International Mathematical Forum, 1, 2006, no. 31, ON JONES POLYNOMIALS OF GRAPHS OF TORUS KNOTS K (2, q ) Tamer UGUR, Abdullah KOPUZLU

Transcription

1 Iteratoal Mathematcal Forum,, 6, o., ON JONES POLYNOMIALS OF RAPHS OF TORUS KNOTS K (, q ) Tamer UUR, Abdullah KOPUZLU Atatürk Uverst Scece Facult Dept. of. Math. 54 Erzurum, Turkey tugur@atau.edu.tr akopuzlu@atau.edu.tr Abstract Let K (, q ) be torus kot. Let K# K#...# K be coected sums of torus kots K (, q ). Algorthms are gve for fdg the Joes polyomals of graphs of torus kot K (, q ) ad graphs of coected sums K# K#...# K Keywords: kot, graph, torus kot, kot graph, Joes polyomals.. Itroducto. I 984, V. F. R. Joes defed a ew polyomal varat for kots or lks. Ths dramatc dscovery opeed a ew era kot theory. The varat was uexpectedly defed through operator algebra, but ts combatoral descrpto dcated that through graph theory kot theory could most beeft from ths varat. I partcularl Murasug has bee appled to kot graphs ths polyomal [],[],[]. A graph s a par V (, E ) of sets satsfyg E [ V ] ; thus, the elemets of E are -elemet subsets of V. We deote the vertex set ad the edge set of a graph by V( ) ad E( ) respectvely. Throughout ths paper, every graph s assumed to be fte, smple ad coected (See [4] for the basc termology of graph theory). A kot s a smple closed curve a space S. The torus kot K( pqof, ) type p, q s the kot whch wraps aroud the stadard sold torus K the logtudal drecto p tmes ad the merdaal drecto q tmes. Where tegers p, q are relatvely prme. Thus the trefol s K(,). (See [5],[6] for may stadard termologes kot theory).. Kot graphs A projecto of a kot or a lk o a -dmesoal plae dvdes the plae to several domas. It s a frequetly used method to separate these domas to two classes, whte domas ad black domas the study of kot theory. Usg ths method, C. Bakwtz [7] troduced the oto of kot graph. Let D be a regular projecto of a kot o a -dmesoal sphere S. If D has double pots D, D,..., D, the t dvdes S to + domas, each of whch s

2 58 Tamer UUR, Abdullah KOPUZLU homeomorphc to a ope dsk. Now, separate these domas to two classes α ad β (Fg. ). Fgure. Startg wth the outermost doma, we ca color the domas ether black or whte (or er ad outer). Now, we shall color the outermost doma black (or whte). We ca color the domas so that eghborg domas are ever the same color. Let W, W,..., Wv be the domas of class α. Take pots c W ( =,,..., v) ad coect these pots by o-tersectg arcs d, d,..., d such a way that each dr correspods to D ( r =,,..., ) ad c ad r c j are coected by dr, f ad oly f, W ad W j have a commo double pot D r o ther boudares. The vertces of graph are the ceters of the whte domas (Fgure ). The domas of class α all ca be cosdered as a projecto of a surface spag K whch s twsted 8 at each double pot of D. However, order for the plae graph to embody some of the characterstcs of the kot, we eed to used the regular dagram rather tha the projecto. So, we eed to cosder the uder - ad over - crossg at a crossg. To ths ed, Fgure s show a way of assgg to each edge of graph ether the sg + or -. A + sg s assged to a edge e f the domas are colored the maer of Fgure, ad sg f they are as Fgure. A sged plae graph that has bee formed by meas of the above process s sad to be the graph of K ad deote t by K ( ). From the same -cosderato about the class β, we get aother graph (K ). We call ths the dual graph of K ( ). + Fgure. -

3 Joes Polyomals 59. Joes polyomal of a graph Let K be a kot ad be ts graph. We fx some otatos, before we defe Joes polyomal J ( x, y, z ) for ay fte sged graph. Throughout ths talk, a graph frequetly represets the geometrc realzato of a fte -dm CW - complex S = R. A vertex correspod to a -smplex ad a edge th correspod to a -smplex. Thus, deote β ( ) the Bett umber of. deotes the cardalty of a set. A graph s sad to be sged f ether + or s assged to each edge. Let H be a subgraph of. Deote ph ( ) ad H ( ), respectvel the umber of postve ad egatve edges H. A subgraph H of has duced sg fucto f = f E( H ). A subgraph H s a spag H subgraph f V( H) = V( ). Let S ( r, s) be the set of all spag subgraphs H such that β ( H) = r+ ad β ( H) s =. Therefore s the set of all spag trees. Defto. p( H ) ( H ) r s J = x y z r, s H S ( r, s) where the frst summato rus over all spag subgraphs H S ( r, s ). J ( x, y, z ) wll be called the Joes polyomals of a graph. Proposto.[] If s a postve graph, the where v= V( ) Example. J (,, ) x y z = r v y s x S r s xz rs, x (,) ( ) Let K(,) be torus kot ( The trefol kot, Fgure ). s ts graph (Fgure ).

4 54 Tamer UUR, Abdullah KOPUZLU The spag subgraphs of K are: Fgure.. = ( ) I, raki, raki =, r =, s = = = J = y = = ( ) I, raki, raki =, r =, s = = J xy =. = = I =, raki =, raki =, r =, s = J = x v. 4 = 4 = I =, raki =, raki =, r =, s = J x z The = 4 (,, ) = J x y z x z x xy y Example. If cossts of oly oe vertex, the J z ) =. Example. Joes polyomals of graphs of some K (, q) torus kots are

5 ) K (,5) 5 4 J x z 5x x y x y 5xy y = Joes Polyomals x z + 7x + x y + 5x y + 5x y + ) K (,7) J z ) = x y + 7xy + y x z+ 9x + 6x y+ 84x y + 6x y + ) K (,9) J z ) = x y + 84x y + 6x y + 9xy + y Proposto. For graph. The, where = E( ). [8] q =,,5,..., Let K (, q ) be torus kot ad also be ts J = x z + k = x k Proof. By Iducto. Proposto. Let K, K,..., K be K (, q ) torus kots ad also K# K#...# K be ts coected sums. If the graph of kot K# K#...# K s, the 4. Refereces k y k (,, ) = J (,, ) = J xyz y xyz [] K. Murasug, O varats of graphs wth applcatos to kot theory. Tras. Amer. Math. Soc., 4: [] K. Murasug, Ivarats of graphs ad ther applcatos to kot theor I: Algebrac topology Poza, 989, pp. 8-97, Lecture Notes Math., 474, Sprger Verlag. 99 [] K. Murasug, Classcal umercal varats kot theor NATO ASI seres, pp.57-94, 99, Kluwer Academc Publ., 99 [4] R. Destel, raph Theor Sprger-Verlag, New York, 997 [5] K. Murasug, Kot Theory ad Its Applcatos, Brkhauser Bosto, 996 [6] D. Rolfse, Kots ad Lks, Publsh or Perch Ic., Wlmgto Delaware, 976 [7] C. Bakwtz Über de Torsoszahle der altererde Kote, Math a., 45-6, 9 [8] T. Ugur, A. Kopuzlu, Applcato of Computer Algebra to Joes Polyomals. Appled Math. ad Computatato. 4, 6 66 (). Receved: December, 5