Generalized crossing changes in satellite knots Cheryl L. Balm Michigan State University Saturday, December 8, 2012
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
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 = =
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 =
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.
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
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).
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?
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?
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)
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.
Generalized crossing changes Notation Notation - η(k) is a regular neighborhood of K in S 3.
Generalized crossing changes Notation Notation - η(k) is a regular neighborhood of K in S 3. - M K L := S 3 η(k L)
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.
Generalized crossing changes Satellite knots Satellite knots f Pattern knot Companion torus
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.
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).
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).
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.
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).
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
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.
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.
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.
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.
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.
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 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.
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.
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.
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
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 ).
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.)
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.
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.
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.
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.
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.)
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
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
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
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.
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 )
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
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.