Cosmetic generalized crossing changes in knots

Transcription

1 Cosmetic generalized crossing changes in knots Cheryl L. Balm Michigan State University Friday, January 11, 2013 Slides and preprint available on my website

2 Background Knot diagrams Knot diagrams A knot is often represented as a knot diagram in the plane. A crossing change in a knot diagram is when the overstrand and the understrand of one crossing in the diagram are switched. We will often show only the portion of the knot diagram where the crossing change is being made.

3 Background Generalized crossing changes Crossing disks and crossing circles We want to be able to talk about crossing changes in a knot K without restricting ourselves to any particular diagram of K.

4 Background Generalized crossing changes Crossing disks and crossing circles We want to be able to talk about crossing changes in a knot K without restricting ourselves to any particular diagram of K. 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

5 Background Generalized crossing changes 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 =

6 Background Generalized crossing changes 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

7 Background Generalized crossing changes 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.

8 Background Generalized crossing changes 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

9 Background Generalized crossing changes 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).

10 Background Main question 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?

11 Background Main question 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?

12 Background Main question 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) The question has also been reduced to the case of prime knots. (Torisu)

13 Background Main question 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) The question has also been reduced to the case of prime knots. (Torisu) Our main goals Study potential cosmetic crossing changes in genus-one knots and satellite knots.

14 Satellite knots Definition Satellite knots f : (K, V ) (K, V ) K K V Pattern knot Companion torus V

15 Satellite knots Results: Part one Lemma 1 (B.) 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: q 5 Any essential torus in S 3 η(k) (e.g. any companion torus for K) can be isotoped to lie in S 3 η(k L)

16 Satellite knots Results: Part one Lemma 1 (B.) 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: q 5 Any essential torus in S 3 η(k) (e.g. any companion torus for K) can be isotoped to lie in S 3 η(k L) Theorem 1 (B.) 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.

17 Satellite knots Corollaries Applications Corollary 1 (B.) 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.

18 Satellite knots Corollaries Applications Corollary 1 (B.) 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 1 (B.) 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.

19 Genus-one knots Seifert surfaces Seifert surfaces A Seifert surface for a knot K is an orientable surface S such that S = K. The genus of a knot K is the minimal genus over all Seifert surfaces for K. L K

20 Genus-one knots Seifert surfaces Seifert surfaces Let S be a minimum-genus Seifert surface for K in S 3 η(l). We may choose S such that S D is a single arc α S, where D is a crossing disk for a non-nugatory crossing C of K. K L α D S

21 Genus-one knots Seifert surfaces Seifert surfaces Suppose L is an order-q cosmetic crossing circle for K. So K = K L (q). Let S L (q) be the Seifert surface for K L (q) obtained via ( 1/q)-surgery at L. K L α D S

22 Genus-one knots Results: Part two Lemma 2 If K = K L (q), then S and S L (q) are minimum-genus Seifert surfaces for K and K L (q), respectively, in S 3.

23 Genus-one knots Results: Part two Lemma 2 If K = K L (q), then S and S L (q) are minimum-genus Seifert surfaces for K and K L (q), respectively, in S 3. Theorem 2 (B., Friedl, Kalfagianni, Powell) If K is a genus-one knot which admits a cosmetic generalized crossing change of any order, then K is algebraically slice. Hence the Alexander polynomial k (t). = f (t)f (t 1 ) for some f (t) Z[t] and the knot determinant det(k) = k ( 1) = n 2.

24 Genus-one knots Application: Pretzel knots Pretzel knots Let K = P(p, q, r) be a pretzel knot with p, q and r odd. p q r P(3, 3, 3)

25 Genus-one knots Application: Pretzel knots Pretzel knots g(p(p, q, r)) 1 det(p(p, q, r)) = pq + qr + pr P(p, q, r) is algebraically slice if and only if pq + qr + pr = m 2, for some odd m Z

26 Genus-one knots Application: Pretzel knots Pretzel knots g(p(p, q, r)) 1 det(p(p, q, r)) = pq + qr + pr P(p, q, r) is algebraically slice if and only if pq + qr + pr = m 2, for some odd m Z Corollary 3 A knot P(p, q, r) with p, q and r odd does not admit a cosmetic generalized crossing change of any order if any of the following are true: (a) pq + qr + pr m 2, for every odd m Z (b) q + r = 0 and gcd(p, q) 1 (c) p + q = 0 and gcd(q, r) 1

27 Genus-one knots Application: Pretzel knots Pretzel knots g(p(p, q, r)) 1 det(p(p, q, r)) = pq + qr + pr P(p, q, r) is algebraically slice if and only if pq + qr + pr = m 2, for some odd m Z Corollary 3 A knot P(p, q, r) with p, q and r odd does not admit a cosmetic generalized crossing change of any order if any of the following are true: (a) pq + qr + pr m 2, for every odd m Z (b) q + r = 0 and gcd(p, q) 1 (c) p + q = 0 and gcd(q, r) 1 Thank you!