THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS

Transcription

1 THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS KATIE WALSH ABSTRACT. Usig the techiques of Gego Masbuam, we povide ad pove a fomula fo the Coloed Joes Polyomial of (1,l 1,2k)-petzel kots to allow us to have a computable fomula fo the coloed Joes polyomial fo a lage class of kots. 1. INTRODUCTION AND BACKGROUND The coloed Joes polyomial is a kot ivaiat that assigs to each kot a sequece of Lauet polyomials idexed by N 2, the umbe of colos. Fo a kot K, deote the Nth tem i this sequece J K,N (q), whee N coespods with the N dimesioal epesetatio, i.e. we use the covetio that whe N = 2, we get the Joes polyomial. We usually thik of the N-coloed Joes polyomial as eithe the Joes polyomial of a liea combiatio of i-cabligs of the kot fo 0 i N 1 o as the evaluatio i the Tempeley-Lieb algeba of the kot diagam decoated with the N 1 st Joes-Wezl idempotet. I what follows, the coloed Joes polyomial is omalized so that its value o the ukot is 1. Oe of the mai ope questios about the coloed Joes polyomial is how to elate it to the geomety of the kot. Oe such elatio is the followig Volume Cojectue. Cojectue 1.1 ([Mu10], Kashaev-Muakami-Muakami). Fo ay hypebolic kot K, log J K,N (e 2πi/N ) 2π lim = vol(s 3 \K) N N whee J K,N (e 2πi/N ) is the omalized Coloed Joes Polyomial of a kot K evaluated at a N th oot of uity ad vol(s 3 \K) is the volume of the uique complete hypebolic Riemaia metic o the kot complemet. The hypebolic volume cojectue has bee poved fo tous kots, the figue-eight kot, Whitehead doubles of tous kots, positive iteated tous kots, Boomea igs, (twisted) Whitehead liks, Boomea double of the figue-eight kot, Whitehead chais, ad fully augmeted liks (see [Mu10]). It is still ope fo othe kots ad liks. I [Das], Dasbach ad Li elated the fist ad last two coefficiets of the oigial Joes polyomial to the the volume of the kot i the followig way: Theoem 1.2 (Dasbach, Li). Volume-ish Theoem: Fo a alteatig, pime, o-tous kot K let J K,2 (q a q + + a m q m be the Joes polyomial of K. The 2v 8 (max( a m 1, a +1 ) 1) Vol(S 3 K) 10v 3 ( a +1 + a m 1 1). Hee, v is the volume of a ideal egula hypebolic tetahedo ad v is the volume of a ideal egula hypebolic octahedo. Date: Octobe 23,

2 2 KATIE WALSH They also poved that the fist two ad last two coefficiets of the Joes Polyomial whee also the fist ad last two coefficiets of the N-coloed Joes polyomial fo all N ad oticed that the fist ad last N coefficets of the N-coloed Joes polyomial seemed to be the same, up to sig, as the fist N coefficiets of the k-coloed Joes polyomial fo all k > N. These types of theoems ecouage us to look moe deeply i to what the coefficiets of the coloed Joes polyomial ca tell us about the kot Pattes i the Coefficiets of the Coloed Joes Polyomial. Whe studyig the coefficiets of the coloed Joes polyomial, I fist looked at pattes i the etie set of coefficiets. To be able to visualize these pattes, I used a fomula iitially poved by Habio ad epoved by Masbaum i [Mas03] to calculate the coloed Joes polyomial of the figue 8 kot ad twist kots ad the plotted the coefficiets of these polyomials. The plot of the coefficiets fo the 95th coloed Joes polyomial of the figue 8 kot is below. (The plot has the degee of the tem o the x axis ad the coefficiet o the y axis. Degees wee shifted by multiplyig by qm fo some M so that all the degees wee positive.) F IGURE 1. Coefficiets of the 95th Coloed Joes Polyomial fo the Figue Eight Kot We see the same basic shape i othe kots as well. Below is a simila plot fo the 30th coloed Joes polyomial of kot 52. This led me to the followig cojectues about the basic shape of the plot of the coefficiets of the N th coloed Joes polyomial. (1) I the middle, the coefficiets of JK,N ae appoximately peiodic with peiod N. (2) Thee is a sie wave like oscillatio with a iceasig amplitude o the fist ad last quate of the coefficiets. (3) We ca see that the oscillatio pesists thoughout the etie polyomial. The amplitude stats small, gow steadily ad the levels off i the middle ad the goes back dow i a simila mae. I ode to be able to look at the coefficiets of the coloed Joes Polyomial fo moe kots, we used the techiques fom [Mas03] to fid a fomula fo the coloed Joes polyomial of petzel kots of the fom (1, 1, 2p 1). This fomula is the mai topic of this ote.

3 THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS FIGURE 2. Coefficiets of the 30 th Coloed Joes Polyomial fo the Kot WHAT IS A PRETZEL KNOT? c 1 c 2 c 3 c FIGURE 3. A (c 1,c 2,...,c ) Petzel Kot. A box with a c i epesets c i half twists. A petzel kot o lik is usually descibed by P(c 1,c 2,...c ) whee each c i is a itege epesetig the umbe of half twists withi that sectio of the kot. These twisted pats ae daw vetically. Positive c i coespod with positive half twists, while egative c i coespod with egative half twists. See Figue 3. I ode fo this to fom a kot (have oly oe compoet), we eed to be odd ad c i to be odd fo each i o fo ay exactly oe eve c i. I othe wods, if all c i ae odd, the we have a kot if is odd ad a two compoet lik is is eve. If at least oe c i is eve, the the umbe of compoets is equal to the umbe of eve c i. I this chapte, we coside petzel kots of the fom P(1,2p 1, 1).

4 4 KATIE WALSH 3. THE FORMULA Theoem 3.1. A petzel kot of the fom K p,l = P(1,2p 1, 1) has the coloed Joes polyomial J N (K p,,a 2 [ ] N + c,p µ N 1 δ(2k;,) 2k ([k]!) 2 {2 + 1}!,,2k [2k]! {}!{1} Coollay 3.2. Whe is eve this educes to J (K p,,a 2 [ ] N + ( 1) N 1 c,p Coollay 3.3. Whe is odd this educes to J (K p,,a 2 [ ] N + {2 + 1}!{}! ( 1) µ 4p c,p N 1 (a a 1 ) 2 {1} {2 + 1}! c,/2 {1}. Coollay 3.4. Oe educed way to wite the fomula the woks fo all is [ ] N + ( 1) c {2 + 1}!{}! 1,p N 1 {1} (a a 1 ) 2 µ 2k 2 [2k + 1] [ + k + 1]![ k]! ( 1) k(+1) [2k + 1] [ + k + 1]![ k]! µ/2 2k The otatio. We thik of the N th -Coloed Joes Polyomial of a kot K as the Kauffma backet of K cabled by ( 1) e. We omalize so that the ukot has coloed Joes Polyomial 1. Fo ow, we tu ou attetio to the Kauffma backet. The Kauffma backet gives a isomophism fom the skei module K(M) of ad oieted 3-maifold M to Z[A ± ]. It is omalized so that the backet of the empty lik is 1. The elemet ω K(M) as defied i [Mas03] has the popety that the backet of a lik with ω liked aoud oe compoet has the same Kauffma backet as the same lik with a positive full twist. We ca exted with to ω p which has the popety of iducig p full ight had twists. (If p is egative, we get left haded twists.) I [Mas03], the fomula fo ω is poved. We will simply estate it hee ad the defie the ecessay pieces of the fomula. ω p = c,pr. Hee, R is a basis fo the skei module of the solid tous which is isomophic to Z[A ± ][z] Specifically, R = (!) 1 1 i=0 (z λ 2i) whee λ i = a i+1 a i 1. The coefficiets ae c,p = 1 (a a 1 ) whee µ i = ( 1) i A i2 +2i ad as usual Also, as ca be expected ( 1) k µ p []! 2k [2k + 1] [ + k + 1]![ k]!, a = A 2,{} = a a,[] = a a {}! = {}{ 1}...{1} []! = [][ 1]...[1] [ ] []! := k [k]![ k]!. a a 1

5 THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS 5 2p-1-1 2p-1-1 2p 2p FIGURE 4. We ca move the leftmost stad i the left-most image ove the middle stad to view the(1,2p 1, 1) petzel kot as a double twist kot 3.2. Poof. The fist step is to edaw the (1,2p 1, 1) petzel kot as a double twist kot. To do this, we pull the ove-stad of the sigle cossig ove the middle cossig sectio ad the otate this twist egio a quate tu i the clockwise diectio. See Figue 4. Now we ae eady to compute the coloed Joes polyomial of this diagam. We will use the defiitio used i [Mas03], amely, J N (K p,,a 2 ( 1) K(e ). We will also use the fact that ad Thus we have e = ω p = c k,p R k [ ] N + ( 1) N 1 J N (K p,,a 2 ( 1) K(e ) = ( 1) 2p R e N-1 1 twist Sice we ae takig the Kauffma backet, ad this is ot ivaiat ude R1, we eed to be caeful to ot emove kiks. We also wat the kot to be zeo-famed. The famig depeds o whethe is eve o odd. At this poit, if is eve, the famig is 2p 2+1. Whe is odd, the famig is 2p + 1. We ow will add 2p + 1 twists. This gives us a 0-famig i the case whee is eve. Fo the case whee is odd we should add 2p + 1 twists. This will give us -2p+ twists. The images below ae fo the eve case. We just eed to elabel the umbe of twists to get the odd case. Thus we have: 2p JN(Kp,,a2 ( 1) e N-1 2p+ twists

6 6 KATIE WALSH Now, we place ω p aoud the pat whee thee is 2p twists. This udoes the 2p full twists but also chages the famig (i.e. it udoes 2p of the exta twists.) I the udaw case whee is odd we ow have 4p + twists. We get e N-1 J N (K p,,a 2 ( 1) ω p twists Now, we use the expasios of e ad ω p fom above. We get J N (K p,,a 2 ( 1) k c k,p [ ] R N + ( 1) R' k N 1 twists Now, followig [Mas03] Sec 5, whe each compoet is a zeo-famed ukot with a spaig disk pieced twice by the othe compoet, the oly tems that ae ozeo ae those whee k =. Thus we have J N (K p,,a 2 [ ] R N + c,p( 1) R' N 1 twists Sice R e has degee less tha we ca eplace R with e ad the do fusio.at this poit, we will emove the exta twists. Each twist chages the coloed Joes polyomial by µ. Thus, the chage depeds o if is eve. Defie: { µ : eve J N (K p,,a 2 µ ( µ 4p+ [ ] N + c,p( 1) N 1 : odd µ R' 2 We will ow daw out the twist ad otate the diagam. J N (K p,,a 2 [ ] N + c,p( 1) N 1 µ R' 2... R' = [ ] N + c,p( 1) N 1 µ 2...

7 THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS 7 J N (K p,,a 2 [ ] N + c,p( 1) N 1 µ δ(2k;,) 2k,,2k ([k]!) 2 [2k]! 2 R' Fially, 2 R' = ( 1) {2 + 1}! {}!{1} so we get J N (K p,,a 2 c,p [ ] N + N 1 µ 3.3. Simplificatio. Fist we eode the tems. J N (K p,,a 2 [ ] N + {2 + 1}! c,p N 1 {}!{1} δ(2k;,) 2k,,2k Now we will simplify some of the pieces. Notice that: (1) (2) ([k]!) 2 [2k]! {2 + 1}! {}!{1} µ δ(2k;,) 2k ([k]!) 2,,2k [2k]! 2k ([k]!) 2 ( 1) 2k [2k + 1][]![]![2k]![k]! 2 =,,2k [2k]! ( 1) +k [ + k + 1]![k]![k]![ k]![2k]! = ( 1)+k []! 2 [2k + 1] [ + k + 1]![ k]! Now, we ll coside the cases whe is eve ad is odd sepaately. Fist, coside the case whee is eve. Fom [Mas03], we kow δ(c;a,b) 2 = µ c µ a µ b. So ( ) /2 (3) δ(2k;,) µ2k = = µ 2k /2 µ µ µ Whe is eve µ ( µ. Thus ou equatio educes to J N (K p,,a 2 [ ] N + {2 + 1}! = c,p c,p N 1 {}!{1} [ ] N + {2 + 1}! N 1 Now, agai fom [Mas03], we have (4) c,ρ = Thus J N (K p,,a 2 {}! (a a 1 ) 2 µ 2k /2 ( 1)+k []! 2 [2k + 1] [ + k + 1]![ k]! {}!{1} ( 1) {}! 2 (a a 1 ) 2 c,p ( 1) k µ ρ 2k [2k + 1] [ + k + 1]![ k]! [ ] N + {2 + 1}! ( 1) c,/2 N 1 {1} ( 1) k µ /2 2k [2k + 1] [ + k + 1]![ k]! as claimed i Coollay 3.2. Now whe is odd, we ca agai use (2) but we eed to simplify δ(2k,,) diffeetly. Fom [?] ( ) δ(2k;, ( 1) k a k2 +k a 2 2 +

8 8 KATIE WALSH so δ(2k;,) = δ(2k;,) 1 δ(2k;,) = µ 1 ( ) 2k 2 µ 1 ( 1) k a k2 +k a = µ 1 2k 2 µ 1 = µ 1 2k 2 µ 1 ( 1) k a k2 +k ( 1) a ( ) ( 1) k a (2k2 +2k)(1/2) ( 1) a = µ 1 2k 2 ( 1) k µ 1/2 2k µ 1 µ = ( 1)k µ 2k 2 µ ( ) Note this oly diffes fom (3) by a facto of ( 1) k. This obsevatio combied with the futhe simplificatio below will lead us to Coollay 3.4. Let s cotiue with the odd case. J N (K p,,a 2 = = = [ ] N + {2 + 1}! c,p N 1 {}!{1} [ ] N + {2 + 1}! c,p N 1 {}!{1} [ ] N + {2 + 1}![]! ( 1) c 2,p N 1 {}!{1} [ ] N + {2 + 1}!{}! ( 1) µ 4p c,p N 1 (a a 1 ) 2 {1} µ δ(2k;,) 2k ([k]!) 2,,2k [2k]! µ 4p+ ( 1) k µ 2k 2 ( 1) +k []! 2 [2k + 1] µ [ + k + 1]![ k]! µ 4p µ 2k 2 [2k + 1] [ + k + 1]![ k]! 4. WHICH KNOTS CAN WE STUDY? µ 2k 2 [2k + 1] [ + k + 1]![ k]! Table 1 list out the kots up to 9 cossigs that have diagams of the fom of a (1,2p, 1)-petzel kot. These fomulas wee coded i to Mathematica to allow us to calculuate out the coloed Joes polyomials fo these kots. The code is available at: k3walsh/eseach.php. Below ae the plots of the coeffciets of the coloed Joes polyomials fo vaious kots ad umbe of colos. REFERENCES [Das] Dasbach, Olive T.,Li, Xiao-Sog. A volumish theoem fo the Joes polyomial of alteatig kots. Pacific Joual of Mathematics, 231. [Mas03] G. Masbaum. Skei-theoetical deivatio of some fomulas of habio. Algeb. Geom. Topol., 3: , [Mu10] H. Muakami. A Itoductio to the Volume Cojectue. AXiv e-pits, Jauay UCSD addess: k3walsh@math.ucsd.edu

9 THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS 9 Kot Twists Petzel Notatio (p,l) (1,3,0) o (1,1,1) (2,1) o (1,2) 4 1 (1,1,2) (1,3) 5 1 (1,5,0) (3,1) (1,3,1) o (1,1,3) (2,2) o (1,4) 6 1 (1,1,4) (1,5) 6 2 (1,3,2) (2,3) 7 1 (1,7,0) (4,1) (1,1,5) o (1,5,1) (1,6) o (3,2) 7 4 (1,3,3) (2,4) 8 1 (1,1,6) (1,7) 8 2 (1,5,2) (3,3) 8 4 (1,3,4) (2,5) 9 1 (1,9,0) (5,1) (1,1,7) o (1,7,1) (1,8) o (4,2) 9 5 (1,3,5) o (1,5,3) (2,6) o (3,4) TABLE 1. Kots with up to 9 cossig that ca be expessed as a (1, 2p, 1)-petzel kot FIGURE 5. Coefficiets of the 25th Coloed Joes Polyomial of the Kot 5 2

10 10 KATIE WALSH F IGURE 6. Coefficiets of the 15th Coloed Joes Polyomial of the Kot F IGURE 7. Coefficiets of the 20th Coloed Joes Polyomial of the Kot 95