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Quantum invariants of knots and 3-manifolds
Clément Maria
The University of Queensland
- I. Topology of knots and manifolds
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I. Topology of knots and manifolds 2 Topological equivalence(s) - - - PowerPoint PPT Presentation
I. Topology of knots and manifolds 2 Topological equivalence(s) - - - PowerPoint PPT Presentation
Quantum invariants of knots and 3 -manifolds Clment Maria The University of Queensland June 2015 I. Topology of knots and manifolds 2 Topological equivalence(s) - Homeomorphism : bijective continuous function with continuous inverse. -
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Topological equivalence(s)
- Homeomorphism: bijective continuous function with continuous
inverse.
- Isotopy: continuous family of homeomorphism (”deformation”).
- Invariant: property invariant under homeomorphism/isotopy.
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Topological equivalence(s)
∼ = ∼ =
- Homeomorphism: bijective continuous function with continuous
inverse.
- Isotopy: continuous family of homeomorphism (”deformation”).
- Invariant: property invariant under homeomorphism/isotopy.
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Topological equivalence(s)
∼ = ̸=
- Homeomorphism: bijective continuous function with continuous
inverse.
- Isotopy: continuous family of homeomorphism (”deformation”).
- Invariant: property invariant under homeomorphism/isotopy.
3
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Topological equivalence(s)
∼ = ̸=
- Homeomorphism: bijective continuous function with continuous
inverse.
- Isotopy: continuous family of homeomorphism (”deformation”).
- Invariant: property invariant under homeomorphism/isotopy.
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Knots, links and ribbons
- Knot: embedding of S1 → R3.
- Link: embedding of S1 × . . . × S1 → R3.
- Ribbon: knot/link with orientation and framing.
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Knots, links and ribbons
- Knot: embedding of S1 → R3.
- Link: embedding of S1 × . . . × S1 → R3.
- Ribbon: knot/link with orientation and framing.
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Knots, links and ribbons
- Knot: embedding of S1 → R3.
- Link: embedding of S1 × . . . × S1 → R3.
- Ribbon: knot/link with orientation and framing.
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Manifolds
- d-manifold: every point is locally homeomorphic to Bd.
- Generalized 3-triangulation: set of tetrahedra with triangle gluings.
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Manifolds
- d-manifold: every point is locally homeomorphic to Bd.
- Generalized 3-triangulation: set of tetrahedra with triangle gluings.
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Quantum invariants of knots
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
g ◦ f: U1 ⊗ . . . ⊗ Uℓ f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
W1 Wm V1 Vn · · · · · ·
f
Wm+1 Wq Vn+1 Vp · · · · · ·
g . = f ⊗ g
W1 Wq V1 · · · Vp · · ·
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
f: V1 ⊗ . . . ⊗ Vn → W1 ⊗ . . . ⊗ Wm
W1 Wm V1 Vn · · · · · · U1 Uℓ · · ·
f g . =
U1 Uℓ · · · W1 Wm · · ·
g ◦ f
f ⊗ g: V1 ⊗ . . . ⊗ Vp → W1 ⊗ . . . Wq g ◦ f: U1 ⊗ . . . ⊗ Uℓ
7
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Construction of the invariant
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V . =
V W W V V W V W V W
. = f cW,V idV ⊗W (cW,V )−1
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Construction of the invariant
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
V W W V V W V W V W
. = f cW,V idV ⊗W (cW,V )−1 . =
7
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Construction of the invariant
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
V W W V V W V W V W
. = f cW,V idV ⊗W (cW,V )−1 . =
V
. = g idV θV
7
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Construction of the invariant
. = idV : V → V V . = idV ∗ V
cV,W : V ⊗ W → W ⊗ V
V W W V
θV : V → V
V V
dv: V ∗ ⊗ V → 1
V
bv: 1 → V ⊗ V ∗ W V
V W W V V W V W V W
. = f cW,V idV ⊗W (cW,V )−1 . =
V
. = g idV θV
7
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Ribbon category and ribbon diagrams
A ribbon category V is a category with:
- tensor product ⊗: V × V → V,
- braiding {cV,W : V ⊗ W → W ⊗ V},
- twist {θV : V → V},
- duality {V∗, bV : 1 → V ⊗ V∗, dV : V∗ ⊗ V → 1},
satisfying a set of natural axioms.
Theorem (Reshetikhin, Turaev)
A ribbon category associates to every V-coloured ribbon diagram a morphism 1 → 1. It is an isotopy invariant. Proof: any isotopy of ribbon diagrams may be described by a sequence of Reidemeister moves.
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Ribbon category and ribbon diagrams
A ribbon category V is a category with:
- tensor product ⊗: V × V → V,
- braiding {cV,W : V ⊗ W → W ⊗ V},
- twist {θV : V → V},
- duality {V∗, bV : 1 → V ⊗ V∗, dV : V∗ ⊗ V → 1},
satisfying a set of natural axioms.
Theorem (Reshetikhin, Turaev)
A ribbon category associates to every V-coloured ribbon diagram a morphism 1 → 1. It is an isotopy invariant. Proof: any isotopy of ribbon diagrams may be described by a sequence of Reidemeister moves.
. = . = . =
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Quantum invariants of 3-manifolds
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Surgery presentation
Let k ⊆ S3. A surgery on the 3-sphere along k consists in ”drilling” k out
- f S3 and glue back a solid torus along the toric boundary.
Theorem (Lickorish-Wallace)
Every 3-manifold may be obtained by surgery on S3 along a link.
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Invariant of 3-manifold
Let M be a 3-manifold, obtained by surgery on S3 along a link k with m components {L1, . . . , Lm}. Let V be a ribbon category 1. For a colouring λ: {L1, . . . , Lm} → V, denote by F(k, λ) the associated ribbon invariant. Finally, sum over all colourings:
τ(M, V) = AV ∑
λ: {L1,...,Lm}→V
Dλ × F(k, λ)
1with an extra notion of ”decomposability” of objects.
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Invariant of 3-manifold
Let M be a 3-manifold, obtained by surgery on S3 along a link k with m components {L1, . . . , Lm}. Let V be a ribbon category 1. For a colouring λ: {L1, . . . , Lm} → V, denote by F(k, λ) the associated ribbon invariant. Finally, sum over all colourings:
τ(M, V) = AV ∑
λ: {L1,...,Lm}→V
Dλ × F(k, λ)
1with an extra notion of ”decomposability” of objects.
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Invariant of 3-manifold
Theorem (Reshetikhin, Turaev)
For a manifold M obtained by surgery on S3 along k, and a ribbon category V,
τ(M, V) = AV ∑
λ: {L1,...,Lm}→V
Dλ × F(k, λ) is a 3-manifold invariant. Proof: Two ribbons leading to the same manifold via surgery on S are related by a sequence of Reidemeister moves and Kirby moves.
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Invariant of 3-manifold
Theorem (Reshetikhin, Turaev)
For a manifold M obtained by surgery on S3 along k, and a ribbon category V,
τ(M, V) = AV ∑
λ: {L1,...,Lm}→V
Dλ × F(k, λ) is a 3-manifold invariant. Proof: Two ribbons leading to the same manifold via surgery on S3 are related by a sequence of Reidemeister moves and Kirby moves.
. = . = . = . =
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Why ”quantum”?
It is easy to find algebraic objects (vector spaces, modules) with the structure of a ribbon category (usual tensor product, duality). These simple examples however lead to trivial knots invariants. Ex: vector spaces cV,W(v ⊗ w) = w ⊗ v V ⊗ W
cV,W
idV⊗W
- ❑
❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ W ⊗ V
cW,V
- V ⊗ W
Quantum groups (in the representation theory of Lie algebras) lead to non-trivial ribbon categories. And powerful invairiants in C.
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Algorithmic aspects of quantum invariants
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Computation of the invariants
Pushing a bit more the construction, we get the Turaev-Viro invariant (== |τ|2) defined directly on the triangulation: Pachner moves Quantum groups lead to invariants parameterised by an integer r ≥ 3.
- r = 3, polynomial time algorithm (reduced to homology),
- r = 4, # P hard,
- fully parameterised algorithm in treewidth: O((r + 1)6k × poly(n))
[Burton, M., Spreer ’15]
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Conclusion
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Take away
Turn a qualitative theory into a quantitative computation via Reidemeister moves, surgery, Kirby moves, Pachner moves, etc.
. = . = . = . =
Pachner moves Interesting complexity theory for the computation of quantum invariants. Thank you!
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Take away
Turn a qualitative theory into a quantitative computation via Reidemeister moves, surgery, Kirby moves, Pachner moves, etc.
. = . = . = . =
Pachner moves Interesting complexity theory for the computation of quantum invariants. Thank you!
17