Generalized crossing changes in satellite knots

Transcription

1 Generalized crossing changes in satellite knots Cheryl L. Balm Michigan State University Saturday, December 8, 2012

2 Generalized crossing changes Introduction Crossing disks and crossing circles Let K be an oriented knot in S 3. A crossing disc for K is an embedded disc D S 3 such that K intersects int(d) twice with zero algebraic intersection number. L = D is a crossing circle. L = D

3 Generalized crossing changes Introduction Crossing changes A crossing change in a knot diagram is equivalent to performing (±1)-Dehn surgery on the corresponding crossing circle. K K L (1) L = =

4 Generalized crossing changes Introduction Generalized crossing changes Since a crossing change is equivalent to adding one full twist at L, we can define an order-q generalized crossing change at L to be ( 1/q)-Dehn surgery on L, which is equivalent to adding q full twists. K K L (2) L =

5 Generalized crossing changes Introduction Nugatory and cosmetic crossing changes Let L be a crossing circle for K, and let K L (q) be the oriented knot obtained from K via an order-q crossing change at L.

6 Generalized crossing changes Introduction Nugatory and cosmetic crossing changes Let L be a crossing circle for K, and let K L (q) be the oriented knot obtained from K via an order-q crossing change at L. L is nugatory if L bounds a disk in S 3 η(k). L K 1 K 2

7 Generalized crossing changes Introduction Nugatory and cosmetic crossing changes Let L be a crossing circle for K, and let K L (q) be the oriented knot obtained from K via an order-q crossing change at L. L is nugatory if L bounds a disk in S 3 η(k). L K 1 K 2 L is cosmetic if L is not nugatory and K L (q) is isotopic to K i.e. there exists an orientation-preserving diffeomorphism f : S 3 S 3 with f (K) = K L (q).

8 Generalized crossing changes Introduction Open questions Nugatory crossing conjecture (Problem 1.58 of Kirby s list): Does there exist a knot K which admits a cosmetic (traditional) crossing change? Conversely, if a crossing change on a knot K yields a knot isotopic to K, must the crossing be nugatory?

9 Generalized crossing changes Introduction Open questions Nugatory crossing conjecture (Problem 1.58 of Kirby s list): Does there exist a knot K which admits a cosmetic (traditional) crossing change? Conversely, if a crossing change on a knot K yields a knot isotopic to K, must the crossing be nugatory? More generally: Does there exist a knot K which admits a cosmetic generalized crossing change of any order?

10 Generalized crossing changes Introduction Known results It has been shown that there are no cosmetic generalized crossing changes of any order for: Unknot (Gabai, Scharleman and Thompson, 1989) 2-bridge knots (Torisu, 1999) Fibered knots (Kalfagianni, 2011) Non-slice genus-one knots (B., Friedl, Kalfagianni and Powell, 2012)

11 Generalized crossing changes Introduction Known results It has been shown that there are no cosmetic generalized crossing changes of any order for: Unknot (Gabai, Scharleman and Thompson, 1989) 2-bridge knots (Torisu, 1999) Fibered knots (Kalfagianni, 2011) Non-slice genus-one knots (B., Friedl, Kalfagianni and Powell, 2012) Our main goal Study potential cosmetic crossing changes in satellite knots.

12 Generalized crossing changes Notation Notation - η(k) is a regular neighborhood of K in S 3.

13 Generalized crossing changes Notation Notation - η(k) is a regular neighborhood of K in S 3. - M K L := S 3 η(k L)

14 Generalized crossing changes Notation Notation - η(k) is a regular neighborhood of K in S 3. - M K L := S 3 η(k L) - M(q) is the 3-manifold obtained from M K L via a ( 1/q) Dehn filling of η(l). So M(q) = M KL (q) and M(0) = M K.

15 Generalized crossing changes Satellite knots Satellite knots f Pattern knot Companion torus

16 Generalized crossing changes Satellite knots Companion tori Fix K and a crossing circle L, and suppose that K L (q) is a satellite knot for some q.

17 Generalized crossing changes Satellite knots Companion tori Fix K and a crossing circle L, and suppose that K L (q) is a satellite knot for some q. Then K L (q) has a companion torus T which is essential in M(q).

18 Generalized crossing changes Satellite knots Companion tori Fix K and a crossing circle L, and suppose that K L (q) is a satellite knot for some q. Then K L (q) has a companion torus T which is essential in M(q). T is Type 1 if T can be isotoped into M K L M(q).

19 Generalized crossing changes Satellite knots Companion tori Fix K and a crossing circle L, and suppose that K L (q) is a satellite knot for some q. Then K L (q) has a companion torus T which is essential in M(q). T is Type 1 if T can be isotoped into M K L M(q). Otherwise T is Type 2.

20 Generalized crossing changes Satellite knots Companion tori Fix K and a crossing circle L, and suppose that K L (q) is a satellite knot for some q. Then K L (q) has a companion torus T which is essential in M(q). T is Type 1 if T can be isotoped into M K L M(q). Otherwise T is Type 2. If T is Type 2, then T corresponds to a punctured torus (P, P) (M K L, η(l)) with boundary-slope ( 1/q).

21 Statements Lemma Let K be a prime satellite knot with a cosmetic crossing circle L of order q. Then at least one of the following must be true: M(q) contains no Type 2 tori q 5

22 Statements Lemma Let K be a prime satellite knot with a cosmetic crossing circle L of order q. Then at least one of the following must be true: M(q) contains no Type 2 tori q 5 Theorem (Gordon, 1998) Let L be a knot or link in S 3 and let Σ be a boundary component of M L. Suppose (P 1, P 1 ) and (P 2, P 2 ) are punctured tori in (M L, Σ) such that the boundary slope of P i on Σ is s i for i = 1, 2. Then (s 1, s 2 ) 5, where (s 1, s 2 ) is the minimal geometric intersection number of the slopes.

23 Statements Lemma Let K be a prime satellite knot with a cosmetic crossing circle L of order q. Then at least one of the following must be true: M(q) contains no Type 2 tori q 5 Main Theorem Suppose K is a satellite knot which admits a cosmetic generalized crossing change of order q with q 6. Then K admits a pattern knot K which also has an order-q cosmetic generalized crossing change.

24 Applications Applications Corollary 1 Suppose K is a fibered knot. Then no prime satellite knot with pattern K admits an order-q cosmetic generalized crossing change with q 6.

25 Applications Applications Corollary 1 Suppose K is a fibered knot. Then no prime satellite knot with pattern K admits an order-q cosmetic generalized crossing change with q 6. Corollary 2 If there exists a knot admitting a cosmetic generalized crossing change of order q with q 6, then there must be such a knot which is hyperbolic. Thus we have reduced the question of cosmetic generalized crossing changes to the cases where either the knot is hyperbolic or the crossing change has order q with q 5.

26 Applications Applications Corollary 3 No Whitehead double of any hyperbolic knot admits a cosmetic generalized crossing change of any order. While this is not a direct corollary, it can be proven using very similar techniques.

27 Proof of Main Theorem Proof of Main Theorem Statement Suppose K is a satellite knot which admits a cosmetic generalized crossing change of order q with q 6. Then K admits a pattern knot K which also has an order-q cosmetic generalized crossing change.

28 Proof of Main Theorem Proof of Main Theorem Statement Suppose K is a satellite knot which admits a cosmetic generalized crossing change of order q with q 6. Then K admits a pattern knot K which also has an order-q cosmetic generalized crossing change. Proof of Theorem Let K be a satellite knot with cosmetic crossing circle L of order q.

29 Proof of Main Theorem Proof of Main Theorem Statement Suppose K is a satellite knot which admits a cosmetic generalized crossing change of order q with q 6. Then K admits a pattern knot K which also has an order-q cosmetic generalized crossing change. Proof of Theorem Let K be a satellite knot with cosmetic crossing circle L of order q. If K is a composite knot K 1 #K 2, then Torisu showed that that the cosmetic crossing occurs in one of the summands, say K 1. Let T be the follow-swallow companion torus for K with core K 2 and K = K 1.

30 Proof of Main Theorem Proof of Theorem, continued So we may assume K is prime. Since q 6, Lemma M(0) contains a Type 1 torus T and, in fact, T is essential in M K L.

31 Proof of Main Theorem Proof of Theorem, continued So we may assume K is prime. Since q 6, Lemma M(0) contains a Type 1 torus T and, in fact, T is essential in M K L. Let D be the crossing disk bounded by L and let V be the solid torus bounded by T. If D cannot be isotoped into int(v ), then T D contains a component C which is homotopically non-trivial and not boundary parallel in D. This means the winding number of K in V is one and either K is the core of V or T is the follow-swallow torus. L K D C

32 Proof of Main Theorem Proof of Theorem, continued So we may assume D V and, hence, T is a companion torus for the link K L. Let K L be the pattern link for K L with f : (K, L, V ) (K, L, V ).

33 Proof of Main Theorem Proof of Theorem, continued So we may assume D V and, hence, T is a companion torus for the link K L. Let K L be the pattern link for K L with f : (K, L, V ) (K, L, V ). Case 1: Assume T is incompressible in V η(k L (q)). (Case 2 has similar proof.)

34 Proof of Main Theorem Proof of Theorem, continued Theorem (Motegi, 1993) Let K be a knot embedded in S 3 and let V 1 and V 2 be knotted solid tori in S 3 such that the embedding of K is essential in V i for i = 1, 2. Then there is an ambient isotopy φ : S 3 S 3 leaving K fixed such that one of the following holds. V 1 φ( V 2 ) =. There exist meridian disks D and D for both V 1 and V 2 such that some component of V 1 cut along (D D ) is a knotted 3-ball in some component of V 2 cut along (D D ). Motegi (plus a little work) there is an isotopy φ : S 3 S 3 such that φ(k L (q)) = K and φ(v ) = V.

35 Proof of Main Theorem Proof of Theorem, continued Hence h := (f 1 φf ) : V V is a homeomorphism with h(k L (q)) = K. So K L (q) = K and either L is an order-q cosmetic crossing for K, as desired, or L is nugatory.

36 Proof of Main Theorem Proof of Theorem, continued Hence h := (f 1 φf ) : V V is a homeomorphism with h(k L (q)) = K. So K L (q) = K and either L is an order-q cosmetic crossing for K, as desired, or L is nugatory. Suppose the L is nugatory. Then L bounds a crossing disk D and L bounds another disk D M K. We may isotope D and D so that D D = S 2 and A := D (D V ) is an annulus and A consists of standard longitudes of V.

37 Proof of Main Theorem Proof of Theorem, continued K V 1 L A V V 2 After some work, we may assume that that h(a) = A. So A gives the same (trivial) decompositions of K and K L (q) and cuts V into two solid tori V 1 and V 2.

38 Proof of Main Theorem Proof of Theorem, continued Subcase A: h maps V 1 V 1 and V 2 V 2. (Subcase B has similar proof.)

39 Proof of Main Theorem Proof of Theorem, continued Subcase A: h maps V 1 V 1 and V 2 V 2. (Subcase B has similar proof.) Up to isotopy we have: 1. K L (q) V 1 = K V 1 2. K L (q) V 2 is obtained from K V 2 via q full twists at L

40 Proof of Main Theorem Proof of Theorem, continued Let X be the 3-manifold obtained from V 2 η(v 2 K ) by attaching to A V 2 a thickened neighborhood of η(k ) V 1. K η(k ) V 1 V 1 L A = L V 2 V 2

41 Proof of Main Theorem Proof of Theorem, continued So h X is a homeomorphism given by q Dehn twists at L X. K η(k ) V 1 V 1 L A = L V 2 V 2

42 Proof of Main Theorem Proof of Theorem, concluded McCullough, 2006 Let N be a compact, orientable 3-manifold that admits a homeomorphism which restricts to Dehn twists on the boundary of N along a simple closed curve in C N. Then C bounds a disk in N.

43 Proof of Main Theorem Proof of Theorem, concluded McCullough, 2006 Let N be a compact, orientable 3-manifold that admits a homeomorphism which restricts to Dehn twists on the boundary of N along a simple closed curve in C N. Then C bounds a disk in N. So L bounds a disk D X V η(k )

44 Proof of Main Theorem Proof of Theorem, concluded McCullough, 2006 Let N be a compact, orientable 3-manifold that admits a homeomorphism which restricts to Dehn twists on the boundary of N along a simple closed curve in C N. Then C bounds a disk in N. So L bounds a disk D X V η(k ) L bounds a disk D V η(k) M K

45 Proof of Main Theorem Proof of Theorem, concluded McCullough, 2006 Let N be a compact, orientable 3-manifold that admits a homeomorphism which restricts to Dehn twists on the boundary of N along a simple closed curve in C N. Then C bounds a disk in N. So L bounds a disk D X V η(k ) L bounds a disk D V η(k) M K L is a nugatory crossing circle So L cannot be nugatory and is, indeed, and order-q cosmetic crossing circle for K.