Link invariants and
Z/
2Z-orbifolds
Or: What makes typesABCDspecial?
Daniel Tubbenhauer
cone strands
Joint work in progress (take it with a grain of salt) with Catharina Stroppel and Arik Wilbert (Based on an idea of Mikhail Khovanov)
January 2018
Khovanov style homologies
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics The type A world
Weyl group♥Quantum group
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics The type A world
Weyl group♥Quantum group Outside of type A
Weyl group side Quantum group side
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics The type A world
Weyl group♥Quantum group Outside of type A
Weyl group side Quantum group side
Homologies!!
for links??
Homologies??
for links!!
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics The type A world
Weyl group♥Quantum group Outside of type A
Weyl group side Quantum group side
Homologies!!
for links??
Homologies??
for links!!
??
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics The type A world
Weyl group♥Quantum group Outside of type A
Weyl group side Quantum group side
Homologies!!
for links??
Homologies??
for links!!
??
Yes Yes
Yes
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
Khovanov style homologies Hecke
algebras Lie theory
(Singular) TQFTs
Quantum groups Commutative
algebra
Geometry More...
Physics The type A world
Weyl group♥Quantum group Outside of type A
Weyl group side Quantum group side
Homologies!!
for links??
Homologies??
for links!!
??
Yes Yes
Yes
My beloved gadget with many connections.
A quantum group of typeE7
is typeA-braided!?
“Homology easy, topology hard”
“Homology hard, topology easy”
1 Tangle diagrams ofZ/2Z-orbifold tangles Diagrams
Tangles inZ/2Z-orbifolds
2 Topology of Artin braid groups The Artin braid groups: algebra Hyperplanes vs. configuration spaces
3 Invariants
Reshetikhin–Turaev-like theory for some coideals Polynomials and homologies for Z/2Z-orbifold tangles
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Tangle diagrams with cone strands
LetcTanbe the monoidal category defined as follows.
Generators. Object generators{+,−,c}, morphism generators
+ +
+ +
,
+ +
+ +
usual crossings
,
− +
,
+ −
,
− +
,
+ −
usual cups and caps
,
+ c
c +
,
+ c
c +
,
+ c
c +
,
+ c
c +
cone crossings
Relations. Reidemeister type relations , and theZ/2Z-relations:
= and =
Examples.
= =
Unknot
, =
Essential unknot
Hopf link
,
Essential Hopf link
Example.
=
Exercise. The relations are actually equivalent.
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
R2
∗ c
cone
R2
∗
× puncture
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
R2
∗ c
cone
R2
∗
× puncture
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial R2
∗ c
cone
R2
∗
× puncture
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
R2
∗
× puncture
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
R2
∗
× puncture
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
R2
∗
× puncture never trivial
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
not trivial R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone
trivial R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone trivial
πOrb 1 =Z/2Z
R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Z
/
2Z-orbifolds
“Definition”. An orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.
Main example. Z/2Z acts onR2by rotation byπaround a fixed pointc:
c1Orb=R2.
Z/2Z )•c*
Z/2Zaction R2
Xc1Orb≈ •c cone point R2/z=−z
)*
Philosophy. cis half-way in between a regular point and a puncture:
R2
∗
· regular
trivial
πOrb 1 = 1
R2
∗ c
cone trivial
πOrb 1 =Z/2Z
R2
∗
× puncture never trivial
πOrb 1 =Z
If we draw tangles inc1Orb×[0,1], then:
=
Pioneers of algebra
Let Γ be a Coxeter graph .
Artin ∼1925, Tits ∼1961++. The Artin braid groups and its Coxeter group quotients are given by generators-relations:
ArΓ=hbi| · · ·bibjbi
| {z }
mijfactors
=· · ·bjbibj
| {z }
mijfactors
i
WΓ=hsi|si2= 1,· · ·sisjsi
| {z }
mijfactors
=· · ·sjsisj
| {z }
mijfactors
i
Artin braid groups generalize classical braid groups, Coxeter groups Weyl groups.
We want to understand these better.
Only algebra: No “interpretation” yet.
Pioneers of algebra
Let Γ be a Coxeter graph .
Artin ∼1925, Tits ∼1961++. The Artin braid groups and its Coxeter group quotients are given by generators-relations:
ArΓ=hbi| · · ·bibjbi
| {z }
mijfactors
=· · ·bjbibj
| {z }
mijfactors
i
WΓ=hsi|si2= 1,· · ·sisjsi
| {z }
mijfactors
=· · ·sjsisj
| {z }
mijfactors
i
Artin braid groups generalize classical braid groups, Coxeter groups Weyl groups.
We want to understand these better.
Only algebra:
No “interpretation” yet.
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
•
•
•
• •
•
1WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2.
Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then: π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
•
•
•
• •
1 s t
•
WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2.
Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then: π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
•
•
•
• •
1 s t st
ts
•
WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2.
Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then: π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
•
•
•
• •
1 s t st ts
sts t=st
•
WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2.
Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then: π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
•
•
•
• •
1 s t st ts
sts t=st
•
WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2.
Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then: π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality.
(Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
∗WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2. Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then:
π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
∗WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2. Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then:
π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
•
∗WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2. Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then:
π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
∗ bs
bt
•
WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2. Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then:
π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek ∼1983. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
I follow hyperplanes
WA2 =hs,tiacts faithfully onR2 by reflecting in hyperplanes (for each reflection):
∗ bs
bt
•
WA2 acts freely onMA
2 =R2\hyperplanes. SetNA
2 =MA
2/WA2. Complexifying the action: R2 C2, MA
2 MCA
2,NA
2 NCA
2. Then:
π1(NCA
2)∼=ArA
2 =hbs,bt |bsbtbs =btbsbti
Coxeter∼1934, Tits∼1961. This works in ridiculous generality. (Up to some minor technicalities in the infinite case.)
Brieskorn ∼1971, van der Lek∼1983. This works in ridiculous generality.
(Up to some minor technicalities in the infinite case.)
Configuration spaces
Artin ∼1925. There is a topological model of ArA via configuration spaces.
Example. TakeConfA
2 = (R2)3\fat diagonal
S3. Thenπ1(ConfA
2)∼=ArA2. Philosophy. Having a configuration spaces is the same as having braid diagrams:
y1 y2 y3
y1 y2 y3
σ=(13)
a usual braid R2 R2
time
Crucial. Note that – by explicitly calculating the equations defining the hyperplanes – one can directly check that:
“Hyperplane picture equals configuration space picture.”
Lambropoulou∼1993, tom Dieck∼1998, Allcock∼2002.
Type A Ae B=C Be Ce D De
Orbifold feature none × × ×,c ×,× c c,c
Additional inside: Works for tangles as well. In those cases one can compute the hyperplanes!
This is very special for (affine) typesABCD. Hope.
The same works for Coxeter diagrams which are “locally typeABCD”, e.g.:
4 4 4
• •
•
•
• •
•
•
• •
• •
• •
•
b+ •
bc b0c
b× b+
c c c
× + + + + ×+ ×+ +
e
B4 eD4 D4 eC3 D4
b+7→ bc7→ bc07→ b×7→
But we can’t compute the hyperplanes...
In words: TheZ/2Z-orbifolds provide the
framework to study Artin braid groups of classical (affine) type and their “glued-generalizations”.
Example.
bib0i =bi0bi, if
•
•
bi b0 i
!
Configuration spaces
Artin ∼1925. There is a topological model of ArA via configuration spaces.
Example. TakeConfA
2 = (R2)3\fat diagonal
S3. Thenπ1(ConfA
2)∼=ArA2. Philosophy. Having a configuration spaces is the same as having braid diagrams:
y1 y2 y3
y1 y2 y3
σ=(13)
a usual braid R2 R2
time
Crucial. Note that – by explicitly calculating the equations defining the hyperplanes – one can directly check that:
“Hyperplane picture equals configuration space picture.”
Lambropoulou∼1993, tom Dieck∼1998, Allcock∼2002.
Type A Ae B=C Be Ce D De
Orbifold feature none × × ×,c ×,× c c,c
Additional inside: Works for tangles as well.
In those cases one can compute the hyperplanes!
This is very special for (affine) typesABCD.
Hope.
The same works for Coxeter diagrams which are “locally typeABCD”, e.g.:
4 4 4
• •
•
•
• •
•
•
• •
• •
• •
•
b+ •
bc b0c
b× b+
c c c
× + + + + ×+ ×+ +
e
B4 eD4 D4 eC3 D4
b+7→ bc7→ bc07→ b×7→
But we can’t compute the hyperplanes...
In words: TheZ/2Z-orbifolds provide the
framework to study Artin braid groups of classical (affine) type and their “glued-generalizations”.
Example.
bib0i =bi0bi, if
•
•
bi b0 i
!
