ON SIGNATURES OF KNOTS

Transcription

1 1 ON SIGNATURES OF KNOTS Andrew Ranicki (Edinburgh) aar For Cherry Kearton Durham, 20 June 2010

2 MR: Publications results for "Author/Related=(kearton, C*)" 2 Matches: 48 Publications results for "Author/Related=(kearton, C*)" MR (2009f:57033) Kearton, Cherry; Kurlin, Vitaliy All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron. Algebr. Geom. Topol. 8 (2008), no. 3, (Reviewer: J. P. E. Hodgson) 57Q37 (57Q35 57Q45) MR (2009k:57039) Kearton, C.; Wilson, S. M. J. New invariants of simple knots. J. Knot Theory Ramifications 17 (2008), no. 3, Q45 (57M25 57M27) MR (2005e:57022) Kearton, C. $S$-equivalence of knots. J. Knot Theory Ramifications 13 (2004), no. 6, (Reviewer: Swatee Naik) 57M25 MR (2004j:57017) Kearton, C.; Wilson, S. M. J. Sharp bounds on some classical knot invariants. J. Knot Theory Ramifications 12 (2003), no. 6, (Reviewer: Simon A. King) 57M27 (11E39 57M25) MR (2004e:57029) Kearton, C.; Wilson, S. M. J. Simple non-finite knots are not prime in higher dimensions. J. Knot Theory Ramifications 12 (2003), no. 2, Q45 MR (2003g:57008) Kearton, C.; Wilson, S. M. J. Knot modules and the Nakanishi index. Proc. Amer. Math. Soc. 131 (2003), no. 2, (electronic). (Reviewer: Jonathan A. Hillman) 57M25 MR (2002a:57033) Kearton, C. Quadratic forms in knot theory. Quadratic forms and their applications (Dublin, 1999), , Contemp. Math., 272, Amer. Math. Soc., Providence, RI, (Reviewer: J. P. Levine) 57Q45 (11E12 57M25) MR (2000g:57040) Kearton, C.; Wilson, S. M. J. Spinning and branched cyclic covers of knots. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1986, (Reviewer: L. Neuwirth) 57Q45 (57M12) MR (99k:57053) Hillman, J. A.; Kearton, C. Simple $4$-knots. J. Knot Theory Ramifications 7 (1998), no. 7, (Reviewer: Washington Mio) 57Q45 MR (99e:57037) Kearton, C.; Saeki, O.; Wilson, S. M. J. Finiteness results for knots and singularities. Kobe J. Math. 14 (1997), no. 2, (Reviewer: J. P. Levine) 57Q45 (32S55) MR (98m:57009) Kearton, C.; Wilson, S. M. J. Alexander ideals of classical knots. Publ. Mat. 41 (1997), no. 2, (Reviewer: Hale F. Trotter) 57M25 MR (99b:57047) Hillman, J. A.; Kearton, C. Algebraic invariants of simple $4$-knots. $-knots. J. Knot Theory Ramifications 6 (1997), no. 3, of 4 04/06/ :49

3 MR: Publications results for "Author/Related=(kearton, C*)" 3 (Reviewer: J. P. Levine) 57Q45 MR (93m:57026) Kearton, C.; Wilson, S. M. J. Cyclic group actions and knots. Proc. Roy. Soc. London Ser. A 436 (1992), no. 1898, (Reviewer: Alexander I. Suciu) 57Q45 (11J86) MR (92j:57015) Kearton, C.; Steenbrink, J. Spherical algebraic knots increase with dimension. Arch. Math. (Basel) 57 (1991), no. 5, (Reviewer: Lee Rudolph) 57Q45 (15A36 32S25 32S55) MR (92a:57007) Kearton, Cherry Knots, groups, and spinning. Glasgow Math. J. 33 (1991), no. 1, (Reviewer: G. Burde) 57M25 MR (90i:57014) Bayer-Fluckiger, E.; Kearton, C.; Wilson, S. M. J. Finiteness theorems for conjugacy classes and branched covers of knots. Math. Z. 201 (1989), no. 4, (Reviewer: G. Burde) 57Q45 MR (90k:11038) Bayer-Fluckiger, E.; Kearton, C.; Wilson, S. M. J. Hermitian forms in additive categories: finiteness results. J. Algebra 123 (1989), no. 2, (Reviewer: R. Parimala) 11E39 (19G38 57M25) MR (90h:57026) Hillman, J. A.; Kearton, C. Stable isometry structures and the factorization of $Q$-acyclic stable knots. J. London Math. Soc. (2) 39 (1989), no. 1, (Reviewer: Don o Dimovski) 57Q45 (55P25 55P42) MR (89e:57001) Kearton, C. Mutation of knots. Proc. Amer. Math. Soc. 105 (1989), no. 1, (Reviewer: Lorenzo Traldi) 57M25 MR (89j:57015) Hillman, J. A.; Kearton, C. Seifert matrices and $6$-knots. $-knots. Trans. Amer. Math. Soc. 309 (1988), no. 2, (Reviewer: Alexander I. Suciu) 57Q45 MR (88f:57036) Bayer-Fluckiger, E.; Kearton, C.; Wilson, S. M. J. Decomposition of modules, forms and simple knots. J. Reine Angew. Math. 375/376 (1987), (Reviewer: Jean-Claude Hausmann) 57Q45 (11E88 15A63) MR (86g:57015) Kearton, C. Integer invariants of certain even-dimensional knots. Proc. Amer. Math. Soc. 93 (1985), no. 4, (Reviewer: John G. Ratcliffe) 57Q45 (57N70) MR (85j:57034) Kearton, C. Doubly-null-concordant simple even-dimensional knots. Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), no. 1-2, (Reviewer: Bradd Evans Clark) 57Q45 MR (85e:57024) Kearton, C. Simple spun knots. Topology 23 (1984), no. 1, (Reviewer: Cameron McA. Gordon) 57Q45 MR (84m:57014) Kearton, C. Some nonfibred $3$-knots. $-knots. Bull. London Math. Soc. 15 (1983), no. 4, (Reviewer: Cameron McA. Gordon) 57Q45 MR (84h:57010) Kearton, C. Spinning, factorisation of knots, and cyclic group actions on spheres. Arch. Math. (Basel) 40 (1983), no. 4, (Reviewer: Jonathan A. Hillman) 57Q45 (57S17) 2 of 4 04/06/ :49

4 MR: Publications results for "Author/Related=(kearton, C*)" 4 MR (84g:57016) Kearton, C. An algebraic classification of certain simple even-dimensional knots. Trans. Amer. Math. Soc. 276 (1983), no. 1, (Reviewer: W. D. Neumann) 57Q45 MR (83i:57014) Kearton, C. A remarkable $3$-knot. Bull. London Math. Soc. 14 (1982), no. 5, (Reviewer: Jonathan A. Hillman) 57Q45 MR (83h:57029) Kearton, C. The factorisation of knots. Low-dimensional topology (Bangor, 1979), pp , London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. Press, Cambridge-New York, Q45 MR (84a:57040) Kearton, C.; Wilson, S. M. J. Cyclic group actions on odd-dimensional spheres. Comment. Math. Helv. 56 (1981), no. 4, (Reviewer: Ian Hambleton) 57S25 (10C02 57Q45 57S17) MR (83h:57009) Kearton, C. Hermitian signatures and doublenull-cobordism of knots. J. London Math. Soc. (2) 23 (1981), no. 3, (Reviewer: W. D. Neumann) 57M25 MR (83e:57005) Bayer, E.; Hillman, J. A.; Kearton, C. The factorization of simple knots. Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 3, (Reviewer: R. Mandelbaum) 57M25 MR (83c:57010) Kearton, C. An algebraic classification of certain simple knots. Arch. Math. (Basel) 35 (1980), no. 4, Q45 MR (82k:57012) Kearton, C. Obstructions to embedding and isotopy in the metastable range. Math. Ann. 243 (1979), no. 2, Q37 (57Q35 57R10) MR (82j:57018) Kearton, C. Factorisation is not unique for $3$-knots. Indiana Univ. Math. J. 28 (1979), no. 3, Q45 MR (81a:57009) Kearton, C. The Milnor signatures of compound knots. Proc. Amer. Math. Soc. 76 (1979), no. 1, (Reviewer: Shin'ichi Suzuki) 57M25 MR (80h:57002) Kearton, C. Signatures of knots and the free differential calculus. Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, (Reviewer: Kenneth A. Perko, Jr.) 57M05 MR (81b:57020) Kearton, C. Attempting to classify knot modules and their Hermitian pairings. Knot theory (Proc. Sem., Plans-sur-Bex, 1977), pp , Lecture Notes in Math., 685, Springer, Berlin, (Reviewer: Hale F. Trotter) 57Q45 (10C05 57M25) MR (56 #1323) Kearton, C. An algebraic classification of some even-dimensional knots. Topology 15 (1976), no. 4, (Reviewer: D. W. L. Sumners) 57C45 MR (52 #6732) Kearton, C. Cobordism of knots and Blanchfield duality. J. London Math. Soc. (2) 10 (1975), no. 4, (Reviewer: J. P. Levine) 57C45 MR (52 #1714) Kearton, C. Simple knots which are doubly- 3 of 4 04/06/ :49

5 MR: Publications results for "Author/Related=(kearton, C*)" 5 null-cobordant. Proc. Amer. Math. Soc. 52 (1975), (Reviewer: W. D. Neumann) 57C45 MR (50 #11255) Kearton, C. Blanchfield duality and simple knots. Trans. Amer. Math. Soc. 202 (1975), (Reviewer: Wilbur Whitten) 57C45 MR (50 #11254) Kearton, C. Presentations of $n$-knots. Trans. Amer. Math. Soc. 202 (1975), (Reviewer: Wilbur Whitten) 57C45 MR (49 #6217) Kearton, C. Noninvertible e knots of codimension $2$. Proc. Amer. Math. Soc. 40 (1973), (Reviewer: R. J. Daverman) 55A25 MR (48 #3056) Kearton, C. Classification of simple knots by Blanchfield duality. Bull. Amer. Math. Soc. 79 (1973), (Reviewer: Roger Fenn) 57C45 MR (46 #9997) Kearton, C.; Lickorish, W. B. R. Piecewise linear critical levels and collapsing. Trans. Amer. Math. Soc. 170 (1972), (Reviewer: D. W. L. Sumners) 57C35 MR (46 #6339) Kearton, C. A theorem on twisted spun knots. Bull. London Math. Soc. 4 (1972), (Reviewer: Roger Fenn) 55A25 MR (45 #1172) Kearton, C. Regular neighbourhoods and mapping cylinders. Proc. Cambridge Philos. Soc. 70 (1971), (Reviewer: R. C. Lacher) 57C45 Copyright 2010, American Mathematical Society Privacy Statement American Mathematical Society 201 Charles Street Providence, RI of 4 04/06/ :49

6 Signatures of knots A knot k : S 1 S 3 determines a finite collection of quadratic forms. Each of these quadratic forms has a signature in Z. Hence k has a finite collection of signatures. These signatures are the main algebraic tools in the cobordism classification of knots and their high-dimensional analogues k : S 2j 1 S 2j+1 for all j 1. The signatures of knots have been studied for 50+ years by many mathematicians, including Kearton. The plan today is to take a walk in the knot garden, and explore some of the more elementary properties of the signatures of knots. A design for a Tudor knot garden: 6

7 Cobordism of manifolds Manifolds M to be oriented, with M the same manifold with the opposite orientation. A cobordism of closed n-dimensional manifolds M 0, M 1 is an (n + 1)-dimensional manifold N with boundary N = M 0 M 1. 7 M 0 N M 1 The set of cobordism classes of closed n-dimensional manifolds is an abelian group Ω n, with addition by disjoint union, and the cobordism class of the empty manifold as the zero element. Thom etc. (1950 s) Computation of Ω n, starting with the signature σ : Ω 4i Z. σ is an isomorphism for i = 1, a surjection for i 2. Each Ω n is finitely generated.

8 8 The signature of a manifold Terminology: the dual of a real vector space V is V = Hom R (V, R). The intersection form of a closed oriented 2j-dimensional manifold M 2j is the nonsingular ( ) j -symmetric form φ = ( ) j φ : H j (M; R) = H j (M; R) = H j (M; R) given by Poincaré duality, with dim R H j (M; R) <. Signature for j even. No signature for j odd. The signature of M 4i is the signature of the intersection form σ(m) = σ(h 2i (M; R), φ) Z. The signature is a cobordism invariant: if M 0 M 1 = N then L = ker(h 2i ( N; R) H 2i (N; R)) H 2i ( N; R) is a lagrangian subspace of (H 2i (M 0 ; R), φ 0 ) (H 2i (M 1 ; R), φ 1 ) L = L H 2i ( N; R) = H 2i (M 0 ; R) H 2i (M 1 ; R) and σ(m 0 ) = σ(m 1 ) Z.

9 An n-knot is an embedding The cobordism of knots k : S n S n+2. k is unknotted if k(s n ) = D n+1 for D n+1 S n+2. A cobordism of n-knots k 0, k 1 : S n S n+2 is an embedding such that l : S n I S n+2 I l(x, i) = (k i (x), i) (x S n, i {0, 1}). k is slice if k(s n ) = D n+1 for D n+1 D n+3. The trivial knot k 0 : S n S n+2 ; (x 0, x 1,..., x n ) (x 0, x 1,..., x n, 0, 0) is both unknotted and slice. (Fox-Milnor, 1957 for n = 1, Kervaire for n 2). The set of cobordism classes of n-knots is an abelian group C n, with addition by connected sum. k = 0 C n if and only if k is slice. 9

10 Where do signatures of knots comes from? The signatures of an n-knot k : S n S n+2 can be defined using two alternative methods: (a) The Seifert surfaces F n+1 S n+2 with F = k(s n ) S n+2. For n = 2i 1 get signatures from the Seifert matrix of linking numbers on H i (F ) (Levine) but F is non-unique. (b) The exterior of an n-knot k : S n S n+2 is the (n + 2)-dimensional manifold with boundary (X, X ) = (closure(s n+2 \k(s n ) D 2 ), S n S 1 ). Unique but H (X ) = H (S 1 ). Canonical surjection p : π 1 (X ) Z ; (S 1 X ) linking no.(s 1, k(s n ) S n+2 ) classifying an infinite cyclic cover X X. For n = 2i 1 get signatures from Poincaré duality on H i (X ; R) (Milnor) or the Blanchfield duality on H i (X ) (Kearton). As X is non-compact dim Z H i (X )/torsion could be infinite, although dim R H i (X ; R) is finite. 10

11 C n is infinitely generated (Kervaire, 1966) (i) For n 2 every n-knot k : S n S n+2 is cobordant to one with p : π 1 (X ) Z an isomorphism. (ii) C 2j = 0 for j 1. (iii) Maps C 2j 1 C 2j+3 : isomorphisms for j 2, surjection for j = 1. (iv) C n = C n+4 for all n 2. Write C n+4 for the high-dimensional groups. (Milnor, Levine 1969) Algebraic computation for j 2 11 C 2j 1 = Z Z 4 Z 2 (countable s). Algebraic number = complex number which is a root of a Z-coefficient polynomial p(z) Z[z, z 1 ]. A (2j 1)-knot k : S 2j 1 S 2j+1 has an Alexander polynomial k (z) Z[z, z 1 ]. k has one Z-valued signature for each root e iθ S 1 of k (z) with 0 < θ < π. For j 2 the cobordism class k C 2j 1 is determined by primary invariants of the signature type.

12 12 C 1 is much bigger than C 1+4 (Casson-Gordon, 1975) The surjection C 1 C 1+4 has non-trivial kernel, detected by the signatures of quadratic forms over cyclotomic fields. Need to consider k : S 1 S 3 with p : π 1 (X ) Z not an isomorphism; k is unknotted if and only if p : π 1 (X ) = Z, by Dehn s lemma. (Cochran-Orr-Teichner, 2003) A geometric filtration F (n+1)/2 F n/2... F 0.5 F 0 C 1 with C 1 /F 0.5 = C 1+4 C 1 /F 0 = Z 2 the Arf invariant map and F 0.5 /F 1.5 partially detected by the Casson-Gordon signatures. The higher quotients F m /F m+1/2 are partially detected by signatures of forms over skewfields and the R-valued L (2) -signatures of quadratic forms over von Neumann algebras. These are all secondary invariants, depending on the vanishing of the invariants in the lower filtration quotients.

13 Fibred knots 13 An n-knot k : S n S n+2 is fibred if p : π 1 (X ) Z is represented by a fibre bundle, meaning that X = T (ζ : F F ) = F I /{(x, 0) (ζ(x), 1) x F } is the mapping torus of the monodromy automorphism ζ : F F of a Seifert surface F n+1 S n+2 such that ζ = 1 : F = k(s n ) F, and p : X = T (ζ) S 1 = I /(0 1) ; [x, t] [t]. For a fibred knot the generating covering translation is A : X = F R F R ; (x, t) (ζ(x), t + 1), A = ζ : H (X ) = H (F ) H (F ), dim Z H (X )/torsion = dim Z H (F )/torsion <. The n-knots arising from function theory and algebraic geometry (e.g. the torus knots T p,q ) are fibred. But there are many non-fibred n-knots.

14 Cherry + ζ 14

15 With R-coefficients every knot k : S n S n+2 is fibred! 15 The infinite cyclic cover X of the knot exterior X is a non-compact (n + 2)-dimensional manifold with boundary S n R. (Milnor, 1968) k has the R-coefficient homology properties of a fibred knot with fibre X and monodromy a generating covering translation A : X X, with The projection dim R H (X ; R) <, H j (X ; R) = H n+1 j (X ; R). T (A : X X ) X /A = X is a homotopy equivalence. Exact sequence H r (X ; R) I A H r (X ; R) H r (X ; R) H r 1 (X ; R)... { with H r (X ; R) = H r (S 1 R if r = 0, 1 ; R) = 0 if r 0, 1. In practice, H (X ; R) and A are computed from a Seifert matrix for k.

16 The Alexander polynomial of k : S 2j 1 S 2j+1 16 Two polynomials p(z), q(z) R[z, z 1 ] are equivalent if p(z) = az b q(z) R[z, z 1 ] with a 0 R, b Z. Written p(z) q(z). The Alexander polynomial of k is the characteristic polynomial of the monodromy on H = H j (X ; R) k (z) = ch z (A) = det(z A : H[z, z 1 ] H[z, z 1 ]) R[z, z 1 ]. Roots of k (z) = complex eigenvalues of A. k (z 1 ) k (z) (Poincaré duality): if ω C is an eigenvalue then so are ω 1, ω C. Seifert matrices show k (z) p(z) for some p(z) Z[z, z 1 ], so roots of k (z) are algebraic numbers ω C\{ 1, 0, 1}. degree k (z) = dim R H, k (1), k ( 1) 0 R. If k is slice then k (z) q(z)q(z 1 ) for some q(z) Z[z, z 1 ].

17 17 Fibred automorphisms An automorphism A : H H is fibred if neither 1 nor 1 is an eigenvalue, so that A A 1 : H H is an automorphism. For any k : S 2j 1 S 2j+1 the Poincaré-Milnor duality of X defines a nonsingular ( 1) j -symmetric form φ = ( ) j φ : H = H j (X ; R) H, dim R H <. The various knot signatures only depend on the fibred automorphism A : (H, φ) (H, φ) with eigenvalues the roots of the Alexander polynomial k (z). The function {fibred automorphisms A of ( 1)-symmetric forms (H, φ )} {fibred automorphisms A of (+1)-symmetric forms (H, φ + )} ; (H, φ, A) (H, φ +, A), φ + = φ (A A 1 ) is a one-one correspondence, so from now on need only consider the case j odd, say j = 1.

18 18 The signature of a knot (Trotter 1962, Murasugi 1965) The signature of k : S 1 S 3 is the signature of the nonsingular symmetric form (H, φ(a A 1 )) σ(k) = σ(h, φ(a A 1 )) Z. Originally defined using a Seifert matrix. The signature is a knot cobordism invariant. The nonsingular symmetric form (H, φ(a A 1 )) related to the linking properties of the 3-manifold M r = X /A r S 1 D 2 (r 2) the r-fold cover of S 3 branched over k. Viro (1984) identified (H, φ(a A 1 )) with the intersection form of a 4-manifold N with boundary N = M 2 which is a double cover of D 4 branched over a Seifert surface for k.

19 The Milnor signatures of k : S 1 S 3 19 Let ω l = e iθ l S 1 (l = 1, 2,..., 2n) be the eigenvalues of A on the unit circle, with 0 < θ 1 < < θ n < π < θ n+1 = 2π θ n < < θ 2n = 2π θ 1 < 2π so that ω l = ω 2n+1 l. The other eigenvalues do not contribute signatures, so may be ignored. (1968) The Milnor signatures of k are knot cobordism invariants { σ(h ω l, φ(a A 1 ) ) if 1 l n τ ωl (k) = τ ω2n+1 l (k) if n + 1 l 2n with H ω l = {x H (A 2 2cos θ l A + I ) s (x) = 0}. s=1 (H, φ, A) = n (H ω l, φ, A ), so σ(k) = n τ ωl (k) Z. l=1 In essence, (H, φ, A) is a sum of 2-dimensional components. l=1

20 20 The Levine-Tristram signatures of k : S 1 S 3 (Levine, Tristram 1969) The Levine-Tristram signatures of k are the knot cobordism invariants defined for non-eigenvalues ω S 1 of A σ ω (k) = σ(c R H, (1 ω)φ(i A) 1 + (1 ω)(i A ) 1 φ ) Z. σ 1 (k) = 0, σ 1 (k) = σ(k). (Levine 1969, Matumoto 1972) (i) The Levine-Tristram signature function S 1 \{eigenvalues of A} Z ; ω σ ω (k) is constant on each of the open arcs between successive eigenvalues. (ii) The jump at an eigenvalue ω = e iθ S 1 of A is the Milnor signature of k at ω lim δ 0 +(σ ωe iδ(k) σ ωe iδ(k)) = τ ω(k) Z.

21 21 Jumping round the circle ω 2 σ ω = τ ω1 ω 1 σ ω = τ ω1 + τ ω2 = σ σ ω = 0 σ ω = τ ω1 ω 4 ω 3

22 The L (2) -signature of k : S 1 S 3 (Reich 2001, Cochran-Orr-Teichner 2003) The L (2) -signature of k is the knot cobordism invariant defined by the average of the Levine-Tristram signatures σ (2) (k) = 1 σ ω (k)dω R. 2π ω S 1 Also called the ρ-invariant of k. Related to Atiyah-Singer index theory. Proposition The L (2) -signature is expressed in terms of the Milnor signatures by σ (2) (k) = 1 2π = n l=1 2n 1 l=1 σ (θl +θ l+1 )/2(k)(θ l+1 θ l ) τ ωl (k)(1 θ l /π) R. In order to capture k C 1 also need to consider the L (2) -signatures of k which arise from factorizations p : π 1 (X ) Γ Z through representations ρ : π 1 (X ) Γ other than p. 22

23 The generic example: elliptic A SL 2 (R) For a 2-dimensional fibred (H, φ, A) can take ( ) 0 1 H = R R, φ =, A = 1 0 ( ) a b c d with A SL 2 (R) (ad bc = 1) and det(i ± A) = 2 ± (a + d) 0. A is elliptic if trace(a) = a + d < 2. Only elliptic A have a signature. Let (a + d)/2 = cos θ (θ (0, π)). Note that c 0. The Alexander polynomial and signatures of (H, φ, A) are given by ch z (A) = z a b c z d = z2 (a + d)z + 1 = (z e iθ )(z e iθ ), ( ) 2c d a σ(h, φ(a A 1 )) = τ e iθ(h, φ, A) = σ = 2 sgn(c) Z, { d a 2b 0 if 0 ψ < θ or 2π θ < ψ < 2π σ e iψ(h, φ, A) = 2 sgn(c) if θ < ψ < 2π θ, σ (2) (H, φ, A) = 1 2π 2π 0 σ e iψ(h, φ, A)dψ = 2 sgn(c)(1 θ/π) R. 23

24 The trefoil knot k : S 1 S 3 : The trefoil knot I. 24 k is a fibred knot with fibre a punctured torus, and H = H 1 (X ; R) = R R. The monodromy and Alexander polynomial of k are ( ) ( ) ( ) a b A = = : (H, φ) = (R R, ) (H, φ), c d k (z) = z 1 1 z 1 = z 2 z + 1.

25 25 The trefoil knot II. Algebraically, k is the generic example with θ = π/3 k (z) = z 2 z + 1 = (z e πi/3 )(z e 5πi/3 ), σ(k) = τ e πi/3(k) = σ(h, φ(a A 1 )) ( ) 2 1 = σ(r R, ) = 2, { if 0 ψ < π/3 or 5π/3 < ψ < 2π σ e iψ(k) = 2 if π/3 < ψ < 5π/3, σ (2) (k) = τ e πi/3(k)(1 θ/π) = 4 3.

26 The m-twist knots K m : S 1 S 3 for m Z 26 K 0 = unknot k 0 m full twists... m full twists... (2m crossings) (2m crossings) K m for m 1 K m for m 1 K 1 = figure 8 knot, K 1 = trefoil knot k. For m 0 K m has ( ) 0 1 (H, φ) = (R R, ), A = 1 0 ( 1 1 ) 1/m 1 + 1/m Km (z) = z 2 2(1 + 1/2m)z + 1 R[z, z 1 ].,

27 27 The m-twist knots K m for m 1 (Milnor 1968) A is elliptic, the generic example with θ = cos 1 (1 + 1/2m) (0, π/2), Km (z) = (z e iθ )(z e iθ ) R[z, z 1 ], ( 2/m 1/m σ(k m ) = τ e iθ(k m ) = σ(r R, 1/m 2 σ e iψ(k m ) = ) ) = 2, { 0 if 0 ψ < θ or 2π θ < ψ < 2π 2 if θ < ψ < 2π θ, σ (2) (K m ) = 2(1 θ/π) 0, K m 0 C 1+4. (Milnor 1968) The twist knots K m (m 1) are linearly independent in C 1, so C 1 is infinitely generated.

28 28 The m-twist knots K m for m 1 A is hyperbolic: trace(a) = 2 + 1/m > 2. The Milnor and Levine-Tristram signatures vanish σ ω (K m ) = σ(k m ) = τ e iθ(k m ) = σ (2) (K m ) = 0 Z. The following conditions are equivalent: (i) K m = 0 C 1+4 (ii) Km (z) q(z)q(z 1 ) for some q(z) Z[z, z 1 ] (iii) m = n(n + 1) for some n 1. Casson-Gordon (1975) used higher signatures to show that for m 2 K m = 0 C 1 if and only if m = 2. Reproved by Collins (2010) using the L (2) -signature of the torus knots T n,n+1. These occur as self-linking 0 simple curves on a Seifert surface F for K n(n+1), so that K n(n+1) 0 C 1 /F (1.5) for n 2.

29 29 The (p, q)-torus knots T p,q : S 1 S 3 For coprime p, q 2 define T p,q to be the composite of S 1 S 1 S 1 ; s (s p, s q ) and S 1 S 1 S 3. Example: T 2,3 = trefoil knot Example: T 2,5 = cinquefoil knot (Solomon s seal)

30 The signatures of the torus knots T p,q I. Studied by many authors, including: Pham 1965 Brieskorn 1966 Hirzebruch-Mayer 1968 Hirzebruch-Zagier 1974 Goldsmith 1975 Matumoto 1977 Kauffman 1978 Kearton 1979 Litherland 1979 Gordon-Litherland-Murasugi 1981 Kirby-Melvin 1994 Nemethi 1998 Borodzik 2009 Collins 2010 Borodzik-Oleszkiewicz

31 The signatures of the torus knots T p,q II. 31 The Alexander polynomial of T p,q Tp,q (z) = (zpq 1)(z 1) (z p 1)(z q 1) has even degree pq + 1 p q = (p 1)(q 1) = 2n. The roots of Tp,q (z) are ω l = e 2πir l/pq S 1 for 1 l 2n with {r 1 < r 2 < < r 2n } = {1, 2,..., pq 1}\{r such that p r or q r}. (Matumoto 1977, Kearton 1979) The Milnor signatures of T p,q are τ ωl (T p,q ) = 2 sgn(1/2 (a l /p + b l /q)) { 2, 2} if r l = a l q + b l p (modpq) with 1 a l < p, 1 b l < q. Number theory related to Dedekind sums. (Litherland 1979) The torus knots T p,q are linearly independent in C 1.

32 The signatures of the torus knots T p,q III. 32 The Levine-Tristram signatures of T p,q at the non-eigenvalues of ω = e iθ S 1 by A are given by l τ ωj (T p,q ) if r l /pq < θ < r l+1 /pq σ ω (T p,q ) = j=1 0 if r 1 /pq < θ < r 1 /pq. (Kirby-Melvin 1994, Nemethi 1998, Borodzik 2009, Collins 2010) The L (2) -signature of T p,q is σ (2) (T p,q ) = (p 1/p)(q 1/q)/3 R. There are inductive procedures for computing σ ω (T p,q ) and σ(t p,q ) = n τ ωj (T p,q ) Z in terms of p, q, but no closed formula. j=1 (Borodzik-Oleszkiewicz 2010) There is no closed formula for σ(t p,q ) as a rational function of p, q.

33 How to photograph birds 33 Plates from With Nature and A Camera (1897) by Richard and Cherry Kearton, the grandfather and great-uncle of our Cherry Kearton.

34 The outfit for descending cliffs 34

35 Knot theory in practice 35