A Recursive Approach to the Kauffman Bracket

Transcription

1 Applied Mathematis, 204, 5, Published Online Otober 204 in SiRes A Reursive Approah to the Kauffman Braet Abdul Rauf Nizami, Mobeen Munir, Umer Saleem, Ansa Ramzan Division of Siene and Tehnology, University of Eduation, Lahore, Paistan arnizami@ueedup, mobeenmunir@gmailom, umerlins@hotmailom, ansaramzan@yahooom Reeived 26 July 204; revised 28 August 204; aepted 0 September 204 Copyright 204 by authors and Sientifi Researh Publishing In This wor is liensed under the Creative Commons Attribution International Liense (CC BY) Abstrat We introdue a simple reursive relation and give an epliit formula of the Kauffman braet of two-strand braid lin Then, we give general formulas of the braet of the sequene of lins = Finally, we give an interesting result that the Kauffman of three-strand braids α ( n) 2 2 braet of the three-strand braid lin m strand braid lins Keywords m and n 2 Reursive Relation, Kauffman Braet, Braid Lin is atually the produt of the braets of the two- Moreover, a reursive relation for a b d 2 2 is also given Introdution The Kauffman braet polynomial was introdued by L H Kauffman in 987 [] in onern with lin invariants The braet polynomial soon beame popular due to its onnetions with the Jones polynomial, dihromati polynomial, and the Potts model While the HOMPLY polynomial and the braet polynomial are distint with different topologial properties, there is a very beautiful relationship between them due to F Jaeger [2], and it is also observed in a speial ase by Reshetihin [3] The Kauffman braet (polynomial) is atually not a lin invariant beause it is not invariant under the first Reidemeister move However, it has many appliations and it an be etended to a popular lin invariant, the Jones polynomial In the present wor we shall onfine ourselves to the Kauffman braet to avoid this wor from unneessary length and to leave it for appliations This paper is organized as follows: In Setion 2 we shall give the basi ideas about nots, braids, and the Kauffman braet In Setion 3 we shall present the main results How to ite this paper: Nizami, AR, Munir, M, Saleem, U and Ramzan, A (204) A Reursive Approah to the Kauffman Braet Applied Mathematis, 5,

2 A R Nizami et al 2 Basi Notions 2 Lins 3 A lin is a disjoint union of irles embedded in A one-omponent lin is alled a not Lins are usually studied via projeting them on a plan; a projetion with etra information of overrossing and underrossing is alled the lin diagram Two lins are isotopi if and only if one of them an be transformed to the other by a diffeomorphism of the ambient spae onto itself A fundamental result by Reidemeister [4] about the isotopi lin diagrams is: Two unoriented lins L and L 2 are equivalent if and only if a diagram of L an be transformed into a diagram of L 2 by a finite sequene of ambient isotopies of the plane and the loal (Reidemeister) moves of the following three types: The set of all lins that are equivalent to a lin L is alled a lass of L By a lin L we shall always mean the lass of L The main question of not theory is Whih two lins are equivalent and whih are not? To address this question one needs a not invariant, a funtion that gives one value on all lins that belong to a single lass and gives different values (but not always) on nots that belong to different lasses The present wor is basially onerned with this question 22 Braids Braids were first studied by Emil Artin in 925 [5] [6], whih now play an important role in not theory, see [7]-[9] for detail An n-strand braid is a set of n non interseting smooth paths onneting n points on a horizontal plane to n points eatly below them on another horizontal plane in an arbitrary order The smooth paths are alled strands of the braid The produt ab of two n-strand braids is defined by putting the braid b above the braid a and then gluing their ommon end points A braid with only one rossing is alled elementary braid The ith elementary braid i on n strands is: 2747

3 A R Nizami et al A useful property of elementary braids is that every braid an be written as a produt of elementary braids For instane, the above 2-strand braid is 3 ( )( )( i = i i i ) The losure of a braid b is the lin ˆb obtained by onneting the lower ends of b with the orresponding upper ends An important result by Aleander [0] onneting nots and braids is: Eah lin an be represented as the losure of a braid This result motivated not theorists to study braids to solve problems of not theory Remar 2 In the last setion, all the onerned lins will be losures of produts of elementary braids 23 The Kauffman Braet Before the definition it is better to understand the two types of splitting of a rossing, the A-type and the B-type splittings: In the following, the symbols and represent respetively the unnot and the disonneted sum Definition 22 The Kauffman braet is the funtion : Lins aa, defined by the aioms: L a L a L = A + B 2 2 ( ) L = a a L = Here L, L A, and as in the figure: L B are three lins whih are isotopi everywhere eept at one rossing where the loo Proposition 23 The Kauffman polynomial is invariant under seond and third Reidemeister moves but not under the first Reidemeister move [] 2748

4 A R Nizami et al 3 Main Results In this setion we shall introdue a reursive relation for the Kauffman braet, shall give an epliit formula of m n, and shall epress m as the produt of and 2 First of all we give the Kauffman braet of the -twist unnot U : Lemma 3 The Kauffman braet of the -twist unnot is U 3 a = Proof We prove it by indution on : The ase = 0 holds by definition as U 0 is the unnot without any rossings Now, with the assumption that the result holds for an arbitrary, we have Theorem 32 (A reursive relation) The following relation holds for any n 2: n n 3n 2 = a + a (3) Proof We prove it using diretly the definition and Lemma 3: From this reursive relation, we get the epliit formula for the 2-strand braid lin Proposition 33 The Kauffman braet of the lin, 2, n is Proof We prove it by indution on n For n = 2, we have n n 2 n+ 2 ( 3n+ 2) 4 = a + a = : 2749

5 A R Nizami et al whih satisfies the reursive relation With the assumption that the relation holds for an arbitrary n, we, using Theorem 32, get This ompletes the proof n+ ( n+ ) 3 ( n+ ) 2 n = a + a = a + a a + a n 3( n+ ) 2 n 3n 2 n 2 3n 6 n n 3( n+ ) 2 n 3n 3 n 2 3n 7 n 3 3n 0 2n 3 n+ 6 n 2 + a + + a a = a + a + a 3 3n 2n 3 n+ 5 n 3 + a + + a a n 3( n+ ) 2 n 3( n+ ) 6 n 2 3( n+ ) 0 = a + a + a n 3 3( n+ ) 4 2 n+ 3 n+ + 6 n+ 2 + a + + a a α n = (n In the following we give the Kauffman braet polynomial of the losure of the braid 2 2 = = : α ( 3 ) = Proposition 34 The Kauffman braet of α ( n) = ( n ) satisfy the reurrene relations: fators); this sequene ontains the powers of the Garside element times 2 6 2( ) = a a a + a + a 2 6 2( ) a a a a a = ( ) a 2 a a a a = + + = a a a + a + a = a a a + a + a = a a a + a + a Proof Simply, apply the definition for different values of, and write reursively eah net braet in terms of the previous one Lemma 35 The Kauffman braets for = 0 are: a 2 a = + + = a + a 0 5 = a 2750

6 A R Nizami et al = a+ a 7 = a + + a = a a + a + a Proof The proofs of first three ases are given (proofs of remaining ases are similar): Theorem 36 For any 0 the Kauffman braet of ( n) ( n ) = 2a + a + a = a + a + a + a a a a = + + = a + a = a + a + a = a a + a + a α = 2 -times is given by: Proof We prove it by indution on The ase = 0 is overed by Lemma 35, and the indutive step an be heed with Proposition 34 For instane, In onneted sum = a a a + a + a = a 2a + a + a a a + a + a = 2a + a + a a a + a + a ( + ) 4 6( + ) 4 6( + ) = 2 a + a + a n # U of the braid lin n with the trivial not U has the diagram: Lemma 37 3 # U = a Proof We prove it by indution on : 275

7 A R Nizami et al For =, we have Now, with the assumption that the result holds for an arbitrary, we have as required The following result onfirms that the Kauffman braet of Theorem 38 For any mn, 2, m n = m n 2 m n is atually the produt 2 m Proof We prove it by indution on n : When n = 2, (32) Suppose the result holds for n Now, using Lemma 37, we have =, that is m = m

8 A R Nizami et al This ompletes the proof Corollary 39 m + m # m = a U + a m m = a ( ) a + a m 3+ = ( ) a + a m = a + a + a m a + + a a 2 3( + ) 2 3( + ) 6 = a + a = 3( + ) ( + ) + 6 ( + ) 2 + a + + m + m n 2 = n m 2 a a Proof It is obvious: Corollary 30 m n = m n = n m = n m 2 2 m n ( m n) deg = and m n = ( m+ n) span 4 2 Proof The result follows immediately from Theorem 38 as For the following, let us fi the notation and rossings of type, 2, and, respetively, and that deg = 3n 2 and span = n 2 L ab for the lin with the understanding that the lin ontains a, b, a b a Lab 2 Proposition 3 The Kauffman braet of the lin ab Proof We prove it by indution on b : For b =, we have L ab is a b b+ ( 2 b 2 ) a b a+ L = + a + a + a 2753

9 A R Nizami et al a 3 a a+ L = + a+ a + a a a a a = a + a+ a + a a a + = a + a Now, with the assumption that the result holds for an arbitrary, we have b = + a # # a = a U + a a # a = a U + a a a 2 2 a a = a a + a + a + a + a 3+ a a + 3 a a+ a a a a a = ( ) a a a a a = a a+ a a a ( + ) + 2 ( + ) 2 + = + a + a + a a + a a+ as required Proposition 32 The Kauffman braet of the lin Proof We prove it by indution on d : For d =, we have a b d is 2 2 d a b d d+ i 3d 4i+ 2 a b 2 2 = ( ) a i= d + b+ 3d b d ( ) a a a + + d d+ i 3d 4i+ 2 b d b a + + ( ) a + a i= d+ i d+ + 3 = ( ) a + ( ) a + a i= a b i a b b b a i 3 4i+ 2 b b a + + ( ) a + a i= a b b+ 3 b a = a + a + a b+ b a+ + a + a Now, with the assumption that the result holds for d =, we have 2754

10 A R Nizami et al as was required + ( + ) = a ( ) a + a + a + a + i a ( ) a + ( ) a + a i= + i 3 4i+ 2 b b a + + ( ) a + a i= 3+ a b 3+ 3 b 3 b a = a + a + a 3 a b b 2 b 2 a b a i a b b b a ( ) 3+ b + i 3 4i+ a b + a + a i= ( ) + 3 b b a + i 3 4i+ b b a+ + a + a + a + a i= ( ) i 3 ( a ) + ( + ) + i 3 ( + ) 4i+ 2 ( + ) + b+ 3 ( + ) b ( + = a + a + a ) i= + + i= a b a i+ 2 b + b a+ + a, Referenes [] Kauffman, LH (987) State Models and the Jones Polynomial Topology, 26, [2] Jaeger, F (990) A Combinatorial Model for the Homy Polynomial European Journal of Combinatoris,, [3] Reshetihin, NY (988) Quantized Universal Enveloping Algebras, the Yang-Bater Equation and Invariants of Lins, I and II LOMI Reprints E-4-87 and E-7-87, Stelov Institute, Leningrad, USSR [4] Reidemeister, K (948) Knot Theory Chelsea Publ and Co, New Yor [5] Artin, E (925) Theorie der Z o pfe Abhandlungen aus dem Mathematishen Seminar der Universität Hamburg, 4, [6] Artin, E (947) Theory of Braids Annals of Mathematis, 48, [7] Birman, JS (974) Braids, Lins, and Mapping Class Groups Prineton University Press, Prineton [8] Manturov, VO (2004) Knot Theory Chapman and Hall/CRC, Boa Raton [9] Murasugi, K (996) Knot Theory and Its Appliations Birh a User, Boston [0] Aleander, J (923) Topologial Invariants of Knots and Lins Transations of the Amerian Mathematial Soiety, 20, [] Adams, CC (994) The Knot Boo W H Freeman and Company, New Yor 2755

11