A Kind of Essential Surfaces in the Complements of Knots

HAN YOU-FA,PAN SHENG-JUN AND WANG SHU-XIN

(School of Mathematics,Liaoning Normal University,Dalian,Liaoning,116029)

Communicated by Lei Feng-chun

A Kind of Essential Surfaces in the Complements of Knots

HAN YOU-FA,PAN SHENG-JUN AND WANG SHU-XIN

(School of Mathematics,Liaoning Normal University,Dalian,Liaoning,116029)

Communicated by Lei Feng-chun

In this paper,by the twist-crossing number of knots,we give an upper bound on the Euler characteristic of a kind of essential surfaces in the complements of alternating knots and almost alternating knots,which improves the estimation of the Euler characteristic of the essential surfaces with boundaries under certain conditions.Furthermore,we give the genus of the essential surfaces.

essential surface,twist-crossing number,almost alternating knot,reduced graph

1 Introduction

Menasco[1]discussed many properties of incompressible surfaces(including closed incompressible surfaces and essential surfaces)which are properly embedded in the complements of knots,and gave the def i nition of incompressible pairwise incompressible surfaces.By Menasco's method,if F is a properly embedded essential surface or incompressible pairwise incompressible surface in the complements of alternating knots,we may assume that F is in a standard position,even if F is not in a standard position,F can also be replaced by another surface F′,where F′is isotopic to F in the complement of K,and lies in a standard position(see[1–2]).Since surfaces lying in standard position have many good properties,the researchers have used them to analyze properties of surfaces which are properly embedded in the complements of knots.Menasco and Thistlethwaite[2]have proved the cabling conjecture and given an upper bound of the Euler characteristic of a kind of essential surfaces in the complements of alternating knots.Meantime,Menasco also has given a geometric proof to that alternating knots are nontrivial.Adams[3]has dealt with closed incompress-ible pairwise incompressible surfaces in the complements of almost alternating knots,and obtained many useful results.Han[4]has studied essential surfaces with meridional boundary in the complements of almost alternating knots,and proved that the essential surfaces with meridional boundary in the complements of almost alternating knots are f i nite under ambient isotopy.So it is meaningful for us to discuss properties of specif i c essential surfaces in the complements of alternating knots and almost alternating knots.

In this paper,we mainly discuss properties of a kind of essential surfaces with boundary by using the triangulation of surfaces and the methods adopted by Menasco.The aim of this paper is to give an estimation of the Euler characteristic of the essential surfaces which are properly embedded in the complements of alternating knots and almost alternating knots. In Sections 2 and 3,we give the main theorems of this paper and their corollaries.

2 A Kind of Essential Surfaces in the Complements of Alternating Knots

Def i nition 2.1The twist-crossing number of a diagram D is the smallest positive integer n such that there exists a sequence a1,b1,···,an,bn,of regular points in cyclic order on the knot K,with the properties that for each i=1,2,···,n,

(1)there is at most one singular point of K between aiand bi;

(2)all singular points between biand ai+1project to crossing points in the same twist of D(an+1is taken to be a1).

Remark 2.1Denote the twist-crossing number of a diagram D by TCN(D).

Def i nition 2.2Let M be a bounded 3-manifold,and F a properly embedded surface in M.If F is incompressible and boundary incompressible,then F is called an essential surface in M.

Let K?S3be a knot,and F be a properly embedded surface with boundary in the complement of K and lie in a standard position.Let C be a cell decomposition of F,andbe the corresponding cell decomposition of?F,where?F is the corresponding capped-of f closed surface of F.If we take the boundary components of F as fat vertices,then we can get the reduced graph?C(see[2]).

Lemma 2.1[4]Let K be a prime alternating knot,which admits a standard alternating diagram D with TCN(D)=n,and let F be an essential surface with f i nite boundary slope in the exterior of K and with β boundary components,each of which has b(b/=0)longitudinal components.Then(1)s=0 if 1≤n≤5,(2)s≤max{0,-χ(F)-bβ}if n≥6,where s is the number of saddle-intersections of F with the crossing-balls of D.

Remark 2.2When 1≤n≤5,all boundary edges and bubbles are good.

Lemma 2.2[5]Every closed surface is a polyhedron of some closed and fake 2-manifold.

Lemma 2.3[5]Let K be a polyhedron of some closed-fake 2-manifold.Then

(1)3r=2e;

(2)e=3(v-χ(K));

where v,e and r denote the number of vertices,edges and regions of K,respectively.

Proposition 2.1Let F be a closed surface which is glued by many polyhedrons,e and r be the numbers of edges,regions respectively in the corresponding polyhedron decomposition of F.If the number of edges of each polygon is greater than or equal to 4,then e≥2r.

Proof.Let K be the polyhedrons decomposition of F,e and r be the numbers of edges and regions of K,respectively.Since each region of K has at least 4 edges,each region of K can be triangulated into many triangles.Based on Lemma 2.2,suppose that K′is another polyhedron decomposition of F,which is obtained from K,and each region of K′is a triangle.Let e′and r′be the numbers of edges and regions of K′,respectively.Let Ridenote the ith region of K,and eRidenote the number of edges of Ri(i=1,2,···,r).Thus

By Lemma 2.2,3r′=2e′,and then

Therefore,e≥2r.

Theorem 2.1Let K be a prime alternating knot,which admits a standard alternating diagram D with TCN(D)=n,and let F be an essential surface with f i nite boundary slope in the exterior of K and with β boundary components,each of which has b(b/=0)longitudinal components.If all boundary edges are good(s=0),and the region which is bounded by c in the reduced graph?C has at least 4 edges,where c is any component of F∩then

Proof.Let e and r denote the numbers of edges and regions of the reduced graph?C, respectively,and N be the smallest valency of any boundary vertex of?C.Then N≥nb(see [2]).By Proposition 2.1,

Remark 2.3Theorem 2.1 improves the estimation of the Euler characteristic of essential surfaces in the complements of alternating knots.

Corollary 2.1Let K be a prime alternating knot,which admits a standard alternating diagram D with TCN(D)≥4,and let F be an essential surface with f i nite boundary slope in the exterior of K and with β boundary components,each of which has b(b=1)longitudinalcomponents.If all boundary edges are good(s=0),and the region which is bounded by c in the reduced graph?C has at least 4 edges,where c is any component of F∩S2±,then g(F)=1.

Proof.By Lemma 2.1,-χ(F)-bβ≤0.Since

one has

By Theorem 3.1,

i.e.,

Therefore,g(F)=1.

3 A Kind of Essential Surfaces in the Complements of Almost Alternating Knots

Def i nition 3.1A projection D of a knot in the 3-sphere is almost alternating,if one crossing change makes the projection alternating.

Def i nition 3.2A knot is almost alternating if it has an almost alternating projection and does not have an alternating projection.

Lemma 3.1[4]Let K be a prime almost alternating knot,which admits a standard almost alternating diagram D with TCN(D)≥4,and let F be an essential surface with fi nite boundary slope in the exterior of K and with β boundary components,each of which has} components.If for any,then s≤max,where s is the number of saddle-intersections of F with the crossing-balls of D,and r is the number of the regions of the reducedC?.

Proposition 3.1Let K be a prime almost alternating knot,which admits a standard almost alternating diagram D with TCN(D)＞1,and let F be an essential surface with fi nite boundary slope in the exterior of K and with β boundary components,each of which has b(b=1)longitudinal components.If for any c,and all boundary edges are good(s=0),then g(F)=0.

Proof.By Lemma 3.1 and χ(F)=2-2g(F)-bβ,we know

and so

Therefore,g(F)=0.

Lemma 3.2[2]Let K be a prime almost alternating knot,which admits a standard almost alternating diagram D with TCN(D)=n,and let F be an essential surface with f i nite boundary slope in the exterior of K,with β boundary components,and with longitudinal b (b/=0),and some vertex of the reduced intersection?C has valency N,then N≥(n-2)b.

Theorem 3.1Let K be a prime almost alternating knot,which admits a standard almost alternating diagram D with TCN(D)=n,and let F be an essential surface with f i nite boundary slope in the exterior of K and with β boundary components,each of which has b (b/=0)longitudinal components.If the region bounded by c in the reduced graphhas at

least 4 edges,where c is any component ofthen

Proof.Let e and r be the numbers of edges and regions of the reduced graph?C,respectively,s be the number of saddle-intersections of F with the crossing-balls of D,r be the number of the regions of the reduced,and N be the smallest valency of any boundary vertex of?C.By Proposition 2.1,

Therefore,e≤2s-2χ(F).By Lemma 3.2 and 2e≥Nβ+4s,one has

Remark 3.1Theorem 3.1 improves Theorem 2.2 in[2].

Corollary 3.1Let K be a prime almost alternating knot,which admits a standard almost alternating diagram D with TCN(D)≥6,and let F be an essential surface with f i nite boundary slope in the exterior of K and with β boundary components,each of which has b (b/=0)longitudinal components.If the region bounded by c in the reduced graph?C has at least 4 edges,where c is any component ofthen g(F)≥1.

Proof.By Theorem 3.1 and TCN(D)≥6,one has g(F)≥1.

[1]Meanasco W.Closed incompressible surface in alternating knot and link complement.Topology Appl.,1984,23(1):37–41.

[2]Menasco W,Thistlethwaite W.Surface with boundary in alternating knot exteriors.J.Reine Angew.Math.,1992,426:47–65.

[3]Adams C C,Brock J F,Bugbee J et al.Almost alternating links.Topology Appl.,1992,46(2): 151–165.

[4]Han Y F.Incompressible boundary incompressible surfaces of knot complements.Acta Sci. Natur.Univ.Jilin,1995,2:21–24.

[5]Chen X.Notes on Algebraic Topology.Beijing:Higer Education Press,1985.

tion:57M15,57M25

A

1674-5647(2013)02-0143-05

Received date:Jan.18,2011.

The NSF(11071106)of China,the Program(LR2011031)for Liaoning Excellent Talents in University.

E-mail address:hanyoufa@sina.com(Han Y F).