Taylor series coefficients of the HP-polynomial as an invariant for links in the solid torus

Transcription

1 Al. Math. Inf. c. 7, No. 1, (213) 23 Aled Mathematcs & Informaton cences An Internatonal Journal c 213 NP Natural cences Publshng Cor. aylor seres coeffcents of the HP-olynomal as an nvarant for lnks n the sold torus K. Bataneh Jordan Unversty of cence and echnology, Jordan and Unversty of Bahran, Bahran Receved: 22 Jul. 212; Revsed 17 Oct. 212; Acceted 27 Oct. 212 Publshed onlne: 1 Jan. 213 Abstract: We show that the coeffcents of a reformulaton of a aylor seres exanson of the Hoste and Pryzytck olynomal are Vasslev nvarants. We also show that many other reformulatons of the aylor seres exanson have coeffcents that are Vasslev nvarants. We charecterze the frst two coeffcents b 1 (t) and b 1 (t) for one of these exansons. Moreover, the second coeffcent b 1 (t), whch s a tye-one Vasslev nvarant, s gven two exlct comutatonal formulas, whch are easy to calculate. b 1 (t) s used to gve a lower bound for the crossng number of a knot of zero wndng number n the sold torus. Keywords: aylor seres; old torus 1. Introducton Dscoverng comutable formulas of Vasslev nvarants started by the aearance of the combnatoral defnton by Brman and n n 5. nce the rchest source of nvarants contnues to be olynomal nvarants, much of work has been done to characterze Vasslev nvarants of a gven tye by exlorng these olynomal nvarants. he dfference relaton n the combnatoral defnton of the Vasslev nvarants s what stands behnd the nterest of the dervatves and hence the aylor seres exanson coeffcents of the aurent olynomal nvarants. ee 5 and 12. he two-varable olynomal for dchromatc lnks by Hoste and Przytyck n 7 was consdered n as an nvarant of lnks n the sold torus. We refer to ths olynomal nvarant by the HP-olynomal. Moreover, the defnton of Vasslev nvarants was extended to knots and lnks n the sold torus. In ths aer we show that the coeffcents of a reformulaton of the aylor seres exanson of the HPolynomal are Vasslev nvarants and we show that many other choces of aylor seres exanson lead to the same result. We characterze the frst two coeffcents of ths exanson b (t) and b 1 (t). he second coeffcent b 1 (t), whch s a tye-one Vasslev nvarant, s gven exlctly by two comutatonal formulas. Both of these two formulas are fnte sum formulas that are easy to calculate. Fnally, as an alcaton, we analyze b 1 (t) to gve a lower bound for one of the most mortant geometrc knot nvarants, the crossng number, whch has secal mortance n natural scences, esecally n studyng knotted DNA. In secton 2 we gve relmnares needed n the later sectons. In secton 3 we show that the coeffcents of a aylor seres exanson of the HP-olynomal are Vasslev nvarants and we show that many other aylor seres exansons of the HP-olynomal have ths roerty too. In secton we show that b 1 (t) can be comuted searately and easly through two dfferent formulas, and we lst some remarks on these formulas. In secton 5 we extract a lower bound for the crossng number of a knot of zero wndng number, and we show how ths bound could be used to comute the crossng number of a knot. 2. Prelmnares nks (knots) and sngular lnks (knots) n the sold torus are usually reresented by what s called unc- Corresondng author: khaledb@just.edu.jo c 213 NP Natural cences Publshng Cor.

2 2 K. Bataneh : aylor seres coeffcents of the HP-olynomal... (III) he fnshng formulas tured dagrams, whch s a usual dagram of the sngular knot or lnk but n R 2 {(, )}. ke n the case of lnks n R 3, we shall call a ont of ntersecton of a sngular lnk a sngular ont of. A sngular lnk n mght also be gven an orentaton for each of ts comonents. Defnton 1. wo sngular knots (lnks) are sad to be sotoy equvalent f we can get from a dagram of one of them to a dagram of the other by a fnte sequence of the known fve generalzed Redemester moves. ee 11. et be the set of sngular lnk tyes wth sngular onts. et = =. Defnton 2. A functon υ : G, where G s an abelan grou, s sad to be a Vasslev nvarant or a fnte-tye nvarant f t satsfes the followng two axoms: () υ( ) = υ( + ) υ( ) and () here exsts n N such that υ() = for any wth n + 1. Moreover, the least such non negatve nteger n s called the tye or order of υ. + and are usually called the ostve and negatve resolutons of, resectvely, and equaton () s usually denoted by resolvng ths double ont n. he two axoms of the defnton above were frst gven by Brman and n 5 for knots, and then the defnton of fnte-tye nvarants was extended to lnks n a smlar way by tanford 12. he defnton of fnte-tye nvarants was extended to knots and lnks n general 3-manfolds by Kalfagann 8, and n the sold torus by Goryunov 6, Acard 1, and Bataneh and Abu Zaytoon 2 n a smlar way. Hoste and Przytyck n 7 ntroduced the twovarable aurent olynomal nvarant d of 1-trval dchromatc lnks wth orented 2-sublnk n R 3. In we vewed ths nvarant as an nvarant of orented lnks n the sold torus nstead of 1-trval dchromatc lnks n R 3. he notaton we use for d as an nvarant n the sold torus s Y. heorem 1. et D be a dagram of a lnk n the sold torus, and let D be determned by the followng formulas: (I) he smoothng formulas (1) = A + A 1 (2) = A + A 1 (II) he reducton formulas (1) D = (A 2 A 2 ) D (2) D = (A 2 A 2 )t D (1) = 1 (2) = t hen Y (A, t) = (A 3 ) w(d) D s a aurent olynomal nvarant n ZA, A 1, t, where w(d) s the sum of the sgns of all crossngs of D. Hoste and Przytyck n 7 gve the followng sken relaton for ths nvarant. emma 1. he Y nvarant satsfes the sken relaton gven by A Y + (A, t) A Y (A, t) = (A 2 A 2 )Y (A, t) he roof of ths emma goes along the lnes of the roof of emma 2.6 n 1. et be a lnk n the sold torus, and n () be the number of coes of the comonent n whose wndng number s. et n() be the number of comonents of. Note that n () = n(). For the roof of the followng theorem see 3. heorem 2. For a lnk n the sold torus Y (1, t) = (2) n()1 (t) n (), where the set { k (t) : k Z} s the set of the extended Chebyshev Polynomals of the Frst Knd gven recursvely by: (t) = 1, 1 (t) = t, k (t) = 2t k1 (t) k2 (t) for k. Moreover k (t) = k (t) for k. he reader can easly deduce the followng corollary. Corollary 1. If K s just a knot n the sold torus, then Y K (1, t) = (t), where s the wndng number of the knot. 3. Power eres Coeffcents of the HP-Polynomal heorem 3. et be a lnk n the sold torus and let Y (A, t) be the HP -olynomal. et Y (e 1 x, t) be obtaned from Y (A, t) by relacng the varable A by e 1 x. et Y (x, t) be obtaned by relacng e 1 x by ts aylor seres exanson about x =, then Y (x, t) = b (t)x, = where b (t) = (2) n()1 (t) n() and each b (t), s a Vasslev nvarant of order. c 213 NP Natural cences Publshng Cor.

3 Al. Math. Inf. c. 7, No. 1, (213) / 25 Proof. et j be a sngular lnk wth j sngular onts one of whch s referred to by. We defne o the coeffcent of x n the sum above s zero for all < j. herefore Y j (A, t) = Y + (A, t) Y (A, t). (1) If we resolve the j crossngs usng (1) we end u wth a sum of 2 j Y olynomals of non-sngular lnks. he sken relaton A Y + (A, t) A Y can be wrtten as Y + (A, t) = A 8 Y (A, t) = (A 2 A 2 )Y (A, t) (A, t) + (A 6 A 2 )Y (A, t) (2) ubsttutng from equaton (2) nto equaton (1) we obtan Y j (A, t) = A 8 Y or equvalently Y j (A, t) = (A 8 1)Y (A, t)+(a 6 A 2 )Y (A, t)y (A, t), (A, t)+(a 6 A 2 )Y (A, t) (3) et j be a sngular lnk wth j sngular onts labeled 1, 2,..., j. Resolvng the j crossngs of j usng (3) we get a sum nvolvng the Y olynomals of the followng 2 j non-sngular knots {K : = ( 1,..., j ), where s ether a mnus sgn or a zero}. hs sum s Y j (A, t) = (A 8 1) (A 6 A 2 ) q Y (A, t), () where s the number of mnus sgns n, and q s the number of zeros n. ubsttutng A = e 1 x we obtan Y j (e 1 x, t) = (e 2x 1) (e 3 2 x e 1 2 x ) q Y (e 1 x, t). Note that the aylor seres exanson about x = gves e 2x 1 = 2x + 2x x3 + e 3 2 x e 1 2 x = x + x x3 +, and hence = (1 + a 1x + ) 1 (1 + 3a 1x + ) Y j (x, t) = (1 + a 1 x + ) q Y (f(x) 2 1, t) (2x + 2x x3 + ) (x + x x3 + ) q Y 1 = a (e x 1 x + 2a 1 x + q, t) = (2x + 2x x3 + ) (x + x x3 + ) q (c (c + c 1 x + ), + c (5) 1 x + ), where c = (2) n()1 (t) n(), because Y (, t) = Y (1, t) = (2) n()1 (t) n(). Note that c. he frst non-zero term of the sum n (5) s Y j (x, t) = b j (t)x, = where b j (t) = for all < j. et be fxed and let j >, then b j (t) =, hence b (t) satsfes the frst condton n the defnton of a Vasslev nvarant of order. Moreover, b (t) s an nvarant that satsfes b (t) = b + (t) b (t) snce Y does. Hence b (t) s a Vasslev nvarant of order. Next we show that we do not have to use the substtuton A = e 1 x n order to rove the revous theorem. In fact we have a wde range of choces. Prooston 1. If f() = 1, f () and f(x) has a convergent aylor seres about x =, then each coeffcent d (t) of the aylor seres exanson of Y (A, t) determned by the substtuton A = f(x) 1 2 s a Vasslev nvarant of order. Proof. Recall equaton () n the roof above Y j (A, t) = (A 8 1) (A 6 A 2 ) q Y (A, t). ubsttutng A = f(x) 1 2 Y j (f(x) 1 2, t) = we obtan (f (x) 1) (f 3 (x) f(x)) q Y (f(x) 1 2, t). et f(x) = a +a 1 x+a 2 x 2 +. Note that f() = 1, f () f(x) = 1 + a 1 x + a 2 x 2 +, such that a 1. Hence, the sum n (6) becomes ((1 + a 1 x + ) 1) ((1 + a 1 x + ) 3 (1 + a 1 x + )) q Y (f(x) 1 2, t) where c = (2) n()1 (t) n(). Note that c. he frst non-zero term of ths sum s (a 1x) (2a 1x) q (c ) = ( 2 q a 1 aq 1 c )x x q = (2x) (x) q (c ) = 2 c x +q = (2 c )x j snce + q = j. (2 j+ a j 1 c)xj snce + q = j. c 213 NP Natural cences Publshng Cor.

4 26 K. Bataneh : aylor seres coeffcents of the HP-olynomal... Note that (2 j+ a j 1 c ). o the coeffcent of x n the sum above s zero for all < j. herefore Y j (x, t) = d j (t)x, = where d j (t) = for all < j. he rest of the roof s smlar to that n the roof of the revous theorem, because f s fxed and j >, then d j (t) =, hence d (t) satsfes the frst condton n the defnton of a Vasslev nvarant of order.. wo formulas for the nvarant b 1 (t) In the revous secton we gave the exlct formula of the tye-zero nvarant b (t). In ths secton we gve two comutatonal exlct formulas for the tye-one aurent olynomal nvarant b 1 (t). he frst one s a sum gven n terms of the extended Chebyshev Polynomals, and the second one s a sum gven n terms of the states of the lnk on Kauffman s way. et be a lnk n the sold torus. If s a crossng n between two strands n the same comonent, then smoothng ths crossng roduces two comonents. We denote the wndng numbers of these two comonents by {m(), l()}. We also defne j() = m() + l(). On the other hand, If q s a crossng n between two strands from two dfferent comonents, we denote the wndng numbers of the two comonents by {m(q), l(q)}. moothng ths crossng connects these two comonents nto one comonent. We defne j(q) = m(q) + l(q) and ths s the total wndng number of the resultng comonent. We refer to the sgn of the crossng by e(), and the sgn of the crossng q by e(q). For the roof of the followng theorem see 3. heorem. A Y (A, t) s gven by A=1 Y (1, t)( e() m() (t) l() (t) + j() (t) e(q) j(q) (t) ). m(q) (t) l(q) (t) q he exlct formula of b 1 (t) s now gven. heorem 5. he tye-one nvarant b 1 (t) s gven by b 1 (t) = b (t)( e() 1 m()(t) l() (t) j() (t) + j(q) (t) e(q) 1 ). q m(q) (t) l(q) (t) Proof. Recall that we have Y (A, t) and A = e 1 x and the ower seres Y (x, t) = b (t)x. = One can easly see that b (t) = Y (, t) = Y (1, t). Moreover b 1 (t) = A Y (x, t) x= = A Y (A, t) da dx x= da = A Y (A, t) A=1 dx x= = A Y (A, t) 1 A=1 = 1 Y (1, t)( e() m()(t) l() (t) j() (t) j(q) (t) e(q) m(q) (t) l(q) (t) ) + q = b 1 (t)( e() 1 m()(t) l() (t) j() (t) + j(q) (t) e(q) 1 ) q m(q) (t) l(q) (t) hs comletes the roof. he followng corollary follows from the last theorem and the fact that when s a knot, we have b (t) = j() (t). Corollary 2. For a knot K n the sold torus, we have b K 1 (t) = e() m()+l() (t) m() (t) l() (t). he other exlct comutatonal formula n ths secton s a formula that we wll derve from the followng state sum formula gven n. heorem 6. et D be a dagram of a lnk n the sold torus. hen Y (A, t) = (A 3 ) w(d) A a()b() (A 2 A 2 ) 1 t where the sum runs over all ossble states of the lnk dagram, s the total number of crcles and dotted crcles n the state, s the number of the dotted crcles, a() s the number of A-slts n the state, and b() s the number of B-slts n the state. Now we gve the state sum formula for the tye-one nvarant b 1 (t). heorem 7. b (1)w(D) 1 (t) = Proof.Note that 3w(D) + b() a() 2 1 t. A Y (A, t) = (A 3 ) w(d) A a()b() ( 1)(A 2 A 2 ) 2 (2A + 2A 3 ) + (a() b())a a()b()1 (A 2 A 2 ) 1 t + w(d)(a 3 ) w(d)1 (3A 2 ) A a()b() (A 2 A 2 ) 1 t c 213 NP Natural cences Publshng Cor.

5 Al. Math. Inf. c. 7, No. 1, (213) / 27 Hence A Y (A, t) = (1) w(d) A=1 (a() b())(2) 1 t + w(d) (1) w(d)1 (3) (2) 1 t = (1) w(d) (a() b())(2) 1 t 3w(D)(2) 1 t = (1) w(d) (a() b() 3w(D))(2) 1 t Now recall from the last roof that b 1 (t) = 1 A Y (A, t) A=1 = 1 (1) w(d) (a() b() 3w(D))(2) 1 t = (1)w(D) hs comletes the roof. 3w(D) + b() a() 2 1 t. We gve the followng useful remarks. Remark.If X (A) s the well-known form of the Jones olynomal dscovered by Kauffman usng hs bracket olynomal, then Y (A, 1) = X (A). hs can be easly seen by defnton of Y (A, t). Remark. If K s a knot n the three sace, then d da X K(A) A=1 =, because a tye-one nvarant of knots n the three sace s of tye zero; that s a constant. Remark. In a smlar ower seres exanson of X K (A), the second coeffcent would be b K 1 (1), and the corresondng state sum formula would be (1) w(d) 3w(D) + b() a() 2 1. However, ths formula s of no use, because t vanshes by the revous remark. herefore, he formula n our last theorem has no corresondng one known for knots n the three sace. But t s useful n comutaton for a tye-one nvarant for knots and lnks n the sold torus. Examle 1.he calculaton of b K 1 (t) for the knot n (Fgure 1) usng both of the two formulas gven n ths secton yelds the same result b K 1 (t) = 2 2t 2. Fgure 1 Knot n the old orus 5. A lower bound on the crossng number For a gven knot K the crossng number of K, denoted by crossng(k), s known to be the mnmal number of crossngs of all dagrams of the knot K. ke all geometrc nvarants of knots and lnks, the crossng number s hard to comute, whle t s so mortant esecally n knotted DNA, where the crossng number s nstrumental n understandng the nformaton coded n the DNA. et K be the set of knots wth zero wndng number n the sold torus. In ths secton we exlore a lower bound for the crossng number of K K. hs lower bound s coded n b K 1 (t) n an mlct way. For the followng extended relaton of Chebyshev olynomals see 3. n(t) = 1 2 ( t ) n+ 1 t ( t + t 2 1) n for all n Z emma 2. If K K, and z = t t 2 1, then b K 1 (z) = 1 e() 1 e()z 2m() + z 2m(). 2 Proof. Note that f K K, then m() + l() =, therefore m()+l() (t) = (t) = 1, m() (t) l() (t) = m() (t) m() (t) Hence = m() (t) m() (t) = 2 m()(t). b K 1 (t) = e()1 2 m()(t). When z = t t 2 1, we have Hence b K 1 (z) = m() (z) = 1 2 zm() zm() 2 m()(z) = z2m() + 1 z2m(). e()1 2 m()(z) = e() z2m() 1 z2m() = 1 e() 1 e()z 2m() + z 2m(). 2 c 213 NP Natural cences Publshng Cor.

6 28 K. Bataneh : aylor seres coeffcents of the HP-olynomal... Note that the revous lemma exresses b K 1 (z) as a symmetrc aurent olynomal n z wth a constant term M 1 2 e(). herefore, we have the followng corollary. Corollary 3. b K 1 (z) can be wrtten as b K 1 (z) = B m1 z 2m B ms z 2m s + B m1 z 2m B ms z 2m s + M, where B m = 1 e( m ), s the sum of sgns m of all crossngs wth wndng numbers {m, m }. heorem 8. If K K, then crossng(k) B m1 + + B ms. Proof. et D be any dagram for a knot K K. nce B m = 1 e( m ), we have e( m ) = B m. m m o we have at least B m = B m crossngs of wndng numbers {m, m }. nce the sets of crossngs of dfferent ars of wndng numbers n D are arwse dsjont, we have at least B m1 + + B ms crossngs n D. Hence crossng(k) B m1 + + B ms. Examle 2. For the knot K n Fgure 1, we have b K 1 (z) = (1)z2 + z 2 + (1)z 2 + z 2 = z2 1 2 z2. Hence crossng(k) 1 2 = 2. We leave t as an easy exercse for the reader to fnd a dagram D 1 of K n FIGURE 1 that has exactly two crossngs. Fndng such a dagram together wth the fact that crossng(k) 2 mles that crossng(k) = 2. 6 V. Goryunov, Vasslev nvarants of knots n R 3 and n a sold torus, Amer. Math. oc. ransl., 19 (1999), J. Hoste and J. Przytyck, An nvarant of dchromatc lnks, Proc. Amer. Math. oc., 15 (1989), E. Kalfagann, Fnte tye nvarants for knots n 3- manfolds, oology, 37 (1998), Kanenobu, Kauffman olynomals as Vasslev lnk nvarants, Proceedngs of Knots 96, (1997), Kauffman, New nvarants n the theory of knots, Amercan Mathematcal Monthly, 95 (1988), A. Kawauch, A survey of knot theory (Brkhauser Verlag, 1996). 12. tanford, Fnte-tye nvarants of knots, lnks, and grahs, oology, 35 (1996), K. Bataneh s resently emloyed as an assstant rofessor; currently at the Unversty of Bahran, and ermanently at Jordan Unversty of cence and echnology. He obtaned hs PhD from New Mexco tate Unversty (UA) n the felds of oology and Knot heory. He s actve n research and teachng and he suervsed a number of master s students, who are followng ther study as PhD students at the UA and Canada. He has ublshed a number of research artcles n reuted nternatonal journals of mathematcs. He has resented many talks n nternatonal conferences. References 1 F. Acard, oologcal nvarants of knots and framed knots n the sold torus, C. R. Acad. c. Pars r. I Math., 321 (1995), K. Bataneh and M. Abu Zaytoon, Vasslev nvarants of tye one for lnks n the sold torus, oology and ts alcatons, 157 (21) K. Bataneh, H. Belkhrat, he dervatves of the Hoste and Przytyck olynomal for orented lnks n the sold torus, Houston Journal of Mathematcs. In ress. K. Bataneh and M. Hajj, Jones olynomal for lnks n the handlebody, Rocky Mountan Journal of Mathematcs. In ress. 5 J. Brman and X-. n, Knot olynomals and Vasslev s nvarants, Invent. Math., 111 (1993), c 213 NP Natural cences Publshng Cor.