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b 2000On Iterated Torus Knots and Transversal Knots William W.Menasco ∗University at Buffalo Buffalo,New York 14214Abstract A knot type is exchange reducible if an arbitrary clod n -braid reprenta-tive K of K can be changed to a clod braid of minimum braid index n min (K )by a finite quence of braid isotopies,exchange moves and ±-destabilizations.(See Figure 1).In the manuscript [BW]a transversal knot in the standard con-tact structure for S 3is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its lf-linking number.Theorem 2of [BW]establishes that exchange reducibility implies transversally simple.Theorem 1,the main result in this note,establishes that iterated torus knots are exchange reducible.It then follows as a Corollary that iterated torus knots are transversally simple.
1Introduction.
Let C ⊂S 3be a knot,let V C be a solid torus neighborhood of C and let ∂V C =T C ⊂S 3be a peripheral torus for C .The simple clod curve on T C that reprents the homotopy class of pm +ql ,where m is meridian homotopy class,l is the preferred longitude homotopy class and p,q ∈Z ,is called the (p,q )cable of C .We will u the notion C (C ,(p,q ))to indicate the resulting knot of this cabling operation .If C is the unknot then the cabling operation produces a (p,q )-torus knot.
We can,of cour,iterate the cabling operation.Starting with an initial knot C 0and a quence of 2-tuples of integers {(p 1,q 1),(p 2,q 2),···,(p h ,q h )},with p i <q i ,1≤quasar
2
William W.Menasco
i ≤h ,we can construct the knot C (C (···C (C (C 0,(p 1,q 1)),(p 2,q 2))···,(p h −1,q h −1)),(p h ,q h )).(1)
If C 0is the unknot then any iteration of the cabling operation produces an iterated torus knot .Letting (P,Q )={(p 1,q 1),(p 2,q 2),···,(p h ,q h )},the final iteration pro-duces a knot,K (P,Q ),which is on the peripheral torus of the next to last knot in the iteration;mainly,T C (C (···C (C (C 0,(p 1,q 1)),(p 2,q 2))···),(p h −1,q h −1)).
老爸老妈浪漫史第九季In §2of
[BW]three moves are discusd which take clod braids to clod braids,prerving knot type:braid isotopy,exchange moves and destabilization.Braid iso-topy means isotopy in the complement of the braid axis.The exchange move is a special type of Reidemeister II move illustrated in Figure 1(a).De
stabilization means reducing the braid index by eliminating a (positive or negative)trivial loop,as shown in Figure 1(b).Notice that braid isotopy and exchange moves prerve both algebraic crossing number and braid index,whereas destabilization changes both.单身插班生
Figure 1:
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For a more extensive treatment of the isotopies,e [BF,BM2,BM3].)
As defined in [BW],a knot type K is exchange reducible if an arbitrary clod n -braid reprentative K of K can be change to a clod braid reprentative of minimum braid index,n min (K ),by a finite quence of braid isotopies,exchange moves,and ±-destabilizations.The main result of [BM1]established the exchange reducibility of the unknot.The main theorem in this paper is an analogous result for iterated torus knots
Theorem 1Oriented iterated torus knots are exchange reducible.
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The proof of Theorem1involves adapting the braid-foliation machinery which was developed in[BM3]to a new situation where there is a torus in S3which is being foliated and a knot is embedded on this torus.It employs the fact[Sch]that an oriented iterated torus knot K(P,Q)has a unique braid reprentative of minimal braid index h1p i.
Theorem1has an immediate application to transversal knots.Letξbe the stan-dard contact structure i
n oriented S3.The structureξcan be thought of as a plane field that is totally non-integrable.A knot K is transversal if and only if K intercts each plane in the planefieldξtransversally.A transversal isotopy of K is an isotopy of K in S3through transversal knots.(See[El].)If K and K′are two transversal knots that are transversally isotopic,then they are reprentatives of the same transversal knot type,T K.
A classical invariant of transversal knot types is a lf-linking number,the Ben-nequin number,β(T K).The lf-linking is defined by pushing the transversal knot offitlf in a direction which is in the contact plane.A well-defined direction exists becau S3is parellelizable.See[BW]for a preci description.A transversal knot type T K is transversally simple if it is determined by its topological knot type K and its lf-linking number.In[El]it wasfirst shown that the unknot is transversally sim-ple.In[Et]it was established that positive transversal torus knots are transversally simple.In[BW]a more general framework for understanding transversally simple knots was established.
Theorem2(e[BW])If T K is a transversal knot type with associated topological knot type K,where K is exchange reducible,then T K is transversally simple.
Combining Theorems1and2,we have the following immediate corollary.
2012年6月四级考试Corollary3Let T K(P,Q)be a transversal knot type with associated topological knot that of the iterated torus knot K(P,Q).Then T K(P,Q)is transversally simple.
The outline for this note is as follows.In§2we review and adapt the braid-foliation machinery for the torus that wasfirst introduced in[BM3].We will be concerned with the situation where we are given a torus which contains a knot K(P,Q)and bounds a solid torus.However,we do not have a natural way of identifying the core curve of the solid torus.Hence,we will u T⊂as notation for the given torus containing
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K(P,Q).The foliation machinery on T⊂will involve understanding the manipulation of three different types of foliations—circular,mixed,and tiled foliations.(The foliations will be defined in§2.)In§3we will prove Theorem1in the special ca where T⊂has a circular foliations.The overriding strategy of the remaining ctions is to reduce the mixed and tiled foliations to circular foliations.In§4we show how destabilizations and exchange moves allow one to replace a mixed foliation with a circular foliations.Similarly,in§5we show how destabilizations and exchange moves allow one to replace a tiled foliation with a circular foliations.
ACKNOWLEDGMENTS—The author wishes to thanks Joan Birman and Nancy Wrinkle for en-couraging him to think about a proof of Theorem1during his brief sabbatical stay at Columbia University.That stay was partially supported by NSF grant DMS-9705019.
2The braid foliation machinery for the torus.
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In this ction we adapt the combinatorics of[BM3]to the pair(K,T⊂),where K= K(P,Q)is an iterated torus knot which lies on the torus T⊂.Since the exposition in[BF]supplies us with a centralized source for most of the previously developed machinery,we will u it almost exclusively as our primary reference.Although the arguments in this note rely heavily on results in the existing literature,a reader need only consult[BF]and[BM3]in almost all cas.
Let K⊂S3be an oriented clod n-braid with axis A.Then there is a choice of afibration of the open solid torus S3−A by2-disc.We will refer to H={Hθ|θ∈[0,2π]}as this2-discfibration of S3−A.Each Hθis a disc with boundary A.We consider the interction of the H′θs with T⊂—the induced singular foliation on T⊂by H.We have a quence of lemmas that begin to standardize this foliation.The lemmas imitate the similar t of lemmas in§1of[BM3]which dealt with an esntial torus in the complement of a clod braid.Since our prent ca is slightly different (the clod braid is actually a homotopic
ally non-trivial curve on the torus),we will only supply the additional details needed to adapt the proofs of[BM3]to this ca.
Lemma4We may assume that:
(i)The interctions of A with T⊂arefinite in number and transver.Also,if
p∈A∩T⊂then p has a neighborhood on T⊂which is radially foliated by its arcs of interction withfibers of H.
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(ii)All butfinitely manyfibers Hθ∈H meet T⊂transversally,and tho which do not(the singularfibers)are each tangent to T⊂at exactly one point in the interior of both T⊂and Hθ.Moreover,the tangencies(which are contained in singular leaves)are either local maxima,or minima,or saddle points.
(iii)A singularfiber contains exactly one singular point.
(iv)Each singular point is either a center or a saddle.
(v)A leaf that does not contain a singular point(a non-singular leaf)is either an arc having endpoints on A or a simple clod curve.
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Proof.We u exactly the same general position argument as in[BM3].♦
We refer to the leaves of the foliation of T⊂as b-arcs and c-circles.Each b-arc and each c-circle lies in both T⊂and in somefiber Hθ∈H.Since K⊂T⊂,for all Hθ∈H,each point of K∩Hθis contained in a b-arc or c-circle.Finally,since K intercts each discfiber of H coherently,K must interct each non-singular leaf coherently.
A b-arc,b⊂T⊂∩Hθ,is esntial if either b∩K=∅,or both sides of Hθsplit along b are intercted by K.A c-circle,c⊂T⊂∩Hθ,is esntial if c∩K=∅.The definition of esntial b-arcs and c-arcs is an adaptation of the definition in[BM3], however inesntial b-arcs(c-circles)are still arcs(resp.circles)splitting offsub-discs (resp.bounding subdiscs)of discfibers that are not intercted by K.
Lemma5Assume that T⊂satisfies(i)-(v)of Lemma4.Then T⊂is isotopic to a cabling torus,T′⊂,such that the foliation of T′⊂also satisfies(i)-(v)and in addition:
1.All b-arcs are esntial.
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2.All c-circles are esntial.
3.Any c-circle in the foliation is homotopically non-trivial on T′⊂.
Moreover,the restriction of the isotopy to K is the identity.grea gun
Proof.The argument for eliminating inesntial b-arcs is exactly the same as the argument which was ud in the proof of Lemma2of[BM3].Similarly,if c is a c-circle
