1

ON SIGNATURES OF KNOTS

Andrew Ranicki (Edinburgh)

http://www.maths.ed.ac.uk/ aar

For Cherry Kearton

Durham, 20 June 2010

2

Matches: 48

Publications results for "Author/Related=(kearton, C*)"

MR2443242 (2009f:57033) Kearton, Cherry; Kurlin, Vitaliy All 2-dimensionallinks in 4-space live inside a universal 3-dimensional polyhedron. Algebr. Geom.Topol. 8 (2008), no. 3, 1223--1247. (Reviewer: J. P. E. Hodgson) 57Q37 (57Q3557Q45)

MR2402510 (2009k:57039) Kearton, C.; Wilson, S. M. J. New invariants ofsimple knots. J. Knot Theory Ramifications 17 (2008), no. 3, 337--350. 57Q45(57M25 57M27)

MR2088740 (2005e:57022) Kearton, C. $S$-equivalence of knots. J. KnotTheory Ramifications 13 (2004), no. 6, 709--717. (Reviewer: Swatee Naik) 57M25

MR2008881 (2004j:57017) Kearton, C.; Wilson, S. M. J. Sharp bounds onsome classical knot invariants. J. Knot Theory Ramifications 12 (2003), no. 6,805--817. (Reviewer: Simon A. King) 57M27 (11E39 57M25)

MR1967242 (2004e:57029) Kearton, C.; Wilson, S. M. J. Simple non-finiteknots are not prime in higher dimensions. J. Knot Theory Ramifications 12 (2003),no. 2, 225--241. 57Q45

MR1933359 (2003g:57008) Kearton, C.; Wilson, S. M. J. Knot modules and theNakanishi index. Proc. Amer. Math. Soc. 131 (2003), no. 2, 655--663 (electronic).(Reviewer: Jonathan A. Hillman) 57M25

MR1803365 (2002a:57033) Kearton, C. Quadratic forms in knot theory.Quadratic forms and their applications (Dublin, 1999), 135--154, Contemp. Math.,272, Amer. Math. Soc., Providence, RI, 2000. (Reviewer: J. P. Levine) 57Q45(11E12 57M25)

MR1702726 (2000g:57040) Kearton, C.; Wilson, S. M. J. Spinning andbranched cyclic covers of knots. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.455 (1999), no. 1986, 2235--2244. (Reviewer: L. Neuwirth) 57Q45 (57M12)

MR1654649 (99k:57053) Hillman, J. A.; Kearton, C. Simple $4$-knots. J. KnotTheory Ramifications 7 (1998), no. 7, 907--923. (Reviewer: Washington Mio)57Q45

MR1612103 (99e:57037) Kearton, C.; Saeki, O.; Wilson, S. M. J. Finitenessresults for knots and singularities. Kobe J. Math. 14 (1997), no. 2, 197--206.(Reviewer: J. P. Levine) 57Q45 (32S55)

MR1485498 (98m:57009) Kearton, C.; Wilson, S. M. J. Alexander ideals ofclassical knots. Publ. Mat. 41 (1997), no. 2, 489--494. (Reviewer: Hale F. Trotter)57M25

MR1457190 (99b:57047) Hillman, J. A.; Kearton, C. Algebraic invariants ofsimple $4$-knots. J. Knot Theory Ramifications 6 (1997), no. 3, 307--318.

MR: Publications results for "Author/Related=(kearton, C*)" http://www.ams.org/mathscinet/search/publications.html?arg3=&co4...

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(Reviewer: J. P. Levine) 57Q45

MR1177575 (93m:57026) Kearton, C.; Wilson, S. M. J. Cyclic group actionsand knots. Proc. Roy. Soc. London Ser. A 436 (1992), no. 1898, 527--535.(Reviewer: Alexander I. Suciu) 57Q45 (11J86)

MR1129530 (92j:57015) Kearton, C.; Steenbrink, J. Spherical algebraic knotsincrease with dimension. Arch. Math. (Basel) 57 (1991), no. 5, 518--520.(Reviewer: Lee Rudolph) 57Q45 (15A36 32S25 32S55)

MR1089960 (92a:57007) Kearton, Cherry Knots, groups, and spinning.Glasgow Math. J. 33 (1991), no. 1, 99--100. (Reviewer: G. Burde) 57M25

MR1004169 (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, 485--493. (Reviewer: G. Burde) 57Q45

MR1000491 (90k:11038) Bayer-Fluckiger, E.; Kearton, C.; Wilson, S. M. J.Hermitian forms in additive categories: finiteness results. J. Algebra 123 (1989),no. 2, 336--350. (Reviewer: R. Parimala) 11E39 (19G38 57M25)

MR0989928 (90h:57026) Hillman, J. A.; Kearton, C. Stable isometry structuresand the factorization of $Q$-acyclic stable knots. J. London Math. Soc. (2) 39(1989), no. 1, 175--182. (Reviewer: Don o Dimovski) 57Q45 (55P25 55P42)

MR0929430 (89e:57001) Kearton, C. Mutation of knots. Proc. Amer. Math. Soc.105 (1989), no. 1, 206--208. (Reviewer: Lorenzo Traldi) 57M25

MR0961617 (89j:57015) Hillman, J. A.; Kearton, C. Seifert matrices and$6$-knots. Trans. Amer. Math. Soc. 309 (1988), no. 2, 843--855. (Reviewer:Alexander I. Suciu) 57Q45

MR0882296 (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), 167--183. (Reviewer: Jean-Claude Hausmann) 57Q45 (11E8815A63)

MR0776214 (86g:57015) Kearton, C. Integer invariants of certaineven-dimensional knots. Proc. Amer. Math. Soc. 93 (1985), no. 4, 747--750.(Reviewer: John G. Ratcliffe) 57Q45 (57N70)

MR0741655 (85j:57034) Kearton, C. Doubly-null-concordant simpleeven-dimensional knots. Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), no. 1-2,163--174. (Reviewer: Bradd Evans Clark) 57Q45

MR0721455 (85e:57024) Kearton, C. Simple spun knots. Topology 23 (1984),no. 1, 91--95. (Reviewer: Cameron McA. Gordon) 57Q45

MR0703762 (84m:57014) Kearton, C. Some nonfibred $3$-knots. Bull. LondonMath. Soc. 15 (1983), no. 4, 365--367. (Reviewer: Cameron McA. Gordon) 57Q45

MR0708598 (84h:57010) Kearton, C. Spinning, factorisation of knots, andcyclic group actions on spheres. Arch. Math. (Basel) 40 (1983), no. 4, 361--363.(Reviewer: Jonathan A. Hillman) 57Q45 (57S17)

MR: Publications results for "Author/Related=(kearton, C*)" http://www.ams.org/mathscinet/search/publications.html?arg3=&co4...

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MR0684492 (84g:57016) Kearton, C. An algebraic classification of certainsimple even-dimensional knots. Trans. Amer. Math. Soc. 276 (1983), no. 1,1--53. (Reviewer: W. D. Neumann) 57Q45

MR0671779 (83i:57014) Kearton, C. A remarkable $3$-knot. Bull. LondonMath. Soc. 14 (1982), no. 5, 397--398. (Reviewer: Jonathan A. Hillman) 57Q45

MR0662429 (83h:57029) Kearton, C. The factorisation of knots.Low-dimensional topology (Bangor, 1979), pp. 71--80, London Math. Soc. LectureNote Ser., 48, Cambridge Univ. Press, Cambridge-New York, 1982. 57Q45

MR0656215 (84a:57040) Kearton, C.; Wilson, S. M. J. Cyclic group actions onodd-dimensional spheres. Comment. Math. Helv. 56 (1981), no. 4, 615--626.(Reviewer: Ian Hambleton) 57S25 (10C02 57Q45 57S17)

MR0616563 (83h:57009) Kearton, C. Hermitian signatures and double-null-cobordism of knots. J. London Math. Soc. (2) 23 (1981), no. 3, 563--576.(Reviewer: W. D. Neumann) 57M25

MR0628832 (83e:57005) Bayer, E.; Hillman, J. A.; Kearton, C. The factorizationof simple knots. Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 3, 495--506.(Reviewer: R. Mandelbaum) 57M25

MR0602582 (83c:57010) Kearton, C. An algebraic classification of certainsimple knots. Arch. Math. (Basel) 35 (1980), no. 4, 391--393. 57Q45

MR0543720 (82k:57012) Kearton, C. Obstructions to embedding and isotopy inthe metastable range. Math. Ann. 243 (1979), no. 2, 103--113. 57Q37 (57Q3557R10)

MR0529677 (82j:57018) Kearton, C. Factorisation is not unique for $3$-knots.Indiana Univ. Math. J. 28 (1979), no. 3, 451--452. 57Q45

MR0534409 (81a:57009) Kearton, C. The Milnor signatures of compound knots.Proc. Amer. Math. Soc. 76 (1979), no. 1, 157--160. (Reviewer: Shin'ichi Suzuki)57M25

MR0534830 (80h:57002) Kearton, C. Signatures of knots and the freedifferential calculus. Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 157--182.(Reviewer: Kenneth A. Perko, Jr.) 57M05

MR0521736 (81b:57020) Kearton, C. Attempting to classify knot modules andtheir Hermitian pairings. Knot theory (Proc. Sem., Plans-sur-Bex, 1977), pp.227--242, Lecture Notes in Math., 685, Springer, Berlin, 1978. (Reviewer: Hale F.Trotter) 57Q45 (10C05 57M25)

MR0442948 (56 #1323) Kearton, C. An algebraic classification of someeven-dimensional knots. Topology 15 (1976), no. 4, 363--373. (Reviewer: D. W.L. Sumners) 57C45

MR0385873 (52 #6732) Kearton, C. Cobordism of knots and Blanchfield duality.J. London Math. Soc. (2) 10 (1975), no. 4, 406--408. (Reviewer: J. P. Levine)57C45

MR0380817 (52 #1714) Kearton, C. Simple knots which are doubly-

MR: Publications results for "Author/Related=(kearton, C*)" http://www.ams.org/mathscinet/search/publications.html?arg3=&co4...

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null-cobordant. Proc. Amer. Math. Soc. 52 (1975), 471--472. (Reviewer: W. D.Neumann) 57C45

MR0358796 (50 #11255) Kearton, C. Blanchfield duality and simple knots.Trans. Amer. Math. Soc. 202 (1975), 141--160. (Reviewer: Wilbur Whitten) 57C45

MR0358795 (50 #11254) Kearton, C. Presentations of $n$-knots. Trans. Amer.Math. Soc. 202 (1975), 123--140. (Reviewer: Wilbur Whitten) 57C45

MR0341466 (49 #6217) Kearton, C. Noninvertible knots of codimension $2$.Proc. Amer. Math. Soc. 40 (1973), 274--276. (Reviewer: R. J. Daverman) 55A25

MR0324706 (48 #3056) Kearton, C. Classification of simple knots by Blanchfieldduality. Bull. Amer. Math. Soc. 79 (1973), 952--955. (Reviewer: Roger Fenn)57C45

MR0310899 (46 #9997) Kearton, C.; Lickorish, W. B. R. Piecewise linear criticallevels and collapsing. Trans. Amer. Math. Soc. 170 (1972), 415--424. (Reviewer:D. W. L. Sumners) 57C35

MR0307219 (46 #6339) Kearton, C. A theorem on twisted spun knots. Bull.London Math. Soc. 4 (1972), 47--48. (Reviewer: Roger Fenn) 55A25

MR0292085 (45 #1172) Kearton, C. Regular neighbourhoods and mappingcylinders. Proc. Cambridge Philos. Soc. 70 (1971), 15--18. (Reviewer: R. C.Lacher) 57C45

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Signatures of knots

I A knot k : S1 ⊂ S3 determines a finite collection of quadratic forms.I Each of these quadratic forms has a signature in Z.I Hence k has a finite collection of signatures.I These signatures are the main algebraic tools in the cobordism

classification of knots and their high-dimensional analoguesk : S2j−1 ⊂ S2j+1 for all j > 1.

I The signatures of knots have been studied for 50+ years by manymathematicians, including Kearton.

I The plan today is to take a walk in the knot garden, and explore someof the more elementary properties of the signatures of knots.

I A design for a Tudor knot garden:

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Cobordism of manifolds

I Manifolds M to be oriented, with −M the same manifold with theopposite orientation.

I A cobordism of closed n-dimensional manifolds M0,M1 is an(n + 1)-dimensional manifold N with boundary ∂N = M0 t −M1.

M0 N M1

I The set of cobordism classes of closed n-dimensional manifolds is anabelian group Ωn, with addition by disjoint union, and the cobordismclass of the empty manifold ∅ as the zero element.

I Thom etc. (1950’s) Computation of Ωn, starting with the signature

σ : Ω4i → Z .

I σ is an isomorphism for i = 1, a surjection for i > 2.I Each Ωn is finitely generated.

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The signature of a manifold

I Terminology: the dual of a real vector space V is V ∗ = HomR(V ,R).I The intersection form of a closed oriented 2j-dimensional manifold

M2j is the nonsingular (−)j -symmetric form

φ = (−)jφ∗ : Hj(M;R)∼= // H j(M;R)

∼= // Hj(M;R)∗

given by Poincare duality, with dimRHj(M;R) <∞.I Signature for j even. No signature for j odd.I The signature of M4i is the signature of the intersection form

σ(M) = σ(H2i (M;R), φ) ∈ Z .

I The signature is a cobordism invariant: if M0 t −M1 = ∂N then

L = ker(H2i (∂N;R)→ H2i (N;R)) ⊂ H2i (∂N;R)

is a lagrangian subspace of (H2i (M0;R), φ0)⊕ (H2i (M1;R),−φ1)

L⊥ = L ⊂ H2i (∂N;R) = H2i (M0;R)⊕ H2i (M1;R)

and σ(M0) = σ(M1) ∈ Z.

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The cobordism of knots

I An n-knot is an embedding

k : Sn ⊂ Sn+2 .

I k is unknotted if k(Sn) = ∂Dn+1 for Dn+1 ⊂ Sn+2.I A cobordism of n-knots k0, k1 : Sn ⊂ Sn+2 is an embedding

` : Sn × I ⊂ Sn+2 × I

such that`(x , i) = (ki (x), i) (x ∈ Sn, i ∈ 0, 1) .

I k is slice if k(Sn) = ∂Dn+1 for Dn+1 ⊂ Dn+3.I The trivial knot

k0 : Sn ⊂ Sn+2 ; (x0, x1, . . . , xn) 7→ (x0, x1, . . . , xn, 0, 0)

is both unknotted and slice.I (Fox-Milnor, 1957 for n = 1, Kervaire for n > 2). The set of cobordism

classes of n-knots is an abelian group Cn, with addition by connectedsum. k = 0 ∈ Cn if and only if k is slice.

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Where do signatures of knots comes from?

I The signatures of an n-knot k : Sn ⊂ Sn+2 can be defined using twoalternative methods:(a) The Seifert surfaces F n+1 ⊂ Sn+2 with

∂F = k(Sn) ⊂ Sn+2 .

For n = 2i − 1 get signatures from the Seifert matrix of linking numberson Hi (F ) (Levine) but F is non-unique.

(b) The exterior of an n-knot k : Sn ⊂ Sn+2 is the (n + 2)-dimensionalmanifold with boundary

(X , ∂X ) = (closure(Sn+2\k(Sn)× D2),Sn × S1) .

Unique but H∗(X ) = H∗(S1). Canonical surjection

p : π1(X )→ Z ; (S1 ⊂ X ) 7→ linking no.(S1, k(Sn) ⊂ Sn+2)

classifying an infinite cyclic cover X → X . For n = 2i − 1 get signaturesfrom Poincare duality on Hi (X ;R) (Milnor) or the Blanchfield duality onHi (X ) (Kearton). As X is non-compact dimZHi (X )/torsion could beinfinite, although dimRHi (X ;R) is finite.

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Cn is infinitely generated

I (Kervaire, 1966) (i) For n > 2 every n-knot k : Sn ⊂ Sn+2 is cobordantto one with p : π1(X )→ Z an isomorphism.(ii) C2j = 0 for j > 1.(iii) Maps C2j−1 → C2j+3: isomorphisms for j > 2, surjection for j = 1.(iv) Cn = Cn+4 for all n > 2.

I Write Cn+4∗ for the high-dimensional groups.I (Milnor, Levine 1969) Algebraic computation for j > 2

C2j−1 =⊕∞

Z⊕⊕∞

Z4 ⊕⊕∞

Z2 (countable ∞′s) .

I Algebraic number = complex number which is a root of aZ-coefficient polynomial p(z) ∈ Z[z , z−1].

I A (2j − 1)-knot k : S2j−1 ⊂ S2j+1 has an Alexander polynomial∆k(z) ∈ Z[z , z−1]. k has one Z-valued signature for each root e iθ ∈ S1

of ∆k(z) with 0 < θ < π.I For j > 2 the cobordism class k ∈ C2j−1 is determined by primary

invariants of the signature type.

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C1 is much bigger than C1+4∗

I (Casson-Gordon, 1975) The surjection C1 → C1+4∗ has non-trivialkernel, detected by the signatures of quadratic forms over cyclotomicfields. Need to consider k : S1 ⊂ S3 with p : π1(X )→ Z not anisomorphism; k is unknotted if and only if p : π1(X ) ∼= Z, by Dehn’slemma.

I (Cochran-Orr-Teichner, 2003) A geometric filtration

· · · ⊂ F(n+1)/2 ⊂ Fn/2 ⊂ . . .F0.5 ⊂ F0 ⊂ C1

with C1/F0.5 = C1+4∗ → C1/F0 = Z2 the Arf invariant map andF0.5/F1.5 partially detected by the Casson-Gordon signatures. Thehigher quotients Fm/Fm+1/2 are partially detected by signatures of

forms over skewfields and the R-valued L(2)-signatures of quadraticforms over von Neumann algebras. These are all secondary invariants,depending on the vanishing of the invariants in the lower filtrationquotients.

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Fibred knots

I An n-knot k : Sn ⊂ Sn+2 is fibred if p : π1(X )→ Z is represented by afibre 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 aSeifert surface F n+1 ⊂ Sn+2 such that ζ| = 1 : ∂F = k(Sn)→ ∂F , and

p : X = T (ζ)→ S1 = I/(0 ∼ 1) ; [x , t]→ [t] .

I For a fibred knot the generating covering translation is

A : X = F × R→ F × R ; (x , t) 7→ (ζ(x), t + 1) ,

A∗ = ζ∗ : H∗(X ) = H∗(F )→ H∗(F ) ,

dimZH∗(X )/torsion = dimZH∗(F )/torsion <∞.

I The n-knots arising from function theory and algebraic geometry (e.g.the torus knots Tp,q) are fibred. But there are many non-fibred n-knots.

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Cherry + ζ

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With R-coefficients every knot k : Sn ⊂ Sn+2 is fibred!

I The infinite cyclic cover X of the knot exterior X is a non-compact(n + 2)-dimensional manifold with boundary Sn × R.

I (Milnor, 1968) k has the R-coefficient homology properties of a fibredknot with fibre X and monodromy a generating covering translationA : X → X , with

dimRH∗(X ;R) <∞ , Hj(X ;R) ∼= Hn+1−j(X ;R)∗ .

I The projectionT (A : X → X )→ X/A = X

is a homotopy equivalence. Exact sequence

· · · → Hr (X ;R)I − A// Hr (X ;R)→ Hr (X ;R)→ Hr−1(X ;R)→ . . .

with Hr (X ;R) = Hr (S1;R) =

R if r = 0, 1

0 if r 6= 0, 1 .

I In practice, H∗(X ;R) and A are computed from a Seifert matrix for k .

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The Alexander polynomial of k : S2j−1 ⊂ S2j+1

I Two polynomials p(z), q(z) ∈ R[z , z−1] are equivalent if

p(z) = azbq(z) ∈ R[z , z−1]

with a 6= 0 ∈ R, b ∈ Z. Written p(z) ∼ q(z).I The Alexander polynomial of k is the characteristic polynomial of the

monodromy on H = Hj(X ;R)

∆k(z) = chz(A) = det(z − A : H[z , z−1]→ H[z , z−1]) ∈ R[z , z−1] .

Roots of ∆k(z) = complex eigenvalues of A.I ∆k(z−1) ∼ ∆k(z) (Poincare duality): if ω ∈ C is an eigenvalue then so

are ω−1, ω ∈ C.I 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.I degree ∆k(z) = dimRH, ∆k(1),∆k(−1) 6= 0 ∈ R.I If k is slice then ∆k(z) ∼ q(z)q(z−1) for some q(z) ∈ Z[z , z−1].

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Fibred automorphisms

I An automorphism A : H → H is fibred if neither 1 nor −1 is aneigenvalue, so that A− A−1 : H → H is an automorphism.

I For any k : S2j−1 ⊂ S2j+1 the Poincare-Milnor duality of X defines anonsingular (−1)j -symmetric form

φ = (−)jφ∗ : H = Hj(X ;R)→ H∗ , dimRH <∞ .

I The various knot signatures only depend on the fibred automorphismA : (H, φ)→ (H, φ) with eigenvalues the roots of the Alexanderpolynomial ∆k(z).

I The function

fibred automorphisms A of (−1)-symmetric forms (H, φ−)→ fibred automorphisms A of (+1)-symmetric forms (H, φ+) ;

(H, φ−,A) 7→ (H, φ+,A) , φ+ = φ−(A− A−1)

is a one-one correspondence, so from now on need only consider thecase j odd, say j = 1.

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The signature of a knot

I (Trotter 1962, Murasugi 1965) The signature of k : S1 ⊂ S3 is thesignature of the nonsingular symmetric form (H, φ(A− A−1))

σ(k) = σ(H, φ(A− A−1)) ∈ Z .

I Originally defined using a Seifert matrix.

I The signature is a knot cobordism invariant.

I The nonsingular symmetric form (H, φ(A− A−1)) related to the linkingproperties of the 3-manifold

Mr = X/Ar ∪ S1 × D2 (r > 2)

the r -fold cover of S3 branched over k .

I Viro (1984) identified (H, φ(A− A−1)) with the intersection form of a4-manifold N with boundary ∂N = M2 which is a double cover of D4

branched over a Seifert surface for k.

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The Milnor signatures of k : S1 ⊂ S3

I Let ω` = e iθ` ∈ S1 (` = 1, 2, . . . , 2n) be the eigenvalues of A on theunit circle, with

0 < θ1 < · · · < θn < π < θn+1 = 2π − θn < · · · < θ2n = 2π − θ1 < 2π

so that ω` = ω2n+1−`.I The other eigenvalues do not contribute signatures, so may be ignored.I (1968) The Milnor signatures of k are knot cobordism invariants

τω`(k) =

σ(Hω` , φ(A− A−1)|) if 1 6 ` 6 n

−τω2n+1−`(k) if n + 1 6 ` 6 2n

with Hω` =∞⋃s=1x ∈ H | (A2 − 2cos θ` A + I )s(x) = 0.

I (H, φ,A) =n∑=1

(Hω` , φ|,A|), so σ(k) =n∑=1

τω`(k) ∈ Z.

I In essence, (H, φ,A) is a sum of 2-dimensional components.

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The Levine-Tristram signatures of k : S1 ⊂ S3

I (Levine, Tristram 1969) The Levine-Tristram signatures of k are theknot cobordism invariants defined for non-eigenvalues ω ∈ S1 of A

σω(k) = σ(C⊗R H, (1− ω)φ(I − A)−1 + (1− ω)(I − A∗)−1φ∗) ∈ Z .

I σ1(k) = 0, σ−1(k) = σ(k).

I (Levine 1969, Matumoto 1972) (i) The Levine-Tristram signaturefunction

S1\eigenvalues of A → Z ; ω 7→ σω(k)

is constant on each of the open arcs between successive eigenvalues.(ii) The jump at an eigenvalue ω = e iθ ∈ S1 of A is the Milnorsignature of k at ω

limδ→0+

(σωe iδ(k)− σωe−iδ(k)) = τω(k) ∈ Z .

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Jumping round the circle

ω1

ω2

ω3

ω4

σω = 0

σω = τω1

σω = τω1 + τω2 = σ

σω = τω1

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The L(2)-signature of k : S1 ⊂ S3

I (Reich 2001, Cochran-Orr-Teichner 2003) The L(2)-signature of k isthe knot cobordism invariant defined by the average of theLevine-Tristram signatures

σ(2)(k) =1

2π

∫ω∈S1

σω(k)dω ∈ R .

I Also called the ρ-invariant of k . Related to Atiyah-Singer index theory.I Proposition The L(2)-signature is expressed in terms of the Milnor

signatures by

σ(2)(k) =1

2π

2n−1∑=1

σ(θ`+θ`+1)/2(k)(θ`+1 − θ`)

=n∑=1

τω`(k)(1− θ`/π) ∈ R .

I In order to capture k ∈ C1 also need to consider the L(2)-signatures ofk which arise from factorizations p : π1(X )→ Γ→ Z throughrepresentations ρ : π1(X )→ Γ other than p.

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The generic example: elliptic A ∈ SL2(R)

I For a 2-dimensional fibred (H, φ,A) can take

H = R⊕ R , φ =

(0 1−1 0

), A =

(a bc d

)with A ∈ SL2(R) (ad − bc = 1) and det(I ± A) = 2± (a + d) 6= 0.

I 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 6= 0.

I The Alexander polynomial and signatures of (H, φ,A) are given by

chz(A) =

∣∣∣∣z − a −b−c z − d

∣∣∣∣ = z2 − (a + d)z + 1 = (z − e iθ)(z − e−iθ),

σ(H, φ(A− A−1)) = τe iθ(H, φ,A) = σ

(2c d − a

d − a −2b

)= 2 sgn(c) ∈ Z ,

σe iψ(H, φ,A) =

0 if 0 6 ψ < θ or 2π − θ < ψ < 2π

2 sgn(c) if θ < ψ < 2π − θ ,

σ(2)(H, φ,A) =1

2π

∫ 2π0 σe iψ(H, φ,A)dψ = 2 sgn(c)(1− θ/π) ∈ R .

24

The trefoil knot I.

I The trefoil knot k : S1 ⊂ S3 :

I k is a fibred knot with fibre a punctured torus, and

H = H1(X ;R) = R⊕ R .

I The monodromy and Alexander polynomial of k are

A =

(a bc d

)=

(1 1−1 0

): (H, φ) = (R⊕ R,

(0 1−1 0

))→ (H, φ) ,

∆k(z) =

∣∣∣∣z −11 z − 1

∣∣∣∣ = z2 − z + 1 .

25

The trefoil knot II.

I Algebraically, k is the generic example with θ = π/3

∆k(z) = z2 − z + 1 = (z − eπi/3)(z − e5πi/3) ,

σ(k) = τeπi/3(k) = σ(H, φ(A− A−1))

= σ(R⊕ R,(−2 11 −2

)) = − 2 ,

σe iψ(k) =

0 if 0 6 ψ < π/3 or 5π/3 < ψ < 2π

−2 if π/3 < ψ < 5π/3 ,

σ(2)(k) = τeπi/3(k)(1− θ/π) =−4

3.

26

The m-twist knots Km : S1 ⊂ S3 for m ∈ Z

I K0 = unknot k0I

m full twists...

(2m crossings)

–m full twists...

(2m crossings)

Km for m > 1 K−m for m > 1

I K1 = figure 8 knot, K−1 = trefoil knot k.I For m 6= 0 Km has

(H, φ) = (R⊕ R,(

0 1−1 0

)) , A =

(1 1

1/m 1 + 1/m

),

∆Km(z) = z2 − 2(1 + 1/2m)z + 1 ∈ R[z , z−1] .

27

The m-twist knots Km for m 6 −1

I (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] ,

σ(Km) = τe iθ(Km) = σ(R⊕ R,(

2/m 1/m1/m −2

)) = − 2 ,

σe iψ(Km) =

0 if 0 6 ψ < θ or 2π − θ < ψ < 2π

−2 if θ < ψ < 2π − θ ,

σ(2)(Km) = − 2(1− θ/π) 6= 0 , Km 6= 0 ∈ C1+4∗ .

I (Milnor 1968) The twist knots Km (m 6 −1) are linearly independentin C1, so C1 is infinitely generated.

28

The m-twist knots Km for m > 1

I A is hyperbolic: |trace(A)| = 2 + 1/m > 2.

I The Milnor and Levine-Tristram signatures vanish

σω(Km) = σ(Km) = τe iθ(Km) = σ(2)(Km) = 0 ∈ Z .

I The following conditions are equivalent:(i) Km = 0 ∈ C1+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.

I Casson-Gordon (1975) used higher signatures to show that for m > 2

Km = 0 ∈ C1 if and only if m = 2 .

Reproved by Collins (2010) using the L(2)-signature of the torus knotsTn,n+1. These occur as self-linking 0 simple curves on a Seifert surfaceF for Kn(n+1), so that Kn(n+1) 6= 0 ∈ C1/F(1.5) for n > 2.

29

The (p, q)-torus knots Tp,q : S1 ⊂ S3

I For coprime p, q > 2 define Tp,q to be the composite of

S1 ⊂ S1 × S1 ; s 7→ (sp, sq) and S1 × S1 ⊂ S3 .

I Example: T2,3 = trefoil knot

I Example: T2,5 = cinquefoil knot (Solomon’s seal)

30

The signatures of the torus knots Tp,q I.

I Studied by many authors, including:

Pham 1965Brieskorn 1966Hirzebruch-Mayer 1968Hirzebruch-Zagier 1974Goldsmith 1975Matumoto 1977Kauffman 1978Kearton 1979Litherland 1979Gordon-Litherland-Murasugi 1981Kirby-Melvin 1994Nemethi 1998Borodzik 2009Collins 2010Borodzik-Oleszkiewicz 2010

31

The signatures of the torus knots Tp,q II.

I The Alexander polynomial of Tp,q

∆Tp,q(z) =(zpq − 1)(z − 1)

(zp − 1)(zq − 1)

has even degree pq + 1− p − q = (p − 1)(q − 1) = 2n.I The roots of ∆Tp,q(z) are

ω` = e2πir`/pq ∈ S1 for 1 6 ` 6 2n

withr1 < r2 < · · · < r2n = 1, 2, . . . , pq − 1\r such that p|r or q|r.

I (Matumoto 1977, Kearton 1979) The Milnor signatures of Tp,q are

τω`(Tp,q) = 2 sgn(1/2− (a`/p + b`/q)) ∈ −2, 2

if r` = a`q + b`p (modpq) with 1 6 a` < p , 1 6 b` < q.Number theory related to Dedekind sums.

I (Litherland 1979) The torus knots Tp,q are linearly independent in C1.

32

The signatures of the torus knots Tp,q III.

I The Levine-Tristram signatures of Tp,q at the non-eigenvalues ofω = e iθ ∈ S1 by A are given by

σω(Tp,q) =

∑j=1

τωj (Tp,q) if r`/pq < θ < r`+1/pq

0 if − r1/pq < θ < r1/pq .

I (Kirby-Melvin 1994, Nemethi 1998, Borodzik 2009, Collins 2010)The L(2)-signature of Tp,q is

σ(2)(Tp,q) = − (p − 1/p)(q − 1/q)/3 ∈ R .

I There are inductive procedures for computing σω(Tp,q) and

σ(Tp,q) =n∑

j=1τωj (Tp,q) ∈ Z in terms of p, q, but no closed formula.

I (Borodzik-Oleszkiewicz 2010) There is no closed formula for σ(Tp,q) asa rational function of p, q.

33

How to photograph birds

Plates from With Nature and A Camera (1897) by Richard and CherryKearton, the grandfather and great-uncle of our Cherry Kearton.

http://gdl.cdlr.strath.ac.uk/keacam

34

The outfit for descending cliffs

35

Knot theory in practice