Satellites and Concordance

Transcription

1 Satellites and Concordance Allison N. Miller Rice University December 8, / 30

2 Concordance of knots Definition Knots K 0, K 1 S 3 are concordant if 2 / 30

3 Concordance of knots Definition Knots K 0, K 1 S 3 are concordant if there is an annulus A: S 1 I S 3 I with A = K 0 {0} K 1 {1}. 3 / 30

4 Concordance of knots Definition Knots K 0, K 1 S 3 are concordant if there is an annulus A: S 1 I S 3 I with A = K 0 {0} K 1 {1}. Warning! Whether we require a smooth or just topological(ly flat) embedding of A matters! 3 / 30

5 A concordance from T 2,3 # T 2,3 to... 4 / 30

6 A concordance from T 2,3 # T 2,3 to... 5 / 30

7 A concordance from T 2,3 # T 2,3 to... 6 / 30

8 A concordance from T 2,3 # T 2,3 to... 7 / 30

9 A concordance from T 2,3 # T 2,3 to... 8 / 30

10 A concordance from T 2,3 # T 2,3 to the unknot! 9 / 30

11 The concordance set Definition C := {knots in S 3 }/, where = sm, top 10 / 30

12 The concordance set monoid Definition C := {knots in S 3 }/, where = sm, top 11 / 30

13 The concordance set monoid Definition C := {knots in S 3 }/, where = sm, top Theorem (Fox-Milnor) The map [K] + [J] := [K#J] is well-defined on C. 11 / 30

14 The concordance set monoid group! Definition C := {knots in S 3 }/, where = sm, top Theorem (Fox-Milnor) The map [K] + [J] := [K#J] is well-defined on C. Moreover, it induces the structure of an abelian group! Known: C contains a Z 2 Z -summand. 12 / 30

15 The concordance set monoid group! Definition C := {knots in S 3 }/, where = sm, top Theorem (Fox-Milnor) The map [K] + [J] := [K#J] is well-defined on C. Moreover, it induces the structure of an abelian group! Known: C contains a Z 2 Z -summand. Unknown: Q C? Z n C, n > 2? 12 / 30

16 The satellite construction Any P : S 1 D 2 S 1 defines a map on the set of knots in S 3 : 13 / 30

17 The satellite construction Any P : S 1 D 2 S 1 defines a map on the set of knots in S 3 : P : K P(K) 13 / 30

18 Satellites induce maps on concordance Proposition Let P be any pattern and K 0 and K 1 be concordant knots. Then P(K 0 ) and P(K 1 ) are concordant. 14 / 30

19 Satellites induce maps on concordance Proposition Let P be any pattern and K 0 and K 1 be concordant knots. Then P(K 0 ) and P(K 1 ) are concordant. Proof. Let A: S 1 I S 3 I be a concordance from K 0 to K / 30

20 Satellites induce maps on concordance Proposition Let P be any pattern and K 0 and K 1 be concordant knots. Then P(K 0 ) and P(K 1 ) are concordant. Proof. Let A: S 1 I S 3 I be a concordance from K 0 to K 1. Consider S 1 I P Id (D 2 S 1 ) I 14 / 30

21 Satellites induce maps on concordance Proposition Let P be any pattern and K 0 and K 1 be concordant knots. Then P(K 0 ) and P(K 1 ) are concordant. Proof. Let A: S 1 I S 3 I be a concordance from K 0 to K 1. Consider S 1 I P Id (D 2 S 1 ) I = D 2 (S 1 I ) 14 / 30

22 Satellites induce maps on concordance Proposition Let P be any pattern and K 0 and K 1 be concordant knots. Then P(K 0 ) and P(K 1 ) are concordant. Proof. Let A: S 1 I S 3 I be a concordance from K 0 to K 1. Consider S 1 I P Id (D 2 S 1 ) I = D 2 (S 1 I ) = ν(a) S 3 I, 14 / 30

23 Satellites induce maps on concordance Proposition Let P be any pattern and K 0 and K 1 be concordant knots. Then P(K 0 ) and P(K 1 ) are concordant. Proof. Let A: S 1 I S 3 I be a concordance from K 0 to K 1. Consider S 1 I P Id (D 2 S 1 ) I = D 2 (S 1 I ) = ν(a) S 3 I, and observe that this is a concordance from P(K 0 ) to P(K 1 )! 14 / 30

24 Satellites and concordance Motivating question: What can we say about P : C C? 15 / 30

25 Satellites and concordance Motivating question: What can we say about P : C C? 1 When does P induce a surjection? injection? bijection? 15 / 30

26 Satellites and concordance Motivating question: What can we say about P : C C? 1 When does P induce a surjection? injection? bijection? 2 When does P induce a group homomorphism? 15 / 30

27 Winding number Definition Given a pattern P, we have [P] = k[{pt} S 1 ] H 1 (D 2 S 1 ) for some k Z. We call k =: w(p) the winding number of P. 16 / 30

28 Winding number Definition Given a pattern P, we have [P] = k[{pt} S 1 ] H 1 (D 2 S 1 ) for some k Z. We call k =: w(p) the winding number of P. w(p) = 0 w(p) = 1 w(p) = 2 16 / 30

29 Winding number Definition Given a pattern P, we have [P] = k[{pt} S 1 ] H 1 (D 2 S 1 ) for some k Z. We call k =: w(p) the winding number of P. w(p) = 0 w(p) = 1 w(p) = 2 16 / 30

30 Satellite maps and surjectivity Proposition (Folklore) If P has w(p) ±1, then P does not induce a surjection. Proof. Easy : Uses classical invariants, e.g. Tristram-Levine signatures. 17 / 30

31 Satellite maps and surjectivity Proposition (Folklore) If P has w(p) ±1, then P does not induce a surjection. Proof. Easy : Uses classical invariants, e.g. Tristram-Levine signatures. Theorem (Levine, 2014) The Mazur pattern does not induce a surjection on C sm. Proof. Difficult: Uses (bordered) Heegaard Floer theory! 17 / 30

32 Satellite maps and injectivity Proposition (M., 2018) For each n N, there exist winding number 0 patterns P which induce nonzero maps on C but for which there are at least n distinct concordance classes K 1,..., K n such that P(K i ) is slice for all i. 18 / 30

33 Satellite maps and injectivity Proposition (M., 2018) For each n N, there exist winding number 0 patterns P which induce nonzero maps on C but for which there are at least n distinct concordance classes K 1,..., K n such that P(K i ) is slice for all i. w(p) = ±1 w(p) > 1 w(p) = 0 Surjective? Not always (sm). Never Never Sometimes. Injective? Sometimes.????? Not always. Always????? Ever? 18 / 30

34 Satellite maps and injectivity Proposition (M., 2018) For each n N, there exist winding number 0 patterns P which induce nonzero maps on C but for which there are at least n distinct concordance classes K 1,..., K n such that P(K i ) is slice for all i. w(p) = ±1 w(p) > 1 w(p) = 0 Surjective? Not always (sm). Never Never Sometimes. Injective? Sometimes.????? Not always. Always????? Ever? Bijective patterns? 18 / 30

35 Satellite maps and injectivity Proposition (M., 2018) For each n N, there exist winding number 0 patterns P which induce nonzero maps on C but for which there are at least n distinct concordance classes K 1,..., K n such that P(K i ) is slice for all i. w(p) = ±1 w(p) > 1 w(p) = 0 Surjective? Not always (sm). Never Never Sometimes. Injective? Sometimes.????? Not always. Always????? Ever? Bijective patterns? Yes! 18 / 30

36 Satellite maps and injectivity Proposition (M., 2018) For each n N, there exist winding number 0 patterns P which induce nonzero maps on C but for which there are at least n distinct concordance classes K 1,..., K n such that P(K i ) is slice for all i. w(p) = ±1 w(p) > 1 w(p) = 0 Surjective? Not always (sm). Never Never Sometimes. Injective? Sometimes.????? Not always. Always????? Ever? Bijective patterns? Yes! But boring: K K#J. 18 / 30

37 Satellite maps and bijectivity Theorem (M.-Piccirillo 2017) There exist patterns P which induce bijective maps on C sm and do not act by connected sum. 19 / 30

38 Satellite maps and bijectivity Theorem (M.-Piccirillo 2017) There exist patterns P which induce bijective maps on C sm and do not act by connected sum. Proof. Step 1: Show that any dualizable P has an inverse. [See also Gompf-Miyazaki 95]. 19 / 30

39 Satellite maps and bijectivity Theorem (M.-Piccirillo 2017) There exist patterns P which induce bijective maps on C sm and do not act by connected sum. Proof. Step 1: Show that any dualizable P has an inverse. [See also Gompf-Miyazaki 95]. Step 2: Compute some HF d-invariants of the dbcs of P(P 1 (U)) and P 1 (U)#P(U). 19 / 30

40 Satellite maps and bijectivity Theorem (M.-Piccirillo 2017) There exist patterns P which induce bijective maps on C sm and do not act by connected sum. Proof. Step 1: Show that any dualizable P has an inverse. [See also Gompf-Miyazaki 95]. Step 2: Compute some HF d-invariants of the dbcs of P(P 1 (U)) and P 1 (U)#P(U). Hard problem: Do any winding number 1 patterns not act by connected sum on C top? 19 / 30

41 Satellite maps and group structure on C Question: Can a pattern induce a homomorphism on C? 20 / 30

42 Satellite maps and group structure on C Question: Can a pattern induce a homomorphism on C? Answer: Yes! 20 / 30

43 Satellite maps and group structure on C Question: Can a pattern induce a homomorphism on C? Answer: Yes! 20 / 30

44 Satellite maps and group structure on C Question: Can a pattern induce a homomorphism on C? Answer: Yes! 20 / 30

45 Satellite maps and group structure on C Question: Can a pattern induce a homomorphism on C? Answer: Yes! K U K K K K rev 20 / 30

46 Satellite maps and group structure on C Question: Can a pattern induce a homomorphism on C? Answer: Yes! K U K K K K rev Conjecture (Hedden) If P induces a homomorphism on C, then the induced map must be K K, K U, or K K rev. 20 / 30

47 Initial observations Conjecture (Hedden) If P induces a homomorphism on C, then the induced map must be K K, K U, or K K rev. First obstruction: If P(U) U, then P does not induce a homomorphism. 21 / 30

48 Initial observations Conjecture (Hedden) If P induces a homomorphism on C, then the induced map must be K K, K U, or K K rev. First obstruction: If P(U) U, then P does not induce a homomorphism. Proposition If P(U) U, then P induces a homomorphism on C alg. 21 / 30

49 Initial observations Conjecture (Hedden) If P induces a homomorphism on C, then the induced map must be K K, K U, or K K rev. First obstruction: If P(U) U, then P does not induce a homomorphism. Proposition If P(U) U, then P induces a homomorphism on C alg. (i.e., the easily computed invariants K (t), σ K (ω) can t help!) 21 / 30

50 Some results Theorem (Gompf; Levine; Hedden) None of the Whitehead pattern, the Mazur pattern, or the (m,1) cable C m,1 for m > 1 induce homomorphisms on C sm. 22 / 30

51 Some results Theorem (Gompf; Levine; Hedden) None of the Whitehead pattern, the Mazur pattern, or the (m,1) cable C m,1 for m > 1 induce homomorphisms on C sm. Proof. Show that P( T 2,3 ) is not smoothly concordant to P(T 2,3 ) via e.g. the τ-invariant of Heegaard Floer homology. 22 / 30

52 Some results Theorem (Gompf; Levine; Hedden) None of the Whitehead pattern, the Mazur pattern, or the (m,1) cable C m,1 for m > 1 induce homomorphisms on C sm. Proof. Show that P( T 2,3 ) is not smoothly concordant to P(T 2,3 ) via e.g. the τ-invariant of Heegaard Floer homology. Problem Given a pattern P with P(U) slice, find an obstruction to P inducing a homomorphism on C top. 22 / 30

53 Winding number 0 case Proposition (M. Pinzón-Caicedo) For any knot J, let P J be the winding number 0 pattern shown. Then P J (U) U. Also, if σ J (e 2πi/3 ) 0, then P J does not induce a homomorphism on C top. P J 23 / 30

54 Winding number 0 case Proposition (M. Pinzón-Caicedo) For any knot J, let P J be the winding number 0 pattern shown. Then P J (U) U. Also, if σ J (e 2πi/3 ) 0, then P J does not induce a homomorphism on C top. Proof. 1 P J (U) U: blue curve. The knot P J (K) with a genus 1 Seifert surface. 24 / 30

55 Winding number 0 case Proposition (M. Pinzón-Caicedo) For any knot J, let P J be the winding number 0 pattern shown. Then P J (U) U. Also, if σ J (e 2πi/3 ) 0, then P J does not induce a homomorphism on C top. Proof. 1 P J (U) U: blue curve. 2 P J (J) U: red curve. The knot P J (K) with a genus 1 Seifert surface. 24 / 30

56 Winding number 0 case Proposition (M. Pinzón-Caicedo) For any knot J, let P J be the winding number 0 pattern shown. Then P J (U) U. Also, if σ J (e 2πi/3 ) 0, then P J does not induce a homomorphism on C top. The knot P J (K) with a genus 1 Seifert surface. Proof. 1 P J (U) U: blue curve. 2 P J (J) U: red curve. 3 P J (# n J) U for n >> 0: Casson-Gordon signatures. 24 / 30

57 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) 25 / 30

58 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) 25 / 30

59 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) = W W 25 / 30

60 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) = W W Theorem (Casson-Gordon) The quantity σ(k, χ) := σ(w ) σ(w ) is an invariant of (K, χ). 25 / 30

61 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) = W W Theorem (Casson-Gordon) The quantity σ(k, χ) := σ(w ) σ(w ) is an invariant of (K, χ). Moreover, if K is slice then for many χ we have σ(k, χ) = / 30

62 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) = W W Theorem (Casson-Gordon) The quantity σ(k, χ) := σ(w ) σ(w ) is an invariant of (K, χ). Moreover, if K is slice then for many χ we have σ(k, χ) = 0. (More precisely, there is a subgroup M H 1 (Σ(K)) such that 1 M 2 = H 1 (Σ(K)). 25 / 30

63 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) = W W Theorem (Casson-Gordon) The quantity σ(k, χ) := σ(w ) σ(w ) is an invariant of (K, χ). Moreover, if K is slice then for many χ we have σ(k, χ) = 0. (More precisely, there is a subgroup M H 1 (Σ(K)) such that 1 M 2 = H 1 (Σ(K)). 2 λ: H 1 (Σ(K)) H 1 (Σ(K)) Q/Z vanishes on M M. 25 / 30

64 Casson-Gordon signatures ( K S 3 χ: H 1 (Σ(K)) Z m ) S 3 0 (K) S 3 0 (K) = W W Theorem (Casson-Gordon) The quantity σ(k, χ) := σ(w ) σ(w ) is an invariant of (K, χ). Moreover, if K is slice then for many χ we have σ(k, χ) = 0. (More precisely, there is a subgroup M H 1 (Σ(K)) such that 1 M 2 = H 1 (Σ(K)). 2 λ: H 1 (Σ(K)) H 1 (Σ(K)) Q/Z vanishes on M M. 3 If χ M = 0, then σ(k, χ) = 0.) 25 / 30

65 Proof of Step / 30

66 Proof of Step 3. We can compute that H 1 (Σ(P J (K))) = H 1 (Σ(P U (U))) = Z 3 a Z 3 b, 26 / 30

67 Proof of Step 3. We can compute that H 1 (Σ(P J (K))) = H 1 (Σ(P U (U))) = Z 3 a Z 3 b, and for any χ we have σ(p J (K), χ) = σ(p U (U), χ) + 2σ J (e 2πi 3 χ(a) ) + 2σ K (e 2πi 3 χ(b) ), 26 / 30

68 Proof of Step 3. We can compute that H 1 (Σ(P J (K))) = H 1 (Σ(P U (U))) = Z 3 a Z 3 b, and for any χ we have σ(p J (K), χ) = σ(p U (U), χ) + 2σ J (e 2πi 3 χ(a) ) + 2σ K (e 2πi 3 χ(b) ), so σ(p J (# n J), χ) = σ(p U (U), χ) 2σ J (e 2πi 3 χ(a) ) + 2nσ J (e 2πi 3 χ(b) ). 26 / 30

69 Proof of Step 3. We can compute that H 1 (Σ(P J (K))) = H 1 (Σ(P U (U))) = Z 3 a Z 3 b, and for any χ we have σ(p J (K), χ) = σ(p U (U), χ) + 2σ J (e 2πi 3 χ(a) ) + 2σ K (e 2πi 3 χ(b) ), so σ(p J (# n J), χ) = σ(p U (U), χ) 2σ J (e 2πi 3 χ(a) ) + 2nσ J (e 2πi 3 χ(b) ). So we can choose n >> 0 so that σ(p J (# n J), χ) = 0 only if χ(b) = 0. But such characters do not vanish on a metabolizer for the torsion linking form. 26 / 30

70 Nonzero winding number case Theorem (M. Pinzón-Caicedo) For each n ±1, there exist a pattern P n of winding number n such that P n (U) U and P n does not induce a homomorphism on C top. 27 / 30

71 Nonzero winding number case Theorem (M. Pinzón-Caicedo) For each n ±1, there exist a pattern P n of winding number n such that P n (U) U and P n does not induce a homomorphism on C top. 27 / 30

72 Nonzero winding number case Theorem (M. Pinzón-Caicedo) For each n ±1, there exist a pattern P n of winding number n such that P n (U) U and P n does not induce a homomorphism on C top. Proof. 1 For p n, observe that H 1 (Σ p (P n (U))) is generated by the lifts of η to Σ p (P n (U)). 27 / 30

73 Nonzero winding number case Theorem (M. Pinzón-Caicedo) For each n ±1, there exist a pattern P n of winding number n such that P n (U) U and P n does not induce a homomorphism on C top. Proof. 1 For p n, observe that H 1 (Σ p (P n (U))) is generated by the lifts of η to Σ p (P n (U)). p 2 σ(p n (K), χ) = σ(p n (U), χ) + σ K (e 2πi mp χ( η i ) ). i=1 27 / 30

74 Nonzero winding number case Theorem (M. Pinzón-Caicedo) For each n ±1, there exist a pattern P n of winding number n such that P n (U) U and P n does not induce a homomorphism on C top. Proof. 1 For p n, observe that H 1 (Σ p (P n (U))) is generated by the lifts of η to Σ p (P n (U)). p 2 σ(p n (K), χ) = σ(p n (U), χ) + σ K (e 2πi mp χ( η i ) ). i=1 3 Analyse the linking form and show that P(K#K) P(K)#P(K) for some K. 27 / 30

75 The concordance set metric space d([k], [J]) := min{g(σ) : Σ S 3 I with Σ = K {0} J {1}}. 28 / 30

76 The concordance set metric space d([k], [J]) := min{g(σ) : Σ S 3 I with Σ = K {0} J {1}}. Question When do P and Q induce roughly the same action on (C, d)? i.e. When does there exist C = C(P, Q) such that d(p(k), Q(K)) C for all K C. When such a C exists, we say P and Q are bounded distance. 28 / 30

77 Winding number and metric structure Proposition (Cochran-Harvey, 2014) If w(p) = w(q) then P and Q are bounded distance. 29 / 30

78 Winding number and metric structure Proposition (Cochran-Harvey, 2014) If w(p) = w(q) then P and Q are bounded distance. Proof idea: When w(p) = w(q), the curves P and Q are homologous in (S 1 D 2 ) I and so cobound some surface F. Take C = g(f ). 29 / 30

79 Winding number and metric structure Proposition (Cochran-Harvey, 2014) If w(p) = w(q) then P and Q are bounded distance. Proof idea: When w(p) = w(q), the curves P and Q are homologous in (S 1 D 2 ) I and so cobound some surface F. Take C = g(f ). Proposition (Cochran-Harvey, 2014) If w(p) w(q), then P and Q are not bounded distance. 29 / 30

80 Winding number and metric structure Proposition (Cochran-Harvey, 2014) If w(p) = w(q) then P and Q are bounded distance. Proof idea: When w(p) = w(q), the curves P and Q are homologous in (S 1 D 2 ) I and so cobound some surface F. Take C = g(f ). Proposition (Cochran-Harvey, 2014) If w(p) w(q), then P and Q are not bounded distance. Proof idea: Show that d(p(# n T 2,3 ), Q(# n T 2,3 )) via Tristram-Levine signatures. 29 / 30

81 Remaining case Question If P has winding number m > 0 and Q has winding number m, are P and Q bounded distance? 30 / 30

82 Remaining case Question If P has winding number m > 0 and Q has winding number m, are P and Q bounded distance? Enough: Consider P = C m,1 and Q = C rev m,1. 30 / 30

83 Remaining case Question If P has winding number m > 0 and Q has winding number m, are P and Q bounded distance? Enough: Consider P = C m,1 and Q = C rev m,1. Theorem (M. 2018) Let m > 0. Then for any M 0 there exists a knot K such that d(c m,1 (K), C rev m,1(k)) = g 4 (C m,1 (K)# C rev m,1(k)) > M. 30 / 30

84 Remaining case Question If P has winding number m > 0 and Q has winding number m, are P and Q bounded distance? Enough: Consider P = C m,1 and Q = C rev m,1. Theorem (M. 2018) Let m > 0. Then for any M 0 there exists a knot K such that d(c m,1 (K), C rev m,1(k)) = g 4 (C m,1 (K)# C rev m,1(k)) > M. Proof. Idea: Casson-Gordon signatures again! 30 / 30