Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture never trivial

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

not trivial R2

∗ 3 order three

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

R2

∗ 3 order three

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

trivial R2

∗ 3 order three

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

trivial

πOrb 1 =^Z/2Z

R2

∗ 3 order three

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

trivial

πOrb 1 =^Z/2Z

R2

∗ 3 order three

not trivial R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

trivial

πOrb 1 =^Z/2Z

R2

∗ 3 order three

not trivial R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

trivial

πOrb 1 =^Z/2Z

R2

∗ 3 order three

trivial

πOrb 1 =^Z/3Z

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Two-dimensional orbifolds

“Definition”. An ^orbifold is locally modeled on the standard Euclidean space modulo an action of some finite group.

Main example. ^Z/_cZ acts onR²by rotation around a fixed pointc, e.g.:

Orb=R².

Z/_2Z )•c*

Z/2Zaction R2

X_Orb≈ •c cone point R²/z=−z

)*

Philosophy. Thec’s are in between regular points and punctures:

R2

∗

· regular

trivial

πOrb 1 = 1

R2

∗ 2 order two

trivial

πOrb 1 =^Z/2Z

R2

∗ 3 order three

trivial

πOrb 1 =^Z/3Z

R2

∗

∞ puncture never trivial

πOrb 1 =Z

If we draw tangles in2Orb, then:

=

2 2

2 2

Im Dokument Link invariants and orbifolds (Seite 30-46)