The Characterization of Jones Polynomial. for Some Knots

Transcription

1 Inernon Mhemc Forum,, 8, no, 9 - The Chrceron of Jones Poynom for Some Knos Mur Cncn Yuuncu Y Ünversy, Fcuy of rs nd Scences Mhemcs Deprmen, 8, n, Turkey m_cencen@yhoocom İsm Yr Non Educon Mnsry, 8, n, Turkey smyr@gmcom bsrc In hs pper hs been ccued he Jones poynoms usng he dfference mehods for some knos so hs been emned he Jones poynoms v brcke poynom of some knos Mhemcs Subjec Cssfcon: 7M Keywords: Brcke poynom, Jones poynom, kno, kno nvrn, nk Inroducon There re some geomerc nvrns h hey re dffcuy ccouned hough hey re esy defned n kno heory Some of hem re mnm crossng number, brd nde nd brdge number Those numerc nvrns cn be compued wh gebrc nvrns ke ender poynoms, bu receny Jones poynoms hve been very usefu n ccounbe of hose numerc nvrns In he begnnng, hs been descrbed ender poynoms for some knos nd nks, see [, 7], nd hs been epned sever propery n hs opcs, see [] er hs been gven prove of heorem used o deermne of Jones poynom,

2 M Cncn nd İ Yr see [], nd ccued o ender poynoms of ws knos, see [], nd defned one new poynom nvrn concernng knos nd nks, see [9] I cn sy h hs nvenon s poneer n kno heory, whch ced Jones poynom Jones poynom hs been defnng by mens of nvrn operor gebrs so prevousy Homfy poynom hs been deermned, see [8] On he oher hnd cn be hnded wh sudy of [] o BM/Ho poynom of one kno Furher my be seen one prcc mehod for ccung of Jones poynom n [] I hs been connng hs mehod n [] In hs mehod s used brcke poynom of kno dgrm I hs desgned one ree beween Homfy nd BM/Ho poynoms, see [] BM/Ho poynom of kno hs nvesged n [] I hs so dscussed one ree beween Homfy nd Jones poynoms, see [] I hs emned one ree beween rf nvrn nd Jones poynom, see [] I s obned some generons usng Jones poynoms of ws knos o he deermnon Conwy poynoms, see [] Kno Invrns nd Sken reon ender poynom s vrbe poynom [] ender poynom of K orened kno s shown wh Δ K or Δ There re he foowng some properes wh respec o Δ : Δ s wys n odd number Δ ±, see [7] I s ccued ender poynoms of ws knos such h Δ Δ Δ n n n n n n Δ n n, see [] I s shown wh h Conwy poynom of K kno s wh one vrbe There s he reon Δ beween Δ nd, see [] I s demonsred wh K o Jones poynom of K kno I s ccued Jones poynoms of ws knos such h for n,, T n n n n k n k k

3 Chrceron of Jones poynoms kn n n n n, T k k see [] s no cssfer consn wh respec o kno ype becuse here re knos n dfference ype whch re equven o Jones poynoms For empe cn be equven o Jones poynoms of hose knos hough 8 nd knos re dfferences, see [] I s defned h uren poynom of K kno s K WK, nd so defned h rf nvrn of K kno s rfk W K, 7 see [9] Homfy poynom of kno s gener cse of Jones poynom of ree kno, nd s shown wh P, m There s reon beween P, m nd such h, P nd here s reon beween P, m nd P, Δ such h 8 Δ, 9 see [] Q Kuffmn BM/Ho poynom of kno s vrbe shown wh poynom of kno cn be shown wh F, ; F F,, see [] By mens of known Q nd vrbes cn be ccued by formus Q, F, see [] I s ssfed he reon mq,, m, m P, m beween P, m nd Q, see [] so here s he reon Q for brdge knos, nd here vue of, see [] I s wren sken reon s ken from ype such h

4 M Cncn nd İ Yr v,, orened nks for Jones poynom, see [] The Deermnon nd s ppcons of Jones Poynom n Some Knos In hs secon ws deermned Jones poynom v brcke poynom so ws ccued Jones poynoms of some knos by hs mehod nd e be D kno dgrm nd O rv kno e be < D > nd < O > respecvey brcke poynoms of D nd O In hs cse s ssfed <O> <DUO>- - - <D> < >< > - < > nd < >< > - < >, see [] Jones Poynom of kno cn be ccued wh w D < D >, w D crossng pons Here my see [] so D dgrm s o of mrks of ken such h Now cn be be o epress he ccuon of brcke poynom nd he ccuon Jones poynom for some knos Empe Brcke poynom ccuon for kno < >< > - < > b Jones poynom ccuon for kno < D > / w D <D>< >

5 Chrceron of Jones poynoms : wd : wd : <D> w D / < D > -7 - / / / / Empe Brcke poynom ccuon for kno < >< > - < > < >< > - < > < >< > - < > b Jones poynom ccuon for kno < D > / w D <D>< > : wd--

6 M Cncn nd İ Yr : wd : - : - w D / < D > 8 8 / 8 8 / / Empe Brcke poynom ccuon for kno < >< > - < > < >< > - < > 7 7 < >< > - < > 7 7 < > < >< > - < > b Jones poynom ccuon for kno < D > / w D 7 < > 9

7 Chrceron of Jones poynoms : - : - wd----- : - : - wd- : - w D / < D > 7 9 / 7 9 / 8 / 8 / Empe Brcke poynom ccuon for kno < >< > - < > < >< > - < > 9 9 < >< > - < > 7 9 b Jones poynom ccuon for kno < D > / w D < >

8 M Cncn nd İ Yr : : : - wd---- : - : - wd- : - / > < D w D / / / 8 8 / 8 Now cn be be o epress he ccuon of Jones poynom usng formu v,, orened nks for some knos Empe Empe / / / /

9 Chrceron of Jones poynoms 7 / / Empe 7 Empe 8 Empe 9 7 9

10 8 M Cncn nd İ Yr 9 Whe s known one from kno poynoms now we cn be be o epress ppcons wh respec o founded oher Empe The Foundon W wh hep W Empe W rf Empe e s ccue of form 8 hen we hve P, nd, m m P from 8 I s founded from s wo epressons Empe e s ccue of Δ from 9 hen we hve, m m P

11 Chrceron of Jones poynoms 9, Δ P Empe The Ccuon Q poynom wh hep F Q,, F Q, Empe The Ccuon poynom by From, F 7, F 7 Empe The Ccuon Q poynom wh hep formu I cn be founded from Thus nd from,, Therefore s wren Q

12 M Cncn nd İ Yr References [] J W ender, Topoogc nv of knos nd nks Trns mer Mh Soc, 98, 7- [] Y un nd M E Bohüyük, Some properes of shepherd s knos n Turksh Dog Ser, 8 98, - [] Y un, The Jones poynom of ws knos In Jour of Mh Gme Theory nd gebr,, - [] J S Brmn, Brds, nks nd Mppng Css Groups Prnceon Unversy Press, US, 97 [] R D Brnd nd W B R ckorsh, K C Me, poynom nvrn for unorened knos nd nks Inven Mh, 8 98, -7 [] J H Conwy, n enumeron of knos nd nks, nd some of her gebrc properes Pergmon, Oford, ondon 97, 9-8 [7] R H Crowe nd R H Fo, Inroducon o Kno Theory Gnn Co Boson, Mss 9, 8 [8] P Freyd, D Yeer, J Hose, W B R ckorsh, K Me, Ocnenu, new poynom nvrn of knos nd nks Bu mer Mh Soc NS, 98, 9- [9] F R Jones, ponom nvrn for knos v on Neumnn gebrs Bu MS, 98, - [] F R Jones, Hecke gebr represenons of brd groups nd nk poynoms nn Mh, 987, -88 [] W B R ckorsh, n Inroducon o Kno Theory Sprnger-erg, Newyork, 997 [] T Knenobu nd T Sum, Poynom nvrns of brdge knos hrough crossngs Mh Comp, 99, [] H Kuffmn, Se modes nd he Jones poynom Topoogy, 987, 9-7

13 Chrceron of Jones poynoms [] H R Moron, Invrns of nks nd -mnfods from sken heory nd from qunum groups: M E BohüyükEdor Topcs n kno heory Kuwer cd Pub, Prned n Nehernds, 99, 7- Receved: Ocober, 7