document: _ dehn surgery _ is an operation to modify a three - manifold by drilling and then regluing a solid torus . 
   denote by @xmath6 the resulting three - manifold via dehn surgery on a knot @xmath0 in @xmath7 along a slope @xmath8 . 
   two dehn surgeries along @xmath0 with distinct slopes @xmath8 and @xmath9 are called _ purely cosmetic _ if @xmath10 as oriented manifolds . in gordon s 1990 icm talk ( 
   * conjecture  6.1 ) and kirby s problem list ( * ? ? ? 
   * problem  1.81 a ) , it is conjectured that two surgeries on inequivalent slopes are never purely cosmetic . we shall refer to this as the _ cosmetic surgery conjecture_.    in the present paper we study purely cosmetic surgeries along knots in the three - sphere @xmath11 . 
   we show that for most knots @xmath0 in @xmath11 , @xmath12 as oriented manifolds for distinct slopes @xmath8 , @xmath9 . 
   more precisely , our main result gives a sufficient condition for a knot @xmath0 that admits no purely cosmetic surgery in terms of its jones polynomial @xmath13 . 
   [ main1 ] if a knot @xmath0 has either @xmath1 or @xmath2 , then @xmath12 for any two distinct slopes @xmath8 and @xmath9 .    here , 
   @xmath14 and @xmath15 denote the second and third order derivative of the jones polynomial of @xmath0 evaluated at @xmath16 , respectively . 
   note that in ( * ? ? ? 
   * proposition 5.1 ) , boyer and lines obtained a similar result for knots @xmath0 with @xmath17 , where @xmath18 is the normalized alexander polynomial . 
   we shall see that @xmath19 ( lemma [ a2 ] ) . 
   hence , our result can be viewed as an improvement of their result ( * ? ? ? 
   * proposition 5.1 ) . 
   previously , other known classes of knots that are shown not to admit purely cosmetic surgeries include the genus @xmath20 knots @xcite and the knots with @xmath21 @xcite , where @xmath22 is the concordance invariant defined by ozsvth - szab @xcite and rasmussen @xcite using floer homology . 
   theorem [ main1 ] along with the condition @xmath21 give an effective obstruction to the existence of purely cosmetic surgery . 
   for example , we used knotinfo @xcite , knot atlas @xcite and baldwin - gillam s table in @xcite to list all knots that have simultaneous vanishing @xmath14 , @xmath15 and @xmath22 invariant . 
   we get the following result :    the cosmetic surgery conjecture is true for all knots with no more than @xmath3 crossings , except possibly @xmath23 @xmath24    in @xcite , ozsvth and szab gave the example of @xmath25 , which is a genus two knot with @xmath26 and @xmath27 . 
   @xmath28 and @xmath29 have the same heegaard floer homology , so no heegaard floer type invariant can distinguish these two surgeries . 
   this example shows that theorem [ main1 ] and those criteria from heegaard floer theory are independent and complementary . 
   the essential new ingredient in this paper is a surgery formula by lescop , which involves a knot invariant @xmath30 that satisfies a crossing change formula ( * ? ? ? 
   * section 7 ) . we will show that @xmath30 is actually the same as @xmath31 . 
   meanwhile , we also observe that @xmath30 is a finite type invariant of order @xmath5 . 
   this enables us to reformulate theorem [ main1 ] in term of the finite type invariants of the knot ( theorem [ cormain ] ) .    as another application of theorem [ main1 ] , we prove the nonexistence of purely cosmetic surgery on certain families of two - bridge knots and whitehead doubles . along the way , 
   an explicit closed formula for the canonically normalized finite type knot invariant of order 3 @xmath32 is derived for two - bridge knots in conway forms @xmath33 in proposition [ v3formula ] , which could be of independent interest . 
   the remaining part of this paper is organized as follows . in section 2 
   , we review background and properties of jones polynomial , and prove crossing change formulae for derivatives of jones polynomial . in section 3 , we define an invariant @xmath34 for rational homology spheres and then use lescop s surgery formula to prove theorem [ main1 ] . in section 4 and section 5 , we study in more detail cosmetic surgeries along two - bridge knots and whitehead doubles .    * acknowledgements . * 
   the authors would like to thank tomotada ohtsuki and ryo nikkuni for stimulating discussions and drawing their attention to the reference @xcite@xcite . the first named author is partially supported by jsps kakenhi grant number 26400100 . 
   the second named author is partially supported by grant from the research grants council of hong kong special administrative region , china ( project no 
   . 14301215 ) . 
   suppose @xmath35 , @xmath36 , @xmath37 is a skein triple of links as depicted in figure [ crossings ] . 
   , @xmath36 , @xmath38 are identical except at one crossing.,title="fig : " ] ( -188,-10)@xmath39 ( -109,-10)@xmath36 ( -30 , -10)@xmath38    recall that the _ jones polynomial _ satisfies the skein relation @xmath40 and the _ conway polynomial _ satisfies the skein relation @xmath41 the _ normalized alexander polynomial _ 
   @xmath42 is obtained by substituting @xmath43 into the conway polynomial .    for a knot @xmath0 , 
   denote @xmath44 the @xmath45-term of the conway polynomial @xmath46 . 
   it is not hard to see that @xmath47 . if one differentiates equations ( [ jones ] ) and ( [ conway ] ) twice and 
   compares the corresponding terms , one can also show that @xmath48 . 
   see @xcite for details . in summary 
   , we have :    [ a2]for all knots @xmath49 , @xmath50    in @xcite , lescop defined an invariant @xmath30 for a knot @xmath0 in a homology sphere @xmath7 . 
   when @xmath51 , the knot invariant @xmath30 satisfies a crossing change formula @xmath52 where @xmath53 is a skein triple consisting of two knots @xmath54 and a two - component link @xmath55 ( * ? ? ? 
   * proposition 7.2 ) . clearly , the values of @xmath56 are uniquely determined by this crossing change formula once we fix @xmath30(@xmath57 ) for the unknot . 
   this gives an alternative characterization of the invariant @xmath30 for knots in @xmath11 . 
   the next lemma relates it to the derivatives of jones polynomial . 
   [ w3 ] for all knots @xmath49 , @xmath58    the main argument essentially follows from nikkuni ( * ? ? ? 
   * proposition 4.2 ) . 
   we prove the lemma by showing that @xmath31 satisfies an identical crossing change formula as equation ( [ w3def ] ) . to this end 
   , we differentiate the skein formula for the jones polynomial ( [ jones ] ) three times and evaluate at @xmath16 . abbreviating the jones polynomial of the skein triple @xmath59 , @xmath60 and @xmath61 by @xmath62 , @xmath63 and @xmath64 , respectively 
   , we obtain    @xmath65    the terms on the right hand side can be expressed as    * @xmath66 * @xmath67 * @xmath68 , @xmath69 * @xmath70 * @xmath71 * @xmath72    here , ( a ) and ( d ) are well - known ; ( b),(c),(e ) and ( f ) are proved by murakami @xcite . after doing substitution and simplification 
   , we have @xmath73 meanwhile , it follows from ( [ conway ] ) and hoste ( * ? ? ? 
   * theorem 1 ) that @xmath74 this enables us to further simplify @xmath75 and reduce it to the same expression as the right hand side of ( [ w3def ] ) . as @xmath76 also equals @xmath77 when @xmath0 is the unknot , @xmath31 must equal @xmath56 for all @xmath49 . 
   we conclude the section by remarking that both lemma [ a2 ] and lemma [ w3 ] can be seen in a simpler way from a more natural perspective . 
   a knot invariant @xmath78 is called a _ finite type invariant _ of order @xmath79 if it can be extended to an invariant of singular knots via a skein relation @xmath80 where @xmath81 is the knot with a transverse double point ( see figure [ crossings2 ] ) , while @xmath78 vanishes for all singular knots with @xmath82 singularities .     with a transverse double point ]    it follows readily from the definition that the set of finite type invariant of order @xmath77 consists of all constant functions . 
   one can also show that @xmath44 and @xmath14 are finite type invariants of order @xmath83 , while @xmath56 and @xmath15 are finite type invariants of order @xmath5 . as the dimension of the set of all finite type invariants of order @xmath84 and @xmath85 are two and three , respectively ( see , e.g. , @xcite ) , there has to be a linear dependence among the above knot invariants , from which one can easily deduce lemma [ a2 ] and lemma [ w3 ] . 
   in fact , if we denote @xmath86 and @xmath87 the finite type invariants of order @xmath83 and @xmath5 respectively normalized by the conditions that @xmath88 and @xmath89 for any knot @xmath0 and its mirror image @xmath90 and that @xmath91 for the right hand trefoil @xmath92 , then it is not difficult to see that @xmath93 and @xmath94 
   the goal of this section is to prove theorem [ main1 ] . 
   recall the following results about purely cosmetic surgery from ( * ? ? ? 
   * proposition 5.1 ) and ( * ? ? ? 
   * theorem 1.2 ) . 
   [ mainlemma ] suppose @xmath0 is a nontrivial knot in @xmath11 , @xmath95 are two distinct slopes such that @xmath96 as oriented manifolds 
   . then the following assertions are true :    * @xmath27 . 
   * @xmath97 . * if @xmath98 , where @xmath99 are coprime integers , then @xmath100 . * 
   @xmath26 , where @xmath22 is the concordance invariant defined by ozsvth - szab @xcite and rasmussen @xcite . 
   our new input for the cosmetic surgery problem is lescop s @xmath34 invariant which , roughly speaking , is the degree @xmath83 part of the kontsevich - kuperberg - thurston invariant of rational homology spheres @xcite . like the famous le - murakami - ohtsuki invariant 
   , the kontsevich - kuperberg - thurston invariant is universal among finite type invariants for homology spheres @xcite@xcite@xcite . 
   see also ohtsuki @xcite for the connection to perturbative and quantum invariants of three - manifolds . 
   we briefly review the construction . 
   jacobi diagram _ is a graph without simple loop whose vertices all have valency @xmath5 . 
   the degree of a jacobi diagram is defined to be half of the total number of vertices of the diagram . if we denote by @xmath101 the vector space generated by degree @xmath79 jacobi diagrams subject to certain equivalent relations as and ihx , then the degree @xmath79 
   part @xmath102 of the kontsevich - kuperberg - thurston invariant takes its value in @xmath101 .    simple argument in combinatorics implies that    * @xmath103 is an @xmath20-dimensional vector space generated by the jacobi diagram @xmath104{tata.pdf } } $ ] * @xmath105 is a @xmath83-dimensional vector space generated by the jacobi diagrams @xmath104{tata.pdf } } \ ; { \includegraphics[bb= 0 50 163 160 , scale=.09]{tata.pdf } } $ ] and @xmath106{tetra.pdf } } $ ]    many interesting real invariants of rational homology spheres can be recovered from the kontsevich - kuperberg - thurston invariant @xmath107 by composing a linear form on the space of jacobi diagrams . in the simplest case , 
   the casson - walker invariant @xmath108 is @xmath109 , where @xmath110{tata.pdf } } ) = 2 $ ] . 
   we shall concentrate on the case of the degree @xmath83 invariant @xmath111 , where @xmath112{tetra.pdf } } ) = 1 $ ] and @xmath113{tata.pdf } } \ ; { \includegraphics[bb= 0 50 163 160 , scale=.09]{tata.pdf } } ) = 0 $ ] . the following surgery formula for @xmath34 
   is proved by lescop and will play a central role in the proof of our main result . 
   * theorem 7.1 ) [ lescopsurgery ] the invariant @xmath34 satisfies the surgery formula @xmath114 for all knots @xmath115 .    here 
   , @xmath44 is the @xmath45-coefficient of @xmath46 , and @xmath116 is the lens space obtained by @xmath117 surgery on the unknot . 
   then @xmath56 is a knot invariant , which was shown earlier in lemma [ w3 ] to be equal to @xmath31 for @xmath49 . 
   the terms @xmath118 and @xmath119 are both explicitly defined in @xcite , but they will not be needed for our purpose . for the moment , we make the following simple observation .    [ w3cosmetic ] suppose @xmath0 is a knot in @xmath11 with @xmath120 , and @xmath121 are nonzero integers satisfying @xmath122 
   . then @xmath123 if and only if @xmath124 . 
   we apply the surgery formula in theorem [ lescopsurgery ] . 
   note that the first and third terms of the right hand side are clearly equal for @xmath117 and @xmath125 surgery . 
   next , recall the well - known theorem that two lens spaces @xmath126 and @xmath127 are equivalent up to orientation - preserving homeomorphisms if and only if @xmath128 . in particular 
   , this implies the lens spaces @xmath129 as oriented manifolds if @xmath130 , so their @xmath34 invariants are obviously the same . 
   consequently , @xmath131 and the statement follows readily . 
   in light of theorem [ mainlemma ] , we only need to consider the case when @xmath27 and @xmath132 , for otherwise , the pair of manifolds @xmath133 and @xmath134 will be non - homeomorphic as oriented manifolds . thus @xmath135 . 
   if we now assume @xmath136 , then lemma [ w3 ] implies that @xmath137 
   . we can then apply proposition [ w3cosmetic ] and conclude that @xmath138 . 
   consequently , @xmath139 .    given ( [ v2 ] ) and ( [ v3 ] ) , theorem [ main1 ] can be stated in the following equivalent way , which is particularly useful in the case where it is easier to calculate the finite type invariant @xmath87 ( or equivalently @xmath30 ) than the jones polynomial . 
   [ cormain ] if a knot @xmath0 has the finite type invariant @xmath140 or @xmath141 , then @xmath12 for any two distinct slopes @xmath8 and @xmath9 . 
   in this section , we derive an explicit formula for @xmath87 and use it to study the cosmetic surgery problem for two - bridge knots . following the presentation of ( * ? ? ? * section 2.1 ) , we sketch the basic properties and notations for two - bridge knots .    every two - bridge knot can be represented by a rational number @xmath142 for some odd integer @xmath143 and even integer @xmath144 . if we write this number as a continued fraction with even entries and of even length @xmath145=2b_1 + \cfrac{1}{2c_1            + \cfrac{1}{\cdots            + \cfrac{1}{2b_m + \cfrac{1}{2c_m } } } } \ ] ] for some nonzero integers @xmath146 s and @xmath147 s , then we obtain the _ conway form _ @xmath148 of the two - bridge knot , which is a special knot diagram as depicted in figure  [ conwayform ] . 
   we will write @xmath149 for the knot of conway form @xmath148 . 
   the genus of @xmath150 is @xmath151 ; and conversely , every two - bridge knot of genus @xmath151 has such a representation .     of a two bridge knot . 
   in the figure , there are @xmath152 positive ( resp . 
   negative ) full - twists if @xmath153 ( resp @xmath154 ) , and there are @xmath155 negative ( resp . 
   positive ) full - twists if @xmath156 ( resp . 
   @xmath157 ) for @xmath158 . ] 
   burde obtained the following formula for @xmath159 , the @xmath45-coefficient of the conway polynomial of @xmath150 . 
   * proposition 5.1)[burde ] for the two - bridge knot @xmath149 , the @xmath45-coefficient of the conway polynomial is given by @xmath160    the above formula can be proved by recursively applying equation ( [ a2crossingchange ] ) . 
   the similar idea can be used to find an analogous formula for @xmath30 , which is the main task of the next few lemmas . 
   [ w3recursivelemma ] the invariant @xmath30 satisfies the recursive formula @xmath161    this follows from a direct application of the crossing change formula ( [ w3def ] ) at the rightmost crossing in figure [ conwayform ] , and the observation that both @xmath162 and @xmath163 are the unknot with @xmath164 . 
   [ w3lemma2 ] the invariant @xmath30 satisfies the recursive formula @xmath165    we first prove the lemma for @xmath166 . 
   we repeatedly apply lemma [ w3recursivelemma ] until @xmath167 is reduced to @xmath77 . 
   note that the knot @xmath168 can be isotoped to @xmath169 by untwisting the far - right @xmath170 full twists . 
   therefore ,    @xmath171    now , the lemma follows from substituting @xmath172 which is an immediate corollary of proposition [ burde ] . 
   the case when @xmath173 is proved analogously . 
   finally , applying lemma [ w3lemma2 ] and induction on @xmath151 , we obtain an explicit formula for @xmath30 , and consequently also for @xmath87 . 
   [ v3formula ]    @xmath174    we use induction on @xmath151 . 
   for the base case @xmath175 , lemma [ w3lemma2 ] readily implies that @xmath176 so @xmath177 satisfies the formula . 
   next we prove that if the formula holds for @xmath178 , then it also holds for @xmath33 . 
   it suffices to show that @xmath179 where @xmath180 the above identity can be verified from tedious yet elementary algebra . 
   we omit the computation here . 
   for the rest of the section , we apply theorem [ cormain ] and proposition [ v3formula ] to study the cosmetic surgery problems for the two - bridge knots of genus @xmath83 and @xmath5 , which correspond to the conway form @xmath181 and @xmath182 , respectively . 
   note that the cosmetic surgery conjecture for genus one knot is already settled by wang @xcite . 
   if a genus @xmath83 two - bridge knot @xmath181 is not of the form @xmath183 for some integers @xmath184 , then it does not admit purely cosmetic surgeries . 
   suppose there are purely cosmetic surgeries for the knot @xmath181 . 
   theorem [ cormain ] implies that @xmath185 and @xmath186 where the formula for @xmath187 and @xmath87 follows from proposition [ burde ] and proposition [ v3formula ] , respectively . from equation ( [ v2genus2 ] ) 
   , we see @xmath188 and @xmath189 , which was then substituted into the second and the third terms of equation ( [ v3genus2 ] ) , and gives @xmath190 hence , @xmath191 . plugging this identity back to equation ( [ v2genus2 ] ) 
   , we see @xmath192 . as a result 
   , the two - bridge knot @xmath181 can be written as @xmath183 for some integers @xmath167 and @xmath193 . 
   we can perform a similar computation for a genus @xmath5 two - bridge knot @xmath182 . by proposition [ v3formula ] , @xmath194 in particular 
   , we see @xmath195 consequently , theorem [ cormain ] implies    the family of two - bridge knots @xmath196 does not admit purely cosmetic surgeries .    as explained in @xcite , both @xmath197 and @xmath198 
   are @xmath77 for the knot @xmath196 . 
   hence , purely cosmetic surgery could not be ruled out by previously known results from theorem [ mainlemma ] . 
   we are devoted to @xmath199 in this section , where @xmath199 denotes the satellite of @xmath0 for which the pattern is a positive - clasped twist knot with @xmath79 twists . 
   the knot @xmath199 is called the _ positive @xmath79-twisted whitehead double _ of a knot @xmath0 . 
   see figure [ whitehead ] for an illustration . 
   we apply the crossing change formula ( [ w3def ] ) at either one of the crossings of the clasps . 
   note that @xmath204 , @xmath205 is the unknot , and @xmath206 . 
   the classical formula for the alexander polynomial of a satellite knot implies that @xmath207 , from which we compute @xmath208 also observe that @xmath209 . 
   therefore , @xmath210 and so @xmath211    since the invariant @xmath212 , the whitehead double @xmath213 does not admit purely cosmetic surgeries if @xmath214 . when @xmath215 , proposition [ v3whitehead ] gives @xmath216 . 
   hence , theorem [ cormain ] immediately implies the following corollary . 
   there is no purely cosmetic surgery for the positive @xmath79-twisted whitehead double @xmath213 for @xmath214 . 
   moreover , if @xmath217 , then there is no purely cosmetic surgery for the untwisted whitehead double @xmath218 .
summary: we show that two dehn surgeries on a knot @xmath0 never yield manifolds that are homeomorphic as oriented manifolds if @xmath1 or @xmath2 . as an application 
   , we verify the cosmetic surgery conjecture for all knots with no more than @xmath3 crossings except for three @xmath4-crossing knots and five @xmath3-crossing knots . 
   we also compute the finite type invariant of order @xmath5 for two - bridge knots and whitehead doubles , from which we prove several nonexistence results of purely cosmetic surgery .