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On the structure of certain semi-groups of spherical knot classes

On the structure of certain semi-groups of spherical knot classes ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES By BARRY MAZUR w L Introduction. The problem of classification of k-sphere knots in r-spheres is the problem of classifying "knot pairs": S= ($1, $2), where $2 is an oriented combinatorial r-sphere, $1 a subcomplex of S 2 (isomorphic to a standard k-sphere), and the pair S is considered equivalent to S' (S.~ S') if there is a combinatorial orientation-preserving homeomorphism of S x onto S' 1 bringing S 2 onto S~. Thus it is the problem of classifying certain relative combinatorial structures. The set of all such, for fixed k and r, will be called Z~, and can be given, in a natural manner, the structure of a semi-group. There is a certain sub-semi-group of Z~ to be singled out -- the semi-group S~ of all pairs S ----- ($t, $2) where $1 is smoothly imbedded in S 2 (locally unknotted). In this paper I shall define a notion of equivalence (which I call .-equivalence) between knot pairs which is (seemingly) weaker than the equivalence defined above. Two knot pairs S and S' are .-equivalent if (again) there is an orientation-preserving homeomorphism ~0 : $2-+S~ bringing $1 onto S~. However q~ is required to be combinatorial (not on all of S~, as before, but) merely on S 2 = $2--(pl, ..., p,), where Pl, 9 -, P,,~ $2, where S 2 is considered as an open infinite complex. Thus .-equivalence neglects some of the combinatorial structure of the pair (S~, $2). The set of all ,-equivalence classes of knot pairs forms a semi-group again, called *Z~. 9 r * r Finally the subsemi-group of smoothly imbedded knots in Z k I call S k. The purpose of this paper is to prove a generalized knot theoretic restatement of lemma 3 in [x]. INVERSE T~EOR~.M: A knot S~ is invertible if and only if it is .-trivial. And in application, derive the following fact concerning the structure of the knot semi-groups: 9 * r There are no inverses m S k. 111 20 BARRY MAZUR w .7. Tet:.~inology. My general use of combinatorial topology terms is as in [~,]. It is clear what is meant by the "usual" or "standard" imbedding of a k-sphere or a k-cell in E ~. Similarly an unknotted sphere or disc in E r means one that may be thrown onto the usual by a combinatorial automorphism of E ~. DEFINITION I. Let M k be a subcomplex (a k-manifold) of E ~. Then M k is locally unknotted at a point m (m~M) if the following condition is met with: i) There is an r-simplex A* drawn about m so that ArnMcSt(m), and A'nM is then a k-cell BkcA r, and OBkcOA ~. 2) There is a combinatorial automorphism of A *, sending B k onto the "standard k-cell in A r''. M is plain locally unknotted if it is locally unknotted at all points. Semi-Groups: All semi-groups to be discussed will be countable, commutative, and possess zero elements. DEFINITION 2. A semi-group F is positive if." X + Y ----- o implies X = o (i.e.if F has no inverses). DEFINITION 3" A minimal base of a semi-group F is a collection J = (ll, ... ) of elements of F such that every element of F is a sum of elements in J, and there is no smaller J' cJ with the same property. DEFINITION 4" A prime element p in the semi-group F is an element for which p=x+y implies either x=o or y=o. Clearly, if a positive semi-group F possesses a minimal base, that minimal base has to be precisely the set of primes in F, and F has the property that every element is expressible as a finite sum of primes. DEFINITION 5" An element x~F is invertible if there is a y~F such that x +y= o. w 3" (*)'h~176176 DEFINITION 6. A (Pl, ..-,P,)-homeomorphism, h:Er--~E" will be an orientation preserving homeomorphism which is combinatorial except at the points pieE r. It is a homeomorphism such that h[ E r- (p~) is a combinatorial map- simplicial with respect to a possibly infinite subdivision of the open complexes involved. When there is no reason to call special attention to the points Pl .... ,P,,, I shall call such: a (.)- homeomorphism. 112 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 2I DEFINITION 7" Two subcomplexes K, K' cE' will be called ,-equivalent (K.~K') if there is a ,-homeomorphism h of E r onto itself bringing K onto K'. (If h is a (pi)- homeomorphism I shall also say K,~K'.) To keep from using too many subscripts, whenever a (,)-equivalence comes up in a subsequent proof, I shall act as if it were a (p)-equivalence for a single point p. This logical gap, used merely as a notation- saving device, can be trivally filled by the reader. I'll say a sphere knot is *-trivial if it is ,-equivalent to the standard sphere. w 4. Knot Addition. There is a standard additive structure that can be put on Z~,, the set of combina- torial k-sphere knots in E r (two k-sphere knots are equivalent if there is an orientation- preserving combinatorial automorphism of E r bringing the one knot onto the other). (For details see [z]). I shall outline the procedure of "adding two knots" So, S1. Separate S o and S 1 by a hyperplane H (possibly after translating one of them). Take a k-simplex A i from each Si, i = o, I. And lead a "tube" from A 0 to A 1 (by "thickening" a polygonal arc joining a point p0eA0 to Pi~Ai, which doesn't intersect the S i except at Ai). Then remove the Ar and replace them by the tube T = S k-1 � I, where one end, S k-1 � o is attached to OA 0 by a combinatorial homeomorphism, and the other S k-a X x is attached to 0 A 1 similarly. The resulting knot is called the sum: S o + S~, and its knot-equivalence class is unique. If one added the point at infinity to E r, to obtain S ~, the hyperplane H would become an unknotted S ~-I c S', separating the knot S O q- S 1 into its components S O and S~. In analytic fashion, then, we can say that a k-sphere knot S c S r is split by an S ~-1 c S ~ if: I) S'-i n S is an unknotted (k-- I)-sphere knot in S. 2) S *-I is unknotted in SL 3) S'-IAS is unknotted in S '-i. Let A o and A 1 be the two complementary components of S*-inS in S, and let B be an unknotted k-disc that S*-lnS bounds in S ~-l. Then S0=AoUB , SI=AluB are knotted spheres again, and clearly S.~ S o -k Sr Thus I'll say: S r-1 splits S into S o q-S 1 ; if E 0 and E i are the complementary regions of S r-1 in S *, I'll refer to S 1 as that <~ part of S ,, lying in E D and similarly for S o. Working in the semi-group Zk, one can be slightly cruder, and say: S ~-1 .-splits S if only ~) and 3) hold. Clearly by [x], every S r-1 is .-trivial in S ~. LEMMA I: If S *-I *-splits S, and So, S 1 are constructed in a manner analogous to the above, then S .~ S o 4- S i. w 5- The Semi-Groups of Spherical Knots. This operation of addition, discussed in the previous section, turns ~ into a commu- tative semi-group with zero. Our object is to study the algebraic structure of the 113 a2 BARRY MAZUR r r * r subsemi-group SkCZ k of locally unknotted k-sphere knots. Let Z k be the semi-group of classes of spherical knots under ,-equivalence. Let G~c Z~ be the maximal subgroup of Z~, that is: the subgroup of invertible knots. INVERSE THEOREM: There is an exact sequence r r * r o--> Gk--~ Sk--> Sk---> 0 r * r (where *S~ is the image of S k in Z~) or, equivalently, a knot in S~ is ,-trivial if and only if it is invertible. w 6. Proof of the Inverse Theorem. a) If S is invertible, then S,~o. The proof is quite as in [x]. Let S+S',-~o. (,) Then consider the knots: S| =S+S'+S+S'+...up| S;,=S'+S+S'+S+...up| (See figure I) and notice: (as was done in detail in [x]) S| cU )o S I ,-~ o (p,0) =s+s" H~ H 2 H3 H4 S S' S S' P~ Fig. i LEMMA 2: There is a (,)-homeomorphism f: E'-+E r such that f: S->S +S'. 114 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 2 3 PROOF: Let D be the k-cell on which the addition of S to S~0 takes place. Since S'_ N o, we may transform figure I to figure 2 by a (p| g which leaves (p| everything to the left of the hyperplane H z fixed, and sends S' to the" standard k-sphere" to the right of H 1. (See figure 2.) H 1 Sk =9 (S--~ Fig. 2 Then, in figure 2, clearly one can construct an automorphism f' which leaves S fixed and sends D onto g(S')--int D. Take f-~g-qf'g, and f has the properties required, and is a (.)-homeomorphism. Therefore, by the above lemma, S,~ S + S" = S| ,~o (,) (*) and finally: S,~o (,) which proves (a). and S~o, then S is invertible. b) If S ~ S~ (v) PROOV: First observe that if k = r-- I, invertibility of knots is generally true (by [I]), and so we needn't prove anything. LEMMA 3: If k<r--I, and SeS~,,S,~o for pr then S,~o. (p) PROOF: There is an r-cell A containing S but not p. Then fl A is combinatorial, and by a standard lemma: LEMMA 4: If g : A~A' is a combinatorial homeomorphism of an r-cell AcE r to an r-cell A'C E r, then g can be extended to a combinatorial automorphism of E' (see [2]). Thus, restrictfto A, and extendf[A to a combinatorial automorphism g ofE r. This g yields the equivalence S,~o. Therefore, assume S,.~o, and psS. (p) 115 BARRY MAZUR Fig. 3 Let B be a small r-cell about p, so that C = BnS is in St(p), and hence an unknotted k-cell, by the local unknottedness of S. 0BiaS = 0C and 0C is unknotted in 0B. Letfbe the (p)-homeomorphism taking S onto the standard S *. Fig. 4 Now let D be an unknotted disc, the image of a perturbation off(C) with the properties: i) O(f(e))-=OD; ii) int Dr B ; iii) f(p) r ; iv) the knot K=Do(Sk--f(C)) is still trivial. Thenf -1 takes K to a knot K' =f-l(K), split by 0B into the sum: K'-----S +S' where S is the knot lying in the exterior component of 0B, and S' in the interior. 116 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 2 5 But K~o, and K'..~K where f(p)r therefore by lemma 3, f(K),~K. t (p) So: S + S' ~f(K)~ K,-~ o, and S' is invertible. S k is a positive semi-group. COROLLARY: * r So we have that S k is precisely S~ ~ modulo units ~>. * r w 7" Infinite Sums in Z k. Let X o i = i, ..., be knots representing the classes )i e E~. Define Y~ X; to be the infinite one point compactified sum of the knots X o in that order (figure 5). X~ X 2 X 3 X 1 9 . . . . .p~ %/-- Fig. 5 As it stands, X = Y- X~ will not represent a knot in X~, because X is not combina- i=1 torially imbedded (at p| ~o DEFINITION 8. ~ Xi= X converges if there is a (p| H : X-+Y, * r where Y is combinatorially imbedded. In that case, the knot class ye E k is uniquely 0o determined by the Xi~ N~, and I shall say ~ Xi =Y. ~ i=1 * r Sk" If ~Xi is in Sk, I'll say that ~)~ converges in * ~ i=l /=1 oo * r THEOREM I. If ~ )4 converges in S~, then it does so finitely. That is, there is i=1 an N such that Xi~o, i>N. * r PROOF: Notice that by the inverse theorem, there are no inverses in S k. oo Let X= ~ X~., and H:X-+Y where Y is a subcomplex of E r and H a i=I (,)-homeomorphism. 117 ~6 BARRY MAZUR Fig. 6 Let B be a ball about p' such that BnY is a disc in St(p'), and by the local unknot- tedness of Y, OB splits Y into two knots, y = yr + yI2) where YI r B is trivial, and Y,-~ Y2. Fig. 7 Now transform the situation by H-< Let B'=H-I(B), and we have that OB' ,-splits X into: X ~ X (t) + X (~I and H yields the ,-equivalences: X (1) ~ Y(1) ~ o X (2) ~.o y(2) Find an i so large that A~cint B'. Then SA~ splits X (*) further: X (1) ~ X(3) + X (4) where X (3t is the part of X (1) lying in A~. But then, by figure 6, X (3) is nothing more than: 118 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 27 ao X/3/~ Y X s. Passing to equivalence classes in Sk, one has: X (3) + Z (4) = o Z (3) ----- .~.Zi * r (where x the ,-equivalence class of X). But repeated application of the fact that S k has no inverses yields Z~ = o for j >1 i, which proves the theorem. REFERENCES [1] B. MAZUR, On Imbeddings of Spheres, Acta Mathematica (to appear). [2] V. K. A. M. GtrGOENgmM, Piece wise Linear Isotopy, Journal of the London Math. Soc., vol. 46. Refu le r6 novembre I959. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

On the structure of certain semi-groups of spherical knot classes

Mazur, Barry
Publications mathématiques de l'IHÉS , Volume 3 (1) – Aug 4, 2007
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Springer Journals
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Copyright © 1959 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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10.1007/BF02684388
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Abstract

ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES By BARRY MAZUR w L Introduction. The problem of classification of k-sphere knots in r-spheres is the problem of classifying "knot pairs": S= ($1, $2), where $2 is an oriented combinatorial r-sphere, $1 a subcomplex of S 2 (isomorphic to a standard k-sphere), and the pair S is considered equivalent to S' (S.~ S') if there is a combinatorial orientation-preserving homeomorphism of S x onto S' 1 bringing S 2 onto S~. Thus it is the problem of classifying certain relative combinatorial structures. The set of all such, for fixed k and r, will be called Z~, and can be given, in a natural manner, the structure of a semi-group. There is a certain sub-semi-group of Z~ to be singled out -- the semi-group S~ of all pairs S ----- ($t, $2) where $1 is smoothly imbedded in S 2 (locally unknotted). In this paper I shall define a notion of equivalence (which I call .-equivalence) between knot pairs which is (seemingly) weaker than the equivalence defined above. Two knot pairs S and S' are .-equivalent if (again) there is an orientation-preserving homeomorphism ~0 : $2-+S~ bringing $1 onto S~. However q~ is required to be combinatorial (not on all of S~, as before, but) merely on S 2 = $2--(pl, ..., p,), where Pl, 9 -, P,,~ $2, where S 2 is considered as an open infinite complex. Thus .-equivalence neglects some of the combinatorial structure of the pair (S~, $2). The set of all ,-equivalence classes of knot pairs forms a semi-group again, called *Z~. 9 r * r Finally the subsemi-group of smoothly imbedded knots in Z k I call S k. The purpose of this paper is to prove a generalized knot theoretic restatement of lemma 3 in [x]. INVERSE T~EOR~.M: A knot S~ is invertible if and only if it is .-trivial. And in application, derive the following fact concerning the structure of the knot semi-groups: 9 * r There are no inverses m S k. 111 20 BARRY MAZUR w .7. Tet:.~inology. My general use of combinatorial topology terms is as in [~,]. It is clear what is meant by the "usual" or "standard" imbedding of a k-sphere or a k-cell in E ~. Similarly an unknotted sphere or disc in E r means one that may be thrown onto the usual by a combinatorial automorphism of E ~. DEFINITION I. Let M k be a subcomplex (a k-manifold) of E ~. Then M k is locally unknotted at a point m (m~M) if the following condition is met with: i) There is an r-simplex A* drawn about m so that ArnMcSt(m), and A'nM is then a k-cell BkcA r, and OBkcOA ~. 2) There is a combinatorial automorphism of A *, sending B k onto the "standard k-cell in A r''. M is plain locally unknotted if it is locally unknotted at all points. Semi-Groups: All semi-groups to be discussed will be countable, commutative, and possess zero elements. DEFINITION 2. A semi-group F is positive if." X + Y ----- o implies X = o (i.e.if F has no inverses). DEFINITION 3" A minimal base of a semi-group F is a collection J = (ll, ... ) of elements of F such that every element of F is a sum of elements in J, and there is no smaller J' cJ with the same property. DEFINITION 4" A prime element p in the semi-group F is an element for which p=x+y implies either x=o or y=o. Clearly, if a positive semi-group F possesses a minimal base, that minimal base has to be precisely the set of primes in F, and F has the property that every element is expressible as a finite sum of primes. DEFINITION 5" An element x~F is invertible if there is a y~F such that x +y= o. w 3" (*)'h~176176 DEFINITION 6. A (Pl, ..-,P,)-homeomorphism, h:Er--~E" will be an orientation preserving homeomorphism which is combinatorial except at the points pieE r. It is a homeomorphism such that h[ E r- (p~) is a combinatorial map- simplicial with respect to a possibly infinite subdivision of the open complexes involved. When there is no reason to call special attention to the points Pl .... ,P,,, I shall call such: a (.)- homeomorphism. 112 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 2I DEFINITION 7" Two subcomplexes K, K' cE' will be called ,-equivalent (K.~K') if there is a ,-homeomorphism h of E r onto itself bringing K onto K'. (If h is a (pi)- homeomorphism I shall also say K,~K'.) To keep from using too many subscripts, whenever a (,)-equivalence comes up in a subsequent proof, I shall act as if it were a (p)-equivalence for a single point p. This logical gap, used merely as a notation- saving device, can be trivally filled by the reader. I'll say a sphere knot is *-trivial if it is ,-equivalent to the standard sphere. w 4. Knot Addition. There is a standard additive structure that can be put on Z~,, the set of combina- torial k-sphere knots in E r (two k-sphere knots are equivalent if there is an orientation- preserving combinatorial automorphism of E r bringing the one knot onto the other). (For details see [z]). I shall outline the procedure of "adding two knots" So, S1. Separate S o and S 1 by a hyperplane H (possibly after translating one of them). Take a k-simplex A i from each Si, i = o, I. And lead a "tube" from A 0 to A 1 (by "thickening" a polygonal arc joining a point p0eA0 to Pi~Ai, which doesn't intersect the S i except at Ai). Then remove the Ar and replace them by the tube T = S k-1 � I, where one end, S k-1 � o is attached to OA 0 by a combinatorial homeomorphism, and the other S k-a X x is attached to 0 A 1 similarly. The resulting knot is called the sum: S o + S~, and its knot-equivalence class is unique. If one added the point at infinity to E r, to obtain S ~, the hyperplane H would become an unknotted S ~-I c S', separating the knot S O q- S 1 into its components S O and S~. In analytic fashion, then, we can say that a k-sphere knot S c S r is split by an S ~-1 c S ~ if: I) S'-i n S is an unknotted (k-- I)-sphere knot in S. 2) S *-I is unknotted in SL 3) S'-IAS is unknotted in S '-i. Let A o and A 1 be the two complementary components of S*-inS in S, and let B be an unknotted k-disc that S*-lnS bounds in S ~-l. Then S0=AoUB , SI=AluB are knotted spheres again, and clearly S.~ S o -k Sr Thus I'll say: S r-1 splits S into S o q-S 1 ; if E 0 and E i are the complementary regions of S r-1 in S *, I'll refer to S 1 as that <~ part of S ,, lying in E D and similarly for S o. Working in the semi-group Zk, one can be slightly cruder, and say: S ~-1 .-splits S if only ~) and 3) hold. Clearly by [x], every S r-1 is .-trivial in S ~. LEMMA I: If S *-I *-splits S, and So, S 1 are constructed in a manner analogous to the above, then S .~ S o 4- S i. w 5- The Semi-Groups of Spherical Knots. This operation of addition, discussed in the previous section, turns ~ into a commu- tative semi-group with zero. Our object is to study the algebraic structure of the 113 a2 BARRY MAZUR r r * r subsemi-group SkCZ k of locally unknotted k-sphere knots. Let Z k be the semi-group of classes of spherical knots under ,-equivalence. Let G~c Z~ be the maximal subgroup of Z~, that is: the subgroup of invertible knots. INVERSE THEOREM: There is an exact sequence r r * r o--> Gk--~ Sk--> Sk---> 0 r * r (where *S~ is the image of S k in Z~) or, equivalently, a knot in S~ is ,-trivial if and only if it is invertible. w 6. Proof of the Inverse Theorem. a) If S is invertible, then S,~o. The proof is quite as in [x]. Let S+S',-~o. (,) Then consider the knots: S| =S+S'+S+S'+...up| S;,=S'+S+S'+S+...up| (See figure I) and notice: (as was done in detail in [x]) S| cU )o S I ,-~ o (p,0) =s+s" H~ H 2 H3 H4 S S' S S' P~ Fig. i LEMMA 2: There is a (,)-homeomorphism f: E'-+E r such that f: S->S +S'. 114 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 2 3 PROOF: Let D be the k-cell on which the addition of S to S~0 takes place. Since S'_ N o, we may transform figure I to figure 2 by a (p| g which leaves (p| everything to the left of the hyperplane H z fixed, and sends S' to the" standard k-sphere" to the right of H 1. (See figure 2.) H 1 Sk =9 (S--~ Fig. 2 Then, in figure 2, clearly one can construct an automorphism f' which leaves S fixed and sends D onto g(S')--int D. Take f-~g-qf'g, and f has the properties required, and is a (.)-homeomorphism. Therefore, by the above lemma, S,~ S + S" = S| ,~o (,) (*) and finally: S,~o (,) which proves (a). and S~o, then S is invertible. b) If S ~ S~ (v) PROOV: First observe that if k = r-- I, invertibility of knots is generally true (by [I]), and so we needn't prove anything. LEMMA 3: If k<r--I, and SeS~,,S,~o for pr then S,~o. (p) PROOF: There is an r-cell A containing S but not p. Then fl A is combinatorial, and by a standard lemma: LEMMA 4: If g : A~A' is a combinatorial homeomorphism of an r-cell AcE r to an r-cell A'C E r, then g can be extended to a combinatorial automorphism of E' (see [2]). Thus, restrictfto A, and extendf[A to a combinatorial automorphism g ofE r. This g yields the equivalence S,~o. Therefore, assume S,.~o, and psS. (p) 115 BARRY MAZUR Fig. 3 Let B be a small r-cell about p, so that C = BnS is in St(p), and hence an unknotted k-cell, by the local unknottedness of S. 0BiaS = 0C and 0C is unknotted in 0B. Letfbe the (p)-homeomorphism taking S onto the standard S *. Fig. 4 Now let D be an unknotted disc, the image of a perturbation off(C) with the properties: i) O(f(e))-=OD; ii) int Dr B ; iii) f(p) r ; iv) the knot K=Do(Sk--f(C)) is still trivial. Thenf -1 takes K to a knot K' =f-l(K), split by 0B into the sum: K'-----S +S' where S is the knot lying in the exterior component of 0B, and S' in the interior. 116 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 2 5 But K~o, and K'..~K where f(p)r therefore by lemma 3, f(K),~K. t (p) So: S + S' ~f(K)~ K,-~ o, and S' is invertible. S k is a positive semi-group. COROLLARY: * r So we have that S k is precisely S~ ~ modulo units ~>. * r w 7" Infinite Sums in Z k. Let X o i = i, ..., be knots representing the classes )i e E~. Define Y~ X; to be the infinite one point compactified sum of the knots X o in that order (figure 5). X~ X 2 X 3 X 1 9 . . . . .p~ %/-- Fig. 5 As it stands, X = Y- X~ will not represent a knot in X~, because X is not combina- i=1 torially imbedded (at p| ~o DEFINITION 8. ~ Xi= X converges if there is a (p| H : X-+Y, * r where Y is combinatorially imbedded. In that case, the knot class ye E k is uniquely 0o determined by the Xi~ N~, and I shall say ~ Xi =Y. ~ i=1 * r Sk" If ~Xi is in Sk, I'll say that ~)~ converges in * ~ i=l /=1 oo * r THEOREM I. If ~ )4 converges in S~, then it does so finitely. That is, there is i=1 an N such that Xi~o, i>N. * r PROOF: Notice that by the inverse theorem, there are no inverses in S k. oo Let X= ~ X~., and H:X-+Y where Y is a subcomplex of E r and H a i=I (,)-homeomorphism. 117 ~6 BARRY MAZUR Fig. 6 Let B be a ball about p' such that BnY is a disc in St(p'), and by the local unknot- tedness of Y, OB splits Y into two knots, y = yr + yI2) where YI r B is trivial, and Y,-~ Y2. Fig. 7 Now transform the situation by H-< Let B'=H-I(B), and we have that OB' ,-splits X into: X ~ X (t) + X (~I and H yields the ,-equivalences: X (1) ~ Y(1) ~ o X (2) ~.o y(2) Find an i so large that A~cint B'. Then SA~ splits X (*) further: X (1) ~ X(3) + X (4) where X (3t is the part of X (1) lying in A~. But then, by figure 6, X (3) is nothing more than: 118 ON THE STRUCTURE OF CERTAIN SEMI-GROUPS OF SPHERICAL KNOT CLASSES 27 ao X/3/~ Y X s. Passing to equivalence classes in Sk, one has: X (3) + Z (4) = o Z (3) ----- .~.Zi * r (where x the ,-equivalence class of X). But repeated application of the fact that S k has no inverses yields Z~ = o for j >1 i, which proves the theorem. REFERENCES [1] B. MAZUR, On Imbeddings of Spheres, Acta Mathematica (to appear). [2] V. K. A. M. GtrGOENgmM, Piece wise Linear Isotopy, Journal of the London Math. Soc., vol. 46. Refu le r6 novembre I959.

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Published: Aug 4, 2007

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