Folding of Trefoil Knot and its Graph

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1 Joural of Mathematcs ad Statstcs 2 (2): , 2006 ISSN Scece Publcatos Foldg of Trefol Kot ad ts Graph 1 M.El-Ghoul, 2 A.I.Elrokh ad 3 M.M. Al-Shamr 1 Mathematcal Departmet, Faculty of Scece, Tata Uversty, Tata, Egypt 2,3 Mathematcal Departmet, Faculty of Scece, Meoufa Uversty, Meoufa, Egypt Abstract: I ths study, we troduce the effect of the foldg, codtoal foldg, retracto ad codtoal retracto o trefol kot ad ts graph. We troduce the sheeted trefol kot ad the foldg ad retracto of ths type ad preset the lmt of foldgs. We study how to obta the types of lk graph by adjacecy matrx for ay umber of vertces ad how to calculate Joes polyomal for ths by ts coecto wth Tutte polyomal. The lk graph whch represets a trefol kot appled as a example. Key words: Foldg, retracto, kot, lk graph, trefol kot Correspodg Author: INTRODUCTION It s dffcult to say who frst showed a mathematcal terest what we ow call kot theory ad whe. However, moder tmes t s kow that the famous C.F. Gauss ( ) had some terest ths feld, the Amerca mathematca J.W. Alexader ( ) was the frst to show that kot theory s extremely mportat the study of 3- dmesoal topology. the Germa mathematca H. Sefert from the late 1920s to the 1930s. I the 1970s kot theory was show, amog other thgs, to be coected to algebrac umber theory, by vrtue of the soluto of Smth's cojecture cocerg perodc mappgs. At the begg of the 1980s, due to the dscovery by V.F.R. Joes of hs epochal kot varat, kot theory moved from the realm of topology to mathematcal physcs. Ths was further uderled whe t was show that kot theory s closely related to the solvable models of statstcal mechacs. As kot theory grows ad develops, ts boudares cotue to shft [1]. A kot s a subset of 3 - space that s homeomrphc to the crcle, a lk s a set of ftely may dsjot kots that are called ts compoets, a sgular kot s a kot wth self-tersectos. If K s a kot (or lk), we shall say that ˆP ( K ) = ˆK s the projecto of K. ˆP The map that projects the pot P ˆ( x, y, z) E 3 oto the pot ˆP ( x. y,0) the xy - plae. However ˆK s ot a smple closed curve lyg o the plae, sce ˆK possesses several pots of tersecto. I fact ˆK s a graph represets K ad has some propertes:- 1. ˆK has at most a fte umber of pots of tersecto. 2. If Q s a pot of tersecto of ˆK the P 1 ( Q) K K has exactly two pots, whch make a cross K [1]. I geeral a graph ot ecessary represets a kot E 3 by embeddg. Now we ca mpose some codto o a graph to represet a kot (or a lk ) we call t a lk graph. G s a lk graph f 1. G s fte coected graph. 2. G s plaer. 3. G has the homogeeous vertex degree 4 [2]. For ths every lk graph G wth vertces has 2 edges ad + 2 coutres. et f be a polygoal embeddg of a lk graph G to E 3. We call the mage f(g) a represetato of G whch s a kot ( geeral a lk ). To obta the types of lk graphs we look for the adjacecy matrx. The adjacecy matrx of the lk graph s square matrx of sze x whch s symmetrc has teger values ad for every row ad every colum the sum s 4 ad the ma dagoal are zeros. For =2 we have oly oe adjacecy matrx also for =3we have oly oe adjacecy matrx. For ay we get all possble adjacecy matrces ad draw the lk graph for all of them ad calculate Joes polyomal Vl ( t ) to kow whch kot ( or lk ) the lk graph s represeted t [2]. Deftos ad backgroud: I ths secto we wll summarze some deftos ad theorems whch we wll be used the ma results. 1. A oreted kot s oe wth a chose drecto alog the strg [3,4]. 2. et K be a oreted kot (or lk) dagram defe the wrthe of K, (K), by the equato ω( K) = ε( P) where P rus over all crossgs M.El-Ghoul, Mathematcal Departmet, Faculty of Scece, Tata Uversty, Tata, Egypt 378 P K ad ε ( P) s the sg of the crossg [1,3,4]. Fg. 1

2 3. Kauffma's braket polyomal: et, ˆ ad ˆ 0 be the ske dagrams gve below Fg. 2 The the followg equalty holds: P 1 ( a) = ap ˆ ( a) + a Pˆ ( a). 0 f s a trval dagram O of a trval kot, tha P 0 ( a ) = 1 f P ( a) s the Kauffma bracket polyomal of the " uoreted " dagram ad ω ( ) s the wrthe of the defe 3 ω ( ) Pˆ ( a ) = ( a ) p ( a ). The Pˆ ( a ) s a varat of a oreted kot (or lk ) deoted by P ˆ K ( a ) [1,3,4]. 4. Joes polyomal: Suppose s a oreted lk, the the Joes polyomal V ( t ) ca be defed (uquely) from the followg two axoms. Axom (1): If s the trval kot,the V ( t) = 1. Axom (2): Suppose +,, 0 are ske dagram gve bellow: 5. f G has at least two edges ad e s a brdge (resp., loop) of G, T ( G; x, y ) = xt ( G / e; x, y ) (resp. T ( G; x, y ) = yt ( G \ e; x, y ). Not that propertes (1),,(5) allow the computato of T ( G; x, y ) for ay graph G [4,2,8]. 6. Kot graph: Every plaar graph gves a decomposto of the plae coected regos the so called coutres. By colorg these coutres chessboard lke maer wth the two colors black ad whte such that the ubouded coutry s whte. From the black coutres (resp. whte coutres ) oe gets the correspodg graph of these black coutres (resp. whte coutres),whch called kot graph [2,8]. Example: Fg. 4 k graph ad graph of black coutres (kot graph). 7. Foldg: et ( X, τ ) be a topologcal space.a map f : ( X, τ ) ( X, τ ) s sad to be foldg of a topologcal space to tself f f ( X ). G τ, F( G) = U G, U τ. X,ad ether or. G τ, f ( G) = U τ [9]. May types of foldgs are dscussed [10-13]. 8. Retracto: et A be a subset of a topologcal space X.A cotuous map r : X A s a retracto f r( a) = a for all a A [14]. Fg. 3 The the followg ske relato holds, 1 1 V ( t) tv ( t) = ( t ) V +. 0 t t The polyomal tself s a auret polyomal t may have terms whch expoet [1,5-7]. t.e t has a egatve 5. Tutte polyomal: let G = ( V, E) be a gve graph ad e ay arbtrary oe of ts edges. The Tutte polyomal of G s a polyomal two varables T ( G; x, y) whch satsfes the followg propertes: 1. T ( G; x, y) = 1 for G = ( V, E) wth E = φ. 2. T ( G; x, y) 3. T ( G; x, y) = x f e s a brdge. = y f e s a loop. 4. T ( G; x, y ) = T ( G \ e; x, y) + T ( G / e; x, y) where G \ e 3. Relato betwee polyomals Theorem (1-3): The Kuafma bracket polyomal ca be computed from Tutte polyomal by V C 4 4 P ( a) = a T ( a, a ), where V s the umber of vertces ad C s the umber of coutres of the kot graph [2,5,7]. Corollary(1-3): The Kaufma bracket polyomal of oreted lk dagram s computg by: ˆ ( ) ( 3 ) ω ( ) P a = a P ( a ) [2,3]. (respectvely G / e ) deotes the graph obtaed V ( t) = t from G by deletg (resp.,cotractg) e. 4 1 T ( t, t ). 379 Theorem (2-3): The Joes polyomal s computed from kaufma brackt polyomal by: 1 4 V ( t) = Pˆ ( t ) [5,7]. Theorem (3-3): The Joes polyomal s computed from Tutte polyomal of kot graph by: c v+ 3 ω ( )

3 Applg to =3, the oly oe adjacecy matrx s: The lk graph correspodg of ths adjacecy matrx s: Theorem (1-1-4): The foldg of a kot s ot ecessary a kot. 2. Crossg foldg: A foldg whch folds a pot of upper arc crossg o a pot of lower crossg s sad to be crossg foldg. I ths case we have: a) Fg. 5 The correspodg graph of black coutres (kot graph) s: Fg. 9 f ( c) = c. b) Fg. 6 Ad ts Tutte polyomal s T ( G; x, y ) = x 2 + x + y. By represet the lk graph G 3 E ad take oretato. Fg. 10 c) f ( b) = b, f ( c) = c Fg. 11 f ( a) = a, f ( b) = b, f ( c) = c. Fg. 7 Hece ω ( ) = 3. c v+ 3 ω ( ) 4 1 V ( t) = t T ( t, t ) = t 4 T ( t, t ) 3 4 V ( t ) = t + t t whch s the Joes polyomal of trefol kot. RESUTS Foldg of trefol kot ad ts graph 1-4. Foldg of trefol kot: et K be a trefol kot ad f be a foldg from K to tself. the we have some types of foldg : (1) f ( a) = f ( b) = c Theorem (2-1-4): Every crossg foldg of a kot to tself gves a sgular kot. Proof: et K be a kot ad f be a crossg foldg from K to tself, the f(k) s ot a smple closed curve 3 E (Fg. 9-11)),sce f(k) possesses at lest oe pot of tersecto..e. f(k) s a sgular kot Theorem (3-1-4): A foldg of a sgular kot ot ecessary a sgular kot. The followg examples show that: 1) Fg. 12 f (2) = f (3) = 1. 2) Fg. 8 Fg

4 f ( c ) = b. 3) Fg. 14 f (1) = 2. 4) Fg Topologcal foldg: Applg topologcal foldg a successve steps (Fg. 16) o a trefol kot gettg a equvalet kots utl reachg the ull kot whch dfferet of the successve kots,t represets a lmt foldg. Fg. 16 Theorem (4-1-4): The lmt of a topologcal foldgs of ay kot to tself s a pot. Proof f1 : K K, f2 : f1( K) f1( K),... f : et f ( f...( K)) f ( f...( K)) We see that lm f ( f 1( f 2 (...( K)...)) = p s a pot. 4. Specal cotracto foldg Defto: The foldg whch cotracts the dstace betwee the crossg s called cotractg foldg. I ths type we fx oe arc ad cotract the other arcs,by successve steps (Fg. 17) gettg a equvalet kots. The lmt of ths process the ull kot,whch s exactly the lmt of foldgs. Fg. 17 Theorem (5-1-4): A lmt of specal cotracto foldgs of ay kot s a loop. Proof: et K be a kot of m- crossgs, the K has m arcs a1, a2,..., a m,ad let be the dstace betwee crossgs,f we are fxed a arc a1 ad cotracted aother by sequece of foldgs f, = 1,2,..., wheever the zero.e. The arc a1 s a loop whch exactly a lmt of the specal cotracto foldgs. See Fg. 17 as specal case. Corollary (1-1-4): If we fx two arcs of a kot, the the lmt of specal cotracto foldg of a kot s two loops commo a vertex. Proof: The proof comes drectly from theorem (5-1-4). Theorem (6-1-4): The retracto of ay kot by removg a pot s a pot. Proof: et r : K p K p be a retracto, f dm r( k p ) = dm K, the t s ot the mmum retracto. ad r( K p ) some topologcal foldg f r, but f dm r( K p ) dm K f r.the mmum retracto Theorem (7-1- 4): The lmt of topologcal foldgs of a kot s equvalet to the retracto of a kot by removg a pot. Proof: et f : K K be a sequece of topologcal foldg of the kot to tself the lm f ( K) = p theorem (4-1-4). Assume r : K q K q a retracto by removg a pot, r( K q) = p theorem (6-1-7)..e. r( K q) = lm f ( K) Foldg of a lk graph: et be a lk graph whch represet a trefol kot (Fg. 5) ad let fa foldg from to tself,the we have some types of foldg: (1) Fg. 18 f ( d) = a, f ( e) = b, f ( g) = c. Ths foldg gves a complete graph K 3, whch s a kot graph of a trefol kot,but ot represet a kot. Theorem (1-2-4): A foldg of a lk graph s ot ecessary a lk graph. (2) (a) Fg

5 f ( g) = c, f (1) = 3 (b) Fg. 20 f (1) = Topologcal foldg: Applg topologcal foldg a successve steps (Fg. 21) of a lk graph, represet a trefol kot gettg a equvalet lk graph utl reachg the ull lk graph whch dfferet of the successve lk graph,t represets the lmt of foldgs. r : K {1} K. Corollary (1-2-4): Every retracto of a lk graph decreases the degree of y Tutte polyomals. Theorem (4-2-4): For ay foldgs of lk graph whch decrease the degree of y Tutte polyomals there exst a retracto s equvalet to t. Proof: et be a lk graph ad f be a foldg from to tself s.t. f decreases the degree of y Tutte polyomals, ths meas oe edge or more was folded to aother ths equvalet to remove ths edge hece there s a retracto r equvalet to f. Fg. 21 Theorem (2-2-4): The lmt of topologcal foldgs for ay lk graph of vertces s a graph wth oly oe vertex havg loops. Fg. 24 Now we see the foldgs ad retracto wth Tutte polyomals: Proof: The proof s clear from the above dscusso. Theorem (3-2-4): There are some types of foldgs of a lk graph decrease the degree of Y Tutte polyomals ad there s aother types crease the degree of Y Tutte polyomals. Proof: et be a lk graph Fg. 20 The Tutte polyomal of s T ( ; x, y) = y + 2y + 3 y + (1 + 3 x) y + x + x The degree of Y s 4 ad the Tutte polyomal of f ( ) s T ( f ( ); x, y) = y + y + y + xy The degree of Y s 5,.e. degree of y was creased. ook Fg. 18 The Tutte polyomal of f ( ) s 2 T ( f ( ); x, y) = y + x + x the degree of Y s 1,.e. degree of y was decreased. We have two types of retractos of a lk graph: The frst types: Fg. 25 From the ch of foldgs ad retractos we get r f = f r ad + 1 T ( K; x, y) = T ( K ; x, y) + T ( K ; x, y) 1 1 T ( K ; x, y) = T ( K ; x, y) + T ( K ; x, y) 1; 2 2 T ( K ; x, y) = T ( K ; x, y) + T ( K ; x, y) T ( K4; x, y) = T ( K4; x, y) + T ( K4; x, y) ths meas T ( K; x, y) = T ( K ; x, y) + T ( K ; x, y) T ( K ; x, y) + T ( K ; x, y) + T ( K ; x, y) Foldg of sheeted trefol kot: Suppose R be a sheeted E 3, Fg. 25, so that we form a trefol kot V ( R ) Fg. 22 r : K { v} K, v g, v 1, v 3 the secod types: Fg. 25 Fg. 26 Fg by t, Fg. 26, called the sheeted trefol kot,hece V ( R ) takes four cases:

6 Case 1: V ( R ) wth two boudares V ( R ) ={,M}, whch s exactly a sheeted boud kots ormal case f : K K, f : f ( K ) f ( K ), f : f ( f ( K )) f ( f ( K )),..., f ( f ( f (...( f ( K ))...) f ( f ( f (...( f ( K ))...) such that f ( ) = ad lm f kot of oe dmeso. = M whch s a trefol Fg. 27 Case 2: V ( R ) wth o boudares V ( R ) =0,we call a sheeted strpe trefol kot. Fg. 28 Case 3: V ( R) wth exteral boudary V ( R ) =.. Fg. 31 b. et K 2 be a sheeted trefol kot wth boudares K 2 ={,M} ad f be a foldg from K 2 to tself such that f twsts M o tself a successve steps utl reachg a boudary so that we gettg a trefol kot whch s the lmt of foldg. Fg. 32,.e. f : K K, f : f ( K ) f ( K ), f : f ( f ( K )) f ( f ( K )),..., f ( f ( f (...( f ( K ))...) f ( f ( f (...( f ( K ))...) such that f ( M ) trefol kot of oe dmeso. = M ad lm f = whch s a Fg. 29 Case 4: V ( R ) wth teral boudary V ( R ) =M. Fg. 30 Now we troduce some types of foldgs for all above cases: Fg. 32 c. et K 3 be a sheeted trefol kot wth boudares K 3 ={,M} ad f be a foldg from K 3 to tself such that f ()= M we gettg a tubular trefol kot, whch fact s a hollow torus. Fg. 33, f : K3 K3, s. t. f ( ) = M ad ay geerator, f ( ) = ad homeomorphc or dffeomorphc to. Ay sequece of foldgs f, f, f,..., f lm f = torus. Also the ufoldgs uf, uf, uf,... uf lm uf = torus For case 1 a. et K 1 be a sheeted trefol kot wth boudares K 1 ={,M} ad f be a foldg from K 1 to tself such that f twsts o tself a successve steps utl reachg a boudary M so that we get a trefol kot whch s the lmt of foldg. Fg. 31,.e. 383 Fg. 33

7 For case 2: et V be a sheeted trefol kot wth out boudares V =0 ad b s a rm le, Fg. 34 ad f be a foldg from V to tself such that f twsts a sheeted aroud the rm le, so we gettg a trefol kot whch exactly the lmt of foldgs (Fg. 35) f : V V, f : f ( V ) f ( V ), f : f ( f ( V )) f ( f ( V )),..., f ( f ( f (...( f ( V ))...) f ( f ( f (...( f ( V ))...) f (rm) = rm, lm f = rm whch s a trefol kot of oe dmeso, Theorem ( ): The lmt of foldgs of a sheeted trefol kot whch twst a boudary o tself s a trefol kot. Proof: The proof s clear. For all cases, f we apply the foldg f from V ( R) to tself such that f (1)= f (2)= 3 we gettg a sheeted ukot heredtary case of boudary, (Fg. 38) Fg. 38 Fg. 34 Fg. 35 For case 3: et V(R) be a sheeted trefol kot wth boudares V(R) = ad f be a foldg from V(R) to tself such that f twsts o tself a successve steps, so that we gettg a trefol kot whch s the lmt of foldgs (Fg. 35). I ths case the lmt of ths codtoal foldg equvalet to the retracto of V(R) to. Theorem (2-3-4): The lmt of foldgs of a sheeted kot V ( R) s equvalet to the retracto f V ( R) s ope. Proof: If f : V ( R) V ( R), f : f (( V ( R) ) f ( V ( R) ), f : f ( f ( V ( R) )) f ( f ( V ( R) )) ,..., f ( f ( f (...( f ( V ( R) ))...) f ( f ( f (...( f ( V ( R) ))...) lm f = rm le Fg. 34 ad by take a retracto r : V ( R) rm le s.t. r(v(r)) = rm le Fg. 36 For case 4: et V(R) be a sheeted trefol kot wth boudares V ( R ) = M ad f be a foldg from V(R) to tself such that f twsts M o tself a successve steps, so that we gettg a trefol kot whch s the lmt of foldgs. Fg. 36. I ths case the lmt of ths codtoal foldg equvalet to the retracto of V(R) to M. Fg Fg. 39 hece we get lm f = rm le = r( V ( R )) REFERENCES 1. Murasug, K., Kot Theory ad ts Applcatos. Brkhaser, Bosto, MA. 2. Elrokh, A.I., Graphs, Kots ad ks. Ph. D. Thess. Uv. Erst-Mortz-Ardt Grefswad. 3. Kauffma,.K., Kot ad Physcs. Sgapore [u.a]:world Scetfc. 4. Seke, K., Algorthm for computg the Tutte polyomal ad ts applcatos. Ph. D. Thess. Uversty of Toky. 5. Chag, S.-C. ad R. Shork, Zero of Joes polyomals for famles of Kots ad lks. Physca, A301:

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