r , or even give similarformulas for Z r ( E K ( s ) ,
0) for any slope s ∈ Q ∪ {∞} . A careful treatment of the anomalyis then needed, and the exact power of u r needed for the correction involves the continuedfraction expansion of s and the Rademacher φ function. Closely related computations canbe found in [11], where the Z r invariant of lens spaces is computed.For our purposes we will not need the exact value of d r . Proof.
Note that the matrix ST − k S sends the meridian of T to the curve of slope k . Thusfor any knot K we get Z r ( E K ( 1 k ) ,
0) = u n r r h RT r ( E K ) , ρ r ( ST − k S ) e i for some integer n r which does not depend on the knot K. Indeed, the integer n r can becomputed as a sum of Maslov indices and depends only on homological data. We compute n r using the fact that when K is the unknot U, the k -surgery is S , and Z r ( S ,
0) = η r . Notethat since J U,i = 1 for any i, the RT r vector of the unknot is RT r ( E U ) = − [2]...( − r − [ r − ] = ρ r ( S ) e . Then we compute h RT r ( E U ) , ρ r ( ST − k S ) e i = h ρ r ( S ) e , ρ r ( ST − k S ) e i = h e , ρ r ( T − k S ) e i = h ρ r ( T k ) e , ρ r ( S ) e i = h e , ρ r ( S ) e i = η r In the above, we used that ρ r ( S ) and ρ r ( T ) are isometries for h , i , and that ρ r ( T ) e = e . Similarly, T S sends the meridian of T to the curve of slope 2 , and T − S sends themeridian to the curve of slope − . Then there exists integers d r and d ′ r , independent of theknot K, such that Z r ( E K (2) ,
0) = u d r r h RT r ( E K ) , ρ r ( T S ) e i and Z r ( E K ( − ,
0) = u d ′ r r h RT r ( E K ) , ρ r ( T − S ) e i . It remains to show that u d ′ r − d r r = κ r . To do this, we use that the +2 and − L (2 , . So the Z r invariants have to be equal when K = U. However, we compute: h RT r ( U ) , ρ r ( T S ) e i = h ρ r ( S ) e , ρ r ( T S ) e i = h e , ρ r ( ST S ) e i However, ρ r ( ST S ) = ρ r ( T − ( T ST )( T ST ) T − ) = κ r ρ r ( T − ( ST − S )( ST − S ) T − ) = κ r ρ r ( T − ( ST − S ) T − )where we used that ρ r ( T ST ) = κ r ρ r ( ST − S ) . Thus h RT r ( U ) , ρ r ( T S ) e i = κ r h e , ρ r ( T − ( ST − S ) T − ) e i = κ r h e , ρ r ( ST − S ) e i = κ r h ρ r ( S ) e , ρ r ( T − S ) e i = κ r h RT r ( U ) , ρ r ( T − S ) e i as ρ r ( S ) , ρ r ( T ) are unitary for h , i and ρ r ( T ) fixes e . Therefore we need to have u d ′ r − d r r = κ r . (cid:3) Proof of the main theorems
Before we move to the proof of Theorem 1.4, we will explicit the finite set F r of vectors in RT r ( T ) . Definition 4.1.
Let r > F r = { ρ r ( ST − k S ) e − ρ r ( ST k S ) e , ≤ k ≤ r − } ∪ { ρ r ( T S ) e − κ r ρ r ( T − S ) e } where ρ r ( T ) , ρ r ( S ) are the matrices defined in Proposition 2.4 and κ r is the constant definedin Proposition 2.5.It is clear from the definition that | F r | ≤ r +12 . The following lemma shows that F r containsonly non-zero vectors: Lemma 4.2.
Let r ≥ be a prime, then ρ r ( ST − k S ) e − ρ r ( ST k S ) e = 0 if and only if k = 0 mod r. Moreover, ρ r ( T S ) e − κ r ρ r ( T − S ) e = 0 . Proof.
Note that as ρ r ( T r ) = Id, if k = 0 mod r, then ρ r ( ST ± k S ) = ρ r ( S ) = Id andthus ρ r ( ST − k S ) e − ρ r ( ST k S ) e = 0 . Assume ρ r ( ST − k S ) e = ρ r ( ST k S ) e , then applying ρ r ( T k S ) on the left we get ρ r ( T k S ) e = ρ r ( S ) e . So ρ r ( S ) e must be an eigenvector of ρ r ( T k ) of eigenvalue 1 . As the order of r is coprime with 2 k, we deduce that ρ r ( S ) e mustbe an eigenvector of ρ r ( T ) of eigenvalue 1 , thus ρ r ( S ) e must be colinear to e . This is notthe case: as all quantum integers [ i ] = A ir − A − ir A r − A − r with 1 ≤ i ≤ m are non-zero, ρ r ( S ) e hasnon-zero coefficient along each e i . Now, by contradiction assume that ρ r ( T S ) e = κ r ρ r ( T − S ) e , then similarly ρ r ( S ) e must be an eigenvector of ρ r ( T ) , and thus an eigenvector of ρ r ( T ) as 4 is coprime to r whichis the order of T. Note that the diagonal coefficients ( − A r ) i − where i = 1 , . . . , r − of ρ r ( T )are all distinct. Thus ρ r ( S ) e would have to be colinear to one of the e i ’s, which is not thecase. (cid:3) We are now ready to prove Theorem 1.4:
QUANTUM OBSTRUCTION TO COSMETIC SURGERIES 11
Proof of Theorem 1.4.
Let K be a knot in S with a purely cosmetic surgery pair s > s ′ . ByTheorem 1.2, the slopes s, s ′ must be opposite and s = k or s = 2 . Let us consider the first case and assume that r does not divide k. We have E K ( k ) ≃ E K ( − k ) , thus the Z r invariants coincide: Z r ( E K ( k ) ,
0) = Z r ( E K ( − k ) ,
0) for all odd prime r ≥ , and by Proposition 3.4 we have h RT r ( E K ) , ρ r ( ST − k S ) e − ρ r ( ST k S ) e i = Z r ( E K ( 1 k , − Z r ( E K ( − k ,
0) = 0 . Note that ρ r ( ST ± k S ) e depends only on k mod r as ρ r ( T r ) = Id.
Let l ∈ {± , ± , . . . , ± r − } such that k = l mod r. Then RT r ( E K ) must be orthogonal to the vector ρ r ( ST − l S ) e − ρ r ( ST l S ) e . Either this vector or its opposite is in the set F r , so the claim is proved in thatcase.Similarly, in the case where E K (2) ≃ E K ( − , applying Proposition 3.4, we get that h RT r ( E K ) , ρ r ( T S ) e − κ r ρ r ( T − S ) e i = Z r ( E K (2) , − Z r ( E K ( − ,
0) = 0 , and thus RT r ( E K ) is orthogonal to ρ r ( T S ) e − κ r ρ r ( T − S ) e , which is in F r by definition. (cid:3) Remark 4.3.
Note that the unknot admits ± k cosmetic surgery pair for any k ∈ Z , andthat ± RT r ( E U )is orthogonal to all of the vectors in F r . As RT r ( E U ) = ρ r ( S ) e is a non-zero vector, all ofthe vectors in F r actually lie in a codimension 1 subspace of RT r ( T ) . The proof of Theorem 1.3 is a simple corollary of Theorem 1.4, Remark 4.3 and the factthat if r = 5 , then the dimension of RT r ( T ) is 2 : Proof of Theorem 1.3.
By Theorem 1.4, if K has a purely cosmetic surgery pair ± s theneither the slope s is of the form k or RT ( E K ) must orthogonal to one of the (non-zero)vectors in F . Note that the vector of the unknot RT ( E U ) = ρ r ( S ) e = 0 is orthogonal toall of the vectors in F by Remark 4.3. As dim( RT r ( T )) = r − = 2 , the vectors in F areall colinear and any vector orthogonal to a vector in F must be colinear to RT r ( E U ) . So if the slope s is not of the form k , the vector RT r ( E K ) = η (cid:18) − [2] J K ( ζ ) (cid:19) must becolinear to RT r ( E U ) = η (cid:18) − [2] (cid:19) . As η and [2] are non-zero, this implies that J K ( ζ ) = 1 . By Galois action, we also have J K ( ζ ) = 1 . (cid:3) References [1] Tetsuya Abe,
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