Annoying trailers: http://www.youtube.com/user/NathanDunfield/ - PowerPoint PPT Presentation

Annoying trailers: http://www.youtube.com/user/NathanDunfield/ http://snappy.computop.org People who heard this talk also viewed: • J. Aaber and N. Dunfield Closed surface bundles of least volume arXiv:1002.3423

Hyperbolically twisted Alexander polynomials of knots Nathan M. Dunfield University of Illinois Stefan Friedl Nicholas Jackson Warwick Jacofest, June 4, 2010 This talk available at http://dunfield.info/ Math blog: http://ldtopology.wordpress.com/

Setup: Hyperbolically twisted Alexander • Knot: K = S 1 ֓ S 3 polynomials of knots ◦ (K) • Exterior: M = S 3 − N Nathan M. Dunfield University of Illinois Stefan Friedl Nicholas Jackson Warwick A basic and fundamental invariant of K its Alexander polynomial (1923): Jacofest, June 4, 2010 ∆ K (t) = ∆ M (t) ∈ Z [t, t − 1 ] This talk available at http://dunfield.info/ Math blog: http://ldtopology.wordpress.com/

Universal cyclic cover: corresponds to the kernel of the unique epimorphism π 1 (M) → Z . Setup: • Knot: K = S 1 ֓ S 3 ◦ (K) • Exterior: M = S 3 − N � M M S A basic and fundamental invariant of K its Alexander polynomial (1923): ∆ K (t) = ∆ M (t) ∈ Z [t, t − 1 ] S

Universal cyclic cover: corresponds to the kernel of the unique epimorphism π 1 (M) → Z . A M = H 1 ( � M ; Q ) is a module over Λ = Q [t ± 1 ] , where � t � is the covering group. � M As Λ is a PID, �� n � M � Λ A M = p k (t) k = 0 Define S � n p k (t) ∈ Q [t, t − 1 ] ∆ M (t) = k = 0 Figure-8 knot: ∆ M = t − 3 + t − 1 S

A M = H 1 ( � M ; Q ) is a module over Λ = Q [t ± 1 ] , where � t � is the covering group. Genus: � � g = min genus of S with ∂S = K As Λ is a PID, � � = min genus of S gen. H 2 (M, ∂M ; Z ) �� n � � Λ A M = p k (t) k = 0 Fundamental fact: Define 2 g ≥ deg ( ∆ M ) � n p k (t) ∈ Q [t, t − 1 ] ∆ M (t) = k = 0 Note deg ( ∆ M ) = dim Q (A M ) . Proof: As A M is generated by H 1 (S ; Q ) ≅ Q 2 g , the inequality fol- Figure-8 knot: lows. ∆ M = t − 3 + t − 1

∆ (t) determines g for all alternating knots and all fibered knots. Genus: Kinoshita-Terasaka knot: ∆ (t) = 1 but g = 2 . � � g = min genus of S with ∂S = K � � = min genus of S gen. H 2 (M, ∂M ; Z ) Fundamental fact: 2 g ≥ deg ( ∆ M ) Note deg ( ∆ M ) = dim Q (A M ) . Proof: As A M is generated by H 1 (S ; Q ) ≅ Q 2 g , the inequality fol- lows. Focus: Improve ∆ M by looking at H 1 ( � M ; V) for some system V of local coefficients.

Assumption: M is hyperbolic, i.e. ◦ = H 3 � ∆ (t) determines g for all alternating knots and for a lattice Γ ≤ Isom + H 3 M Γ all fibered knots. Thus have a faithful representation Kinoshita-Terasaka knot: ∆ (t) = 1 but g = 2 . where V = C 2 . α : π 1 (M) → SL 2 C ≤ Aut (V) Hyperbolic Alexander polynomial: � t ± 1 � coming from H 1 ( � τ M (t) ∈ C M ; V α ) . Examples: • Figure-8: τ M = t − 4 + t − 1 • Kinoshita-Terasaka: τ M ≈ ( 4 . 417926 + 0 . 376029 i)(t 3 + t − 3 ) − ( 22 . 941644 + 4 . 845091 i)(t 2 + t − 2 ) + ( 61 . 964430 + 24 . 097441 i)(t + t − 1 ) − ( − 82 . 695420 + 43 . 485388 i) Focus: Improve ∆ M by looking at H 1 ( � M ; V) for some system V of local coefficients. Really best to define τ M (t) as torsion, a la Reide- meister/Milnor/Turaev.

Assumption: M is hyperbolic, i.e. ◦ = H 3 � for a lattice Γ ≤ Isom + H 3 M Γ Thus have a faithful representation Basic Properties: where V = C 2 . • Can be normalized so τ M (t) = τ M (t − 1 ). α : π 1 (M) → SL 2 C ≤ Aut (V) • Then τ M is an actual element of C [t ± 1 ] , in Hyperbolic Alexander polynomial: � � [t ± 1 ] . tr ( Γ ) fact of Q � t ± 1 � coming from H 1 ( � τ M (t) ∈ C M ; V α ) . • τ M = τ M (t) Examples: • M amphichiral ⇒ τ M (t) ∈ R [t ± 1 ] . • Figure-8: τ M = t − 4 + t − 1 • τ M (ζ) ≠ 0 for any root of unity ζ . • Kinoshita-Terasaka: • Genus bound: τ M ≈ ( 4 . 417926 + 0 . 376029 i)(t 3 + t − 3 ) 4 g − 2 ≥ deg τ M (t) − ( 22 . 941644 + 4 . 845091 i)(t 2 + t − 2 ) + ( 61 . 964430 + 24 . 097441 i)(t + t − 1 ) For the KT knot, g = 2 and deg τ M (t) = 3 so this is sharp, unlike with ∆ M . − ( − 82 . 695420 + 43 . 485388 i) Really best to define τ M (t) as torsion, a la Reide- meister/Milnor/Turaev.

Knots by the numbers: Basic Properties: 313,231 number of prime knots with at most 15 crossings. [HTW 98] • Can be normalized so τ M (t) = τ M (t − 1 ). • Then τ M is an actual element of C [t ± 1 ] , in number where 2 g > deg ( ∆ M ) . 8,834 � � [t ± 1 ] . tr ( Γ ) fact of Q 22 number which are non-hyperbolic. • τ M = τ M (t) • M amphichiral ⇒ τ M (t) ∈ R [t ± 1 ] . number where 4 g − 2 > deg (τ M ) . 0 • τ M (ζ) ≠ 0 for any root of unity ζ . Conj. τ M determines the genus for any hyper- • Genus bound: bolic knot in S 3 . 4 g − 2 ≥ deg τ M (t) Computing τ M : Approximate π 1 (M) → SL 2 C to For the KT knot, g = 2 and deg τ M (t) = 3 so 250 digits by solving the gluing equations asso- this is sharp, unlike with ∆ M . ciated to some ideal triangulation of M to high precision.

Knots by the numbers: Basic Properties: 313,231 number of prime knots with at most 15 crossings. [HTW 98] • Can be normalized so τ M (t) = τ M (t − 1 ). • Then τ M is an actual element of C [t ± 1 ] , in number where 2 g > deg ( ∆ M ) . 8,834 � � [t ± 1 ] . tr ( Γ ) fact of Q 22 number which are non-hyperbolic. • τ M = τ M (t) • M amphichiral ⇒ τ M (t) ∈ R [t ± 1 ] . number where 4 g − 2 > deg (τ M ) . 0 • τ M (ζ) ≠ 0 for any root of unity ζ . Conj. τ M determines the genus for any hyper- • Genus bound: bolic knot in S 3 . 4 g − 2 ≥ deg τ M (t) Computing τ M : Approximate π 1 (M) → SL 2 C to For the KT knot, g = 2 and deg τ M (t) = 3 so 250 digits by solving the gluing equations asso- this is sharp, unlike with ∆ M . ciated to some ideal triangulation of M to high precision.

Many properties of M 3 are algorithmically com- putable, including Knots by the numbers: [Haken 1961] Whether a knot K in S 3 is unknot- 313,231 number of prime knots with ted. More generally, can find the genus of K . at most 15 crossings. [HTW 98] [Jaco-Oertel 1984] Whether M contains an incom- number where 2 g > deg ( ∆ M ) . 8,834 pressible surface. 22 number which are non-hyperbolic. [Rubinstein-Thompson 1995] Whether M is S 3 . number where 4 g − 2 > deg (τ M ) . 0 [Haken-Hemion-Matveev] Whether two Haken 3- Conj. τ M determines the genus for any hyper- manifolds are homeomorphic. bolic knot in S 3 . All of these plus Perelman, Thurston, Casson- Computing τ M : Approximate π 1 (M) → SL 2 C to Manning, Epstein et. al., Hodgson-Weeks, and oth- 250 digits by solving the gluing equations asso- ers give: ciated to some ideal triangulation of M to high precision. Thm. There is an algorithm to determine if two compact 3-manifolds are homeomorphic.

Many properties of M 3 are algorithmically com- Normal surfaces meet each tetrahedra in a trian- gulation T of M in a standard way: putable, including [Haken 1961] Whether a knot K in S 3 is unknot- ted. More generally, can find the genus of K . [Jaco-Oertel 1984] Whether M contains an incom- pressible surface. and correspond to certain lattice points in a finite polyhedral cone in R 7 t where t = # T : [Rubinstein-Thompson 1995] Whether M is S 3 . [Haken-Hemion-Matveev] Whether two Haken 3- manifolds are homeomorphic. All of these plus Perelman, Thurston, Casson- Manning, Epstein et. al., Hodgson-Weeks, and oth- ers give: Thm. There is an algorithm to determine if two compact 3-manifolds are homeomorphic.

Normal surfaces meet each tetrahedra in a trian- gulation T of M in a standard way: Meta Thm. In an interesting class of surfaces, there is one which is normal. Moreover, one lies on a vertex ray of the cone. E.g. The class of minimal genus surfaces whose boundary is a given knot. and correspond to certain lattice points in a finite polyhedral cone in R 7 t where t = # T : Problem: There can be exponentially many ver- tex rays, typically ≈ O( 1 . 6 t ) [Burton 2009]. In practice, limited to t < 40 . [Agol-Hass-Thurston 2002] Whether the genus of a knot K ⊂ M 3 is ≤ g is NP-complete. [Agol 2002] When M = S 3 the previous question is in co-NP.

Practical Trick: Finding the simplest surface rep- resenting some φ ∈ H 1 (M ; Z ) ≅ H 2 (M, ∂M ; Z ) . Meta Thm. In an interesting class of surfaces, there is one which is normal. Moreover, one lies Take a triangulation with only one vertex (cf. Jaco- on a vertex ray of the cone. Rubinstein, Casson). Then φ comes from a unique 1-cocycle, which realizes φ as a piecewise affine E.g. The class of minimal genus surfaces whose map M → S 1 . boundary is a given knot. Problem: There can be exponentially many ver- tex rays, typically ≈ O( 1 . 6 t ) [Burton 2009]. In practice, limited to t < 40 . [Agol-Hass-Thurston 2002] Whether the genus of a knot K ⊂ M 3 is ≤ g is NP-complete. [Agol 2002] When M = S 3 the previous question is in co-NP. Power of randomization: Trying several differ- ent T usually yields the minimal genus surface.