Madeti Prabhakar and Rama Mishra- Polynomial Representation for Links

8/3/2019 Madeti Prabhakar and Rama Mishra- Polynomial Representation for Links

1/12

Polynomial Representation for Links

Madeti PRABHAKAR Rama MISHRA

ABSTRACT

In this paper we introduce the concept of a degree sequence and theminimal degree sequence for link-types. The main focus is on finding adegree sequence for an arbitrary torus link. We conclude the paper bygiving polynomial representations of all links up to 7 crossings.

Keywords: Torus Links, Quasitoric braids

Mathematics Subject Classification 2000: 57M25, 14P25

1. Introduction

In 1992, Shastri ([10]) created polynomial parametrizations ofseveral knots, and showed that every knot-type has a polynomialrepresentation. In 1996, Mishra ([3]) introduced the concepts of adegree sequence and the minimal degree sequence for knot-types,which occurs naturally from polynomial representation of knots.In our earlier papers [5], [6], [7], [8], we studied the propertiesof rational knots, torus knots, quasitoric braids and explored theminimal degree sequence for 2-bridge knots and torus knots, whichpaved the way to find a degree sequence for a general knot type.

It is evident that every link-type has a polynomial represen-tation. So, naturally one can define a degree sequence and the

minimal degree sequence for a link-type. The main aim of this ar-ticle is to explore a way to find out a degree sequence to a generallink-type. Giving a polynomial representation to a knot-type laysthe foundation to the concept of a degree sequence and the minimaldegree sequence for knots.

In this paper, our goal is to understand the coefficient space ofpolynomial links, and to determine a degree sequence for a generallink-type. We outlined this paper as follows:

1. Introducing the concepts of polynomial representation of links,degree sequence and the minimal degree sequence of link types.

2. Finding a degree sequence for a torus link-type.

1


2/12

3. Regular projection of links up to 7 crossings with a polynomialrepresentations and their degree sequence.

The main part of this paper will focus on finding a degree se-quence for torus links. Finding a degree sequence for a generallink-type is then a self-explanatory.

2. Preliminaries

In this section we present the basic definitions, background and

the known results which serve as the prerequisite for the main re-sult.

Definition 2.1 An isotopy class of a proper smooth embedding :X1 X2 . . . Xn R

3 where each Xi = R such that |Xi ismonotone outside some closed interval is called a non-compact n-component link-type or an open n-component link-type.

In Shastris theorem [10], the existence of a polynomial repre-sentation for a given knot type K is shown by using Weierstrassapproximation. As each n-link type L can be obtained by n-copies

of knots, the existence of a polynomial representation for a givenlink type L can also be shown by using Weierstrass approximation.In this case also estimating the degrees of the polynomials is notclear. For a knot type K ift (f(t), g(t), h(t)) is a polynomial rep-resentation and deg(f(t)) = l, deg(g(t)) = m and deg(h(t)) = n,then we say that (l ,m,n) is a degree sequence of K. We define(l ,m,n) to be the minimal degree sequence of K if (l ,m,n) is min-imal amongst all the degree sequences of K with respect to theusual lexicographic ordering of N3. Note that a degree sequenceof K need not be unique. Using the same technique, we will for-mulate the definition of a degree sequence and the minimal degree

sequence for an n-link-type L.

Definition 2.2 Let L be a k-component open link. Let : R R . . . R R3, such that(t1, t2, . . . , tk) ((f1(t1), g1(t1), h1(t1)), (f2(t2), g2(t2), h2(t2)),

. . . , (fk(tk), gk(tk), hk(tk))) be a polynomial representation of L. Ifdeg(fi(ti)) = li, deg(gi(ti)) = mi and deg(hi(ti)) = ni then we saythat ((l1, m1, n1), (l2, m2, n2), . . . , (lk, mk, nk)) is a degree sequence

for a given link-type.

Definition 2.3 We say that

((l1, m1, n1), (l2, m2, n2), . . . , (lk, mk, nk))

2


3/12

is the minimal degree sequence for a link-type L if

((l1, m1, n1), (l2, m2, n2), . . . , (lk, mk, nk))

is a degree sequence for L and whenever

((p1, q1, r1), (p2, q2, r2), . . . , (pk, qk, rk))

is another degree sequence for L then (li, mi, ni) (pi, qi, ri) i,with respect to lexicographic ordering inN3.

One can observe that only finitely many link-types can be re-alized with a given degree sequence. Hence, for a given link-typeestimating a degree sequence and eventually the minimal degreesequence is an important aspect for polynomial link-types.

Here our main objective is to discuss the polynomial parametriza-tion of torus link-types and find some bound on the degrees ofpolynomials for a given link-type L.

In our earlier papers ([9],[2],[3], [5],[6], [7]) we have estimated adegree sequence and the minimal degree sequence for torus knots,2-bridge knots. Using quasitoric braid representation we had givena degree sequence for a general knot type also.

Theorem 2.4 A torus knot of type (2, 2n+1) has a degree sequence(3, 4n, 4n + 1).

Theorem 2.5 A torus knot of type (p, q), 2 < p < q, has a degreesequence (2p 1, 2q 1, 2q).

Remark 2.6 These are not the minimal degree sequence for torusknots.

For minimal degree sequence we have the following:

Theorem 2.7 [3] The minimal degree sequence for torus knot oftype (2, 2n + 1) for n = 3m; 3m + 1 and 3m + 2 is (3, 2n +2, 2n + 4); (3, 2n + 2, 2n + 3) and (3, 2n + 3, 2n + 4) respectively.

Theorem 2.8 [6] The minimal degree sequence for a 2-bridge knothaving minimal crossing number N is given by

(3, N + 1, N + 2) when N 0 (mod 3);

(3, N + 1, N + 3) when N 1 (mod 3);

(3, N + 2, N + 3) when N 2 (mod 3)

3


4/12

Theorem 2.9 [5] The minimal degree sequence for a torus knot oftype (p, 2p1), p 2 denoted byKp,2p1 is given by(2p1, 2 p, d),where d lies between 2p + 1 and 4p 3.

Remark 2.10 In order to represent a link-type by a polynomialembedding we require a suitable link diagram. For example for toruslink of type (p,q) we use its representation as closure of a p-braidnamely (1.2 . . . p1)

q. For a general link-type there may not besuch systematic nice diagram available.

Definition 2.11 For any two positive integers p and q the p-braid(1 . . . p1)

q is called the toric braid of type (p, q).

Remark 2.12 Closure of a toric braid gives a torus link of type(p,q). In particular if (p,q) = 1, then we obtain the torus knot oftype (p,q), and it is denoted by Kp,q.

Definition 2.13 A braid is said to be quasitoric of type (p,q)if it can be expressed as 1. . . . . q, where each j =

ej,11

. . . ej,p1p1 ,

with ej,k is either 1 or1.

Remark 2.14 A quasitoric braid of type (p, q) is a braid obtained from the standard diagram of the toric (p,q) braid by switchingsome of the crossing types.

Theorem 2.15 (Manturovs Theorem) [1] Each knot isotopy classcan be obtained as a closure of some quasitoric braid.

By using these results we proved the following in [8].

Theorem 2.16 LetK be a knot which is the closure of a quasitoricbraid obtained from a toric braid (1 2 . . . p1)

q, where (p, q) =1, by making r crossing changes. Then(2p1, q + r0, d) is a degree

sequence forK

wherer0

is the least positive integer such that (2p1, q + r0) = 1 and d 2q 1 + 4r.

In [8], we had given polynomial representations for all knottypes up to 8 crossings.

3. A Degree Sequence for Torus Links

In order to prove the main result of this section, we first provethe following lemmas.

Lemma 3.17 Let

(t1, . . . , tk) ((f1(t1), g1(t1)), . . . , (fk(tk), gk(tk)))

4


5/12

represent a regular projection of a k-component link L. LetNi be thenumber of variations in the nature of the crossings as we move alongthe ith component of the linkL. Then there exists a polynomialhi(t)of degree Ni such that the embedding

(t1, . . . , tk) ((f1(t1), g1(t1), h1(t1)), . . . , (fk(tk), gk(tk)), hk(tk))

is a representation for L.

Proof. Let s1 < s2 < . . . < sNi be such that all crossings correspond

to parameter values t R \ {s1, s2, . . . , sNi} and in any of the openintervals (, s1), (s1, s2), . . . , (sNi ,) all the crossings are ofthe same type (either over or under), also in successive intervals,the crossings are of opposite type. Now define

hi(t) =

Nij=1

(t sj).

It is easy to observe that hi(t) has constant sign on each intervaland opposite sign on consecutive intervals. Once after determiningall hi(t)s they provide an over/under crossing data for the link L.

Hence


is a representation for L.

Lemma 3.18 If L is a k-component alternating link representedby a polynomial embedding


and if the ith component of the link L has ni number of self inter-

sections and n

i number of intersections with other components ofthe link L, then deg(hi(t)) = 2ni + ni 1.

The proof directly follows from the above lemma.

Theorem 3.19 LetL(P, Q) denote a torus link of type (P, Q) withd = gcd(P, Q). Thus, each component is a torus knot of type (p,q)such that P = dp, Q = dq. Then a degree sequence for L(P, Q) isgiven by

((2p 1, q + r, 2q), (2p 1, q + r, 2q), . . . , (2p 1, q + r, 2q))

where r is the least positive integer satisfying the following:

5


6/12

(i) (2p 1, q + r) = 1

(ii) r(2p 1) q is a non-negative even integer.

Proof. Let L(P, Q) be a d component torus link of type (P, Q).Then a braid representation ofL(P, Q) is given by (12 . . . P1)

Q.Since L(P, Q) is a d component torus link, there exist integers p, qsuch that P = dp and Q = dq. Hence

L(P, Q) = K1p,q

K2p,q

. . .

Kdp,q,

where each Kip,q is a torus knot of type (p, q). Also observe that

#Kip,q Kjp,q = 2pq.

Since Kip,q has bridge number p, there are p relative maxima andp relative minima. In the non-compact version, one relative maximaand one relative minima can be destroyed. Thus there are at least2p2 relative extrema. Hence the degree of fi(t) (representing theX-coordinate) is 2p 1. For a detailed proof on the existence offi(t), gi(t) and the degree of fi(t) one can refer to [6].

From [6] the degree of a polynomial gi(t) such that (Xi(t), Yi(t)) =(fi(t), gi(t)) is a regular projection ofK

ip,q , must be at least q + r,

where r is the least positive integer such that (2p 1, q + r) = 1.Now if we consider the curve Ci : (Xi(t), Yi(t)) = (fi(t), gi(t)) withdegrees of fi(t) is equal to 2p 1 and degree of gi(t) is equal toq + r, then the Milnor number of this curve is (2p 1)(q + r 1)and hence the delta invariant is (p 1)q + (p 1)(r 1). Since thetotal number of double points in the link is (P 1)Q = (dp 1)dqand the linking number between any two components is 2pq, bytranslation of each component curve to get the desired number ofdouble points, we choose the integer r such that it satisfy the fol-lowing:(i) (2p 1, q + r) = 1(ii) r(2p 1) q is a non-negative even integer.

By using lemma it is easy to construct the third polynomialhi(t) which gives over/under crossing data. Since the number ofcrossings in L(P, Q) is (P 1)Q = (dp 1)dq and as L(P, Q) is a(P 1) = (dp 1) alternating, we have degree of hi(t) is equal to2q.

Now by using quasitoric braid representation of any link L, itis easy to find a degree sequence for L. If the quasitoric braidrepresentation ofL is L(P, Q), then the regular projection of L andL(P, Q) is same and hence by above theorem we can give a degree

sequence for L. Since the crossing pattern is different for both L

6


7/12


8/12

-2 - 1. 5 - 1 -0.5 0.5 1

-2

-1.5

-1

-0.5

0.5

1

-10 -5 5 10 15 20

-100

-50

50

100

L5a1 L6a1

L6a2: (t1, t2) ((t1 + 3.5,(t21 3.025)(t1 2.5)(t1 .0415), (t +

2.337045)(t+2.072685)(t+1.63653)(t0.01967)(t1.58398)), (t22

, t2,(t+2.072685)(t + 2.337045)(t + 1.90094)(t 0.01967)(t 1.58398)))

L6a3: (t1, t2) ((102(t2

1 3)(t2

1 1.5)(t2

1 0.25), 100t1, (t

21

1.4905952)(t210.8550892)t1), (10

2(t223)(t2

21.5)(t2

20.25), 100t2,(t

22

1.4905952)(t22 0.8550892)t2))

2 4 6 8

-15

-10

-5

5

-400 -200 200 400

-200

-100

100

200

L6a2 L6a3

L6a4: (t1, t2, t3) ((t1, t210.25,(t1+0.20798995)(t10.29201005)(t10.82175)), (t2

2+ 0.025, t2,(t2 0.6932205)(t2 0.16348055)(t2 +

0.48678495)), (t3,t23

+ 0.25, (t3 + 0.1285295)(t3 0.3714705)(t3 0.278795)))

L6a5: (t1, t2, t3) ((t1, (.34t210.5),(t1+0.2723875)(t11.0437375)(t1

1.72035)), (t22

+1.195, 1.4t20.25,(t21.72035)(t20.522664)(t2+0.680401)), (1.9t30.75,0.7t

23

+0.25, (t3+0.2723875)(t30.3355415)(t3+0.680401)))

8


9/12

-2 -1 1 2

-1

-0.5

0.5

1

-6 -4 -2 2 4 6

-2

-1

1

L6a4 L6a5

L6n1: (t1, t2, t3) ((t1, t21 0.25,(t1)), (t

22

+ 0.025, t2,(t2 0.61376005)), (t3,t

23

+ 0.25, t3))

L7a1: (t1, t2) ((t1(t1 1)(t1 + 1), t21

(t1 1.09)(t1 + 1.07),(t1 +0.57852485)(t1 + 0.54183035)(t1 + 0.30542775)(t10.1008705)(t1 +0.019425)(t1+0.1264315)(t10.316296)(t10.5329602)(t10.5650752)), ((t2),(t

22

+0.095), (t2 + 0.23753885)(t2 + 0.02087465)(t2 0.2155282)))

-2 -1 1 2

-1

-0.5

0.5

1

-1.5 -1 -0.5 0.5 1 1.5

-0.8

-0.6

-0.4

-0.2

0.2

0.4

L6n1 L7a1

L7a2: (t1, t2) ((t1(t1 1)(t1 + 1), t21(t1 1.09)(t1 + 1.09), (t1 +1.037945)(t1+0.4666504)(t10.2027426)(t10.081977)t1(t1+0.081977)(t1+0.2027426)(t10.4666504)(t11.037945)), ((t2),(t

22

+0.235), (t2+0.2027426)t2(t2 0.2027426)))

L7a3: (t1, t2) ((t1(t1 2)(t1 + 2), t21

(t1 2.28)(t1 + 2.39), (t1 +2.12)(t1+1.25)(t10.85)(t10.8)(t10.12)(t11.35)(t10.99)(t10.85)(t11.95)), ((t2+1.8),(t

22

+ 1.9), (t2+0.265)(t20.85)(t21.895)))

9


10/12


11/12

-3 -2 -1 1 2

0.5

1

1.5

2

2.5

-3 -2 -1 1 2

-1

1

2

3

L7a6 L7a7

L7n1: (t1, t2) ((t1(t1 1)(t1 + 1), t21

(t1 1.09)(t1 + 1.07),(t1 +0.57852485)(t1 0.14052315)(t1 0.2566305)(t1 0.5740625)(t1 1.043905)), ((t2),(t

22

+ 0.095),(t2 + 0.1625337)))

L7n2: (t1, t2) ((t1(t1 1)(t1 + 1), t21

(t1 1.09)(t1 + 1.07),(t1 +0.57852485)(t1 0.14052315)(t1 0.2566305)(t1 + 0.1214315)(t1 +0.1625337)), ((t2),(t

22

+0.095), (t2+0.23753885)(t20.14052315)))

-1.5 -1 -0.5 0.5 1 1.5

-0.8

-0.6

-0.4

-0.2

0.2

0.4

-1.5 -1 -0.5 0.5 1 1.5

-0.8

-0.6

-0.4

-0.2

0.2

0.4

L7n1 L7n2

Acknowledgements

The authors would like to thank Prof. Akio Kawauchi for his sup-port during their stay in OCU, where they carried out this research.

References

[1] V. O. Manturov, A combinatorial representation of links by quasitoricbraids, pp. 207212, European J. Combin. 23 (2002).

[2] R. Mishra, Polynomial representation of torus knots of type (p,q), pp.667700, J. Knot Theory Ramifications 8 (1999).

[3] R. Mishra, Minimal degree sequence for torus knots, pp. 759769, J. Knot

Theory Ramifications 9 (2000).

11


12/12

[4] Norbert ACampo, Le Groupe de Monodromie du Deploiement des Singu-larites Isolees de Courbes Planes I, pp. 132, Math. Ann. 213 (1975).

[5] P. Madeti and R. Mishra, Minimal degree sequence for torus knots of type(p, 2p 1), J. Knot Theory Ramifications Vol. 15, (9) (2006).

[6] P. Madeti and R. Mishra, Minimal degree sequence for 2-bridge knots, pp.191210, Fund. Math. 190 (2006).

[7] P. Madeti and R. Mishra, Minimal degree sequence for torus knots of type(p, q), Preprint, http://paniit.iitd.ac.in/ rama/publi.html.

[8] R. Mishra and P. Madeti, Degree sequence for polynomial knots, commu-nicated.

[9] R. Shukla and A. Ranjan, On polynomial representation of torus knots,

pp. 279294, J. Knot Theory Ramifications 5 (1996).[10] A. R. Shastri, Polynomial representations of knots, pp. 1117, Tohoku

Math. J. (2) 44 (1992).[11] R. Shukla, On polynomial isotopy of knot-types, pp. 543548, Proc. Indian

Acad. Sci. Math. Sci. 104 (1994).

Prabhakar Madeti: Department of Mathematics, Indian Instituteof Technology Guwahati, Guwahati - 781039, [email protected]

Rama Mishra: Department of Mathematics, Indian Institute ofTechnology Delhi, New Delhi - 110016, India

[email protected]

12

Madeti Prabhakar and Rama Mishra- Polynomial Representation for Links

Documents

Transcript of Madeti Prabhakar and Rama Mishra- Polynomial Representation for Links