ON THE DEGREES OF RATIONAL KNOTS

Transcription

1 ON THE DEGREES OF RATIONAL KNOTS DONOVAN MCFERON, ALEXANDRA ZUSER Absrac. In his paper, we explore he issue of minimizing he degrees on raional knos. We se a bound on hese degrees using Bézou s heorem, define minimal degree sequences for raional funcions, and explore he degree sequences of he refoil and figure-eigh knos. This research was conduced a he Moun Holyoke College REU program during he summer of 00. Special hanks go ou o our advisors, Alan Durfee and Donal O Shea, and he oher members of our research group, David Clark, Virginia Peerson and Craig Phillips.. Inroducion Obaining raional parameerizaions of knos, wheher by convering polynomial [5, 6] or rigonomeric [] parameerizaions o raional ones or by consrucing raional parameerizaions from scrach [4, 6], yields funcions wih very high degrees on he polynomials in he numeraor and denominaor. In many cases, hese degrees can be reduced o as lile as one fourh of heir original size. This phenomenon leads o he quesion, wha are he minimum degrees wih which a kno ype can be parameerized raionally? Having he minimum degrees of kno ypes would also be useful in rying o classify raional knos according o heir degrees.. Bounds on Raional Knos Based on Degree Le us define he degree of a raional funcion by he following: Definiion.. The Degree of a raional funcion f() = p() q(), where p() and q() are relaively prime polynomials of degree p and degree q is defined o be p q Le α() = (f(x), g(x), h(x)) be a raional funcion and denoe by ( m n, p q, r s ) he degrees of (x(), y(), z()). If α() parameerizes a kno, hen we can find an upper bound on is crossing number by considering he degrees of x(), y() and z(). Theorem.. Le c be he crossing number of a raionally parameerized kno and ( m n, p q, r s ) be he degrees of is componens. Then c (m + n )(p + q ). Proof. Since he crossing number of a kno, c, is he leas number of crossings, or self-inersecions, in any projecion of he kno [, page 67], we can pu an upper bound on c by considering he x y, x z, and y z projecions. Le us firs consider he x y projecion. Wihou loss of generaliy, we can assume ha x() = m +... n +..., Key words and phrases. Raional Knos, Minimal Degree, figure-eigh Knos, Trefoils. This research is suppored by he NSF, gran DMS-9738.

2 On he Degrees of Raional Knos y() = p +... q +..., where... denoes lower-order erms. A self-inersecion occurs when here exis and such ha and x( ) = x( ), y( ) = y( ). This gives us he equaions m +... n +... = m +... n +... and p +... q +... = p +... q +... for x and y, respecively. By cross-muliplying each of he above equaions and collecing all erms on he lef hand side, we ge a polynomial of he form for x, and a polynomial of he form m n n m +... = 0 p q q p +... = 0 for y. These are polynomials in degree m + n and degree p + q, respecively. = is a soluion o boh of hese equaions, bu an unwaned one. To remove i, we can divide each of hem by o obain m n n m +... = 0 p q q p +... = 0, which are polynomial equaions wih degrees m + n and p + q, respecively. By Bézou s heorem, he number of common soluions is a mos he produc of he degrees of hese wo polynomials, (m + n )(p + q ) [3, page 9]. Since each crossing of he projecion in fac gives wo soluions, he number of crossings is a mos (m + n )(p + q ). To find he lowes number for he bound, we use he projecion wih he lowes degrees, i.e. if (m + n) < (p + q) < (r + s), hen (m + n )(p + q ) < (m + n )(r + s ) < (p + q )(r + s ), which gives c (m + n )(p + q ) as he mos accurae bound. A bound of his ype is useful for choosing a saring poin when rying o deermine he minimal degree sequences of a kno, as we will do in Secions 3, 4, and 5... Example. Le us consider an example where deg(x(), y(), z()) = (,, ). Here, c 4.5, so any kno parameerized by such a funcion would have a mos four crossings. Thus, a raional funcion wih degrees (,, ) can only parameerize an unkno, a refoil or a figure-eigh kno, since hese are he only knos ha have crossing number 4 [7, 6]. Unforunaely, as will be proven in secion??, i is acually impossible o parameerize a refoil or a figure-eigh kno wih degrees as low as (,, ). This means ha he only kno wih degree (,, ) is he unkno, suggesing ha his mehod ses a raher high bound on he crossing number. Making he bound lower and more accurae would be a good goal for fuure research.

3 On he Degrees of Raional Knos 3 3. The Minimal Degree Sequence of a Compac Raional Parameerizaion Definiion 3.. A riple ( p q, p q, p 3 ) Q 3 is said o be a degree sequence for a kno-ype K if () p i q i i () q q (3) If q i = q i+, hen p i p i+ (4) here exis real raional funcions f(), g(), and h() of degree p q, p q, and p 3 respecively such ha he embedding (f(), g(), h()) represens K. Noe: q i N. Definiion 3.. A degree sequence ( p q, p q, p 3 ) Q 3 is said o be minimal for a kno-ype K if for any oher degree sequence, ( m n, m n, m 3 n 3 ), for K, hen ( p q, p q, p 3 ) ( m n, m n, m 3 n 3 ). Here is he lexicographic ordering in N 6, on he sexuples (q, q,, p, p, p 3 ) and (n, n, n 3, m, m, m 3 ). Example 3.. ( 4, 4, 3 6 ) < ( 4, 3 4, 6 ) Example 3.. ( 4, 3 4, 3 4 ) < ( 4, 4, 6 ) 4. The Minimal Degree Sequence of he Compac Raional Trefoil According o Bézou s heorem, he firs degree sequence ha can produce hree crossings is (, ). The x y projecion of he refoil, (f(), g()), has hree double poins. Tha is, here mus be < < 3 < 4 < 5 < 6 ) such ha: f( i ) = f( i+3 ) g( i ) = g( i+3 ) i =,, 3 In order ha (f(), g(), h()) race ou a refoil, we need h( ) < h( 4 ) ( )h( ) > h( 5 ) h( 3 ) < h( 6 ) So, if he degree sequence (, ) can produce he x y projecion of he refoil, hen i is possible o arrange he 3 double poins on he graphs of f and g. We ask wheher we can choose < < 3 < 4 < 5 < 6 ) on he -axis so ha h saisfies ( ) if h has he following graphs. The firs hree figures and heir reflecions show he only possible forms of a degree raional funcion. I is clear ha his requires four or more monoone regions. Therefore, he refoil canno be consruced using a raional funcion of degree or lower. The fourh figure is he simples graph on which he hree double poins of he refoil can be arranged. I is easy o show ha i canno be formed by a raional funcion of degree or lower. In fac, wih a lile work i can be shown ha 4 is he lowes degree capable of producing such a graph. Therefore, ( 4, 4 ) is he firs possible minimal degree sequence for he x y projecion of he refoil.

4 4 On he Degrees of Raional Knos Figure. A Degree raional funcion Figure. A Degree raional funcion Figure 3. A Degree raional funcion Figure 4. Simples Graph for he Trefoil s Double Poins We define f () and g () as follows: f () = g () = Figure 4 shows he race of (f (), g ()), which is clearly a refoil projecion. I seems ha ( 4, 4, 4 ) is he minimal degree sequence for he refoil. However, an h() of degree 4 has no been found ha will saisfy ( ) for he given x y projecion. So, we can only conclude ha he minimal degree sequence for he compac raional refoil is greaer han ( 4, 4, 4 ).

5 On he Degrees of Raional Knos Figure 5. Trace of (f (), g ()) On he oher hand, le us show ha i is possible o parameerize a refoil by a ( 3 4, 3 4, 3 4 ) degree sequence. This will give us an upper bound on he minimal degree sequence for a refoil. Le ψ : C C 3 be defined by: g () = h () = f () = ( +.53)( +.04)( ) ( +.8)(.)(.8) Figure 6. x y projecion of (f (), g (), h ()) A series of Reidemeiser moves can ransform his refoil ino one ha resembles he sandard refoil. We conclude ha he minimal degree sequence of he compac raional refoil is greaer han or equal o ( 4, 4, 4 ) and less han or equal o ( 3 4, 3 4, 3 4 ). 5. The Minimal Degree Sequence of he Compac Raional figure-eigh Kno According o Bézou s heorem, he firs degree sequence ha can produce four crossings is (, ). However, i seems inuiive ha he minimal degree sequence for he figure-eigh kno would be greaer han or equal o ha of he refoil. In order ha (f(), g(), h()) race ou a figure-eigh kno, we need

6 6 On he Degrees of Raional Knos h( ) < h( 6 ) ( )h( ) > h( 5 ) h( 3 ) < h( 8 ) h( 4 ) < h( 7 ). Therefore, he firs possible degree sequence for he figure-eigh is he same as he refoil, ( 4, 4, 4 ). Unforunaely, we canno deermine wheher one can parameerize he figureeigh kno wih a ( 4, 4, 4 ) degree sequence. However, an upper bound for he degree sequence has been deermined. I is possible o parameerize he figure-eigh kno by a ( 3 4, 3 4, 3 4 ) degree sequence. Le ψ 3 : C C 3 be defined by: g 3 () = f 3 () = ( +.83)(.4)(.65). + 4 ( + )(.3)(.65) h 3 () = + 4 Once again, a series of Reidemeiser moves can ransform his figure-eigh kno ino one ha resembles he sandard one Figure 7. XZ projecion of (f 3 (), g 3 (), h 3 ()) We conclude ha he minimal degree sequence of he compac raional figureeigh is greaer han or equal o ( 4, 4, 4 ) and less han or equal o ( 3 4, 3 4, 3 4 ). 6. Conjecures The minimal degree sequence of he compac raional refoil is ( 4, 4, 4 ), The minimal degree sequence of he compac raional figure-eigh is sricly less han ( 3 4, 3 4, 3 4 ), There exiss a condiion sronger han Bézou s heorem for reducing degree sequences for a kno wih n crossings.

7 On he Degrees of Raional Knos 7 References [] C. Adams, The Kno Book, W.H. Freeman and Company, New York, 994. [], D. Clark, Transforming Trigonomeric Kno Parameerizaions Ino Raional Kno Parameerizaions, Moun Holyoke REU, 00. [3] D. Cox, J. Lile, D. O Shea, Using Algebraic Geomery, Springer-Verlag, New York, 998. [4] D. McFeron, Algorihm For Consrusing Compac Raional Parameerizaions, Moun Holyoke REU, 00. [5] V. Peerson, Mirror Images of Knos, Moun Holyoke REU, 00. [6] C. Phillips, Three Mehods of Consrucing Raional Represenaions of Knos, Moun Holyoke REU, 00. [7] P. Tai, On Knos, XXVIII, Transacions of he Royal Sociey of Edinburgh, Edinburgh, 879, Donovan McFeron: Universiy of Nore Dame, Souh Bend IN, Zuser: Marlboro College, Marlboro VT, address: dmcferon@nd.edu, az@marlboro.edu Alexandra