Twisting of Torus Knots Speaker: Logan Godkin · Twisting of Torus Knots Speaker: Logan Godkin Supervisor: Dr. Mohamed Ait Nouh [email protected] Department of Mathematics University

Twisting of Torus Knots

Speaker: Logan Godkin

Supervisor: Dr. Mohamed Ait Nouh

[email protected]

Department of Mathematics

University of California, Riverside

Riverside, CA 92521 USA

Twisting of Torus Knots – p.1/30

http://prosper.sourceforge.net/

Plan

twisted knots: Definition.


Plan


Connection between twisting of knots and 4-manifolds.


Plan



Signature of (p, p + 6)-torus knots.


Plan



Signature of (p, p + 6)-torus knots.

(p, p + 6)-torus knots is not twisted for p ≥ 11.


What is a Twisting operation?

If ωg = 2:

C

(+1)−twisting

(−1)−twisting

C


What is a Twisting operation?

If ωg = 2:

C

(+1)−twisting

(−1)−twisting

C

If ωg = 3:

(+1)−twisting

(−1)−twisting

C

C


What is a Twisted Knot?

Trefoil Knot T(3, 2)

Unknot

U(+1)−twist

+ 1

C

Unknotting

(+1)−twisting

twist knotC


Yves Mathieu 1990: Is any knot in S3 twisted ?

Y. Ohyama: Any knot can be untied by at most twotwisting operations.


Yves Mathieu 1990: Is any knot in S3 twisted ?

Y. Ohyama: Any knot can be untied by at most twotwisting operations.

Ait Nouh-Yasuhara(2000): T (p, p + 4) (p ≥ 9) can not beuntied by a single twisting.


Twisting of torus knots

Miyazaki-Motegi(1999):

If T (p, q) (0 < p < q and q 6= kp ± 1) is n-twisted, thenn = ±1.



Miyazaki-Motegi(1999):

If T (p, q) (0 < p < q and q 6= kp ± 1) is n-twisted, thenn = ±1.

Ait Nouh- Yasuhara(2004):

If T (p, q) (0 < p < q and q 6= kp ± 1) is n-twisted, thenn = +1.



Ait Nouh-Godkin(2009):

T (p, p + 6) (p ≥ 11) can not be untied by a single twisting.

p strands half−twistsp+6

T(p,p+6)



Ait Nouh-Yasuhara old conjecture (2009): T (p, q)(0 < p < q) can not be untied by a single twisting forq 6= np ± 1.

Goda-Hayashi-Song: T (p, p + 2) is untied by(−1, p + 1)-twisting !



Ait Nouh-Godkin conjecture (2009): T (p, q) (0 < p < q) can

not be untied by a single twisting for q 6= np±1 and q 6= p+2.


4-manifolds

X4 is a simply-connected (π1(X4, Z) = {1}), orientable

and closed 4-manifold.


4-manifolds



H2(X4, Z) =< γ1, γ2, ..., γn > free abelian group of n

generators.


4-manifolds




generators.

The intersection matrix (form) is: M = (γi.γj)1≤i,j≤n


4-manifolds




generators.


The signature of M is called the signature of X4, and isgiven by the formula: σ(X4) = b+

2 − b−2


4-manifolds




generators.


The signature of M is called the signature of X4, and isgiven by the formula: σ(X4) = b+

2 − b−2

Example: H2(CP 2, Z) =< γ1 >, where γ1 =< CP 1 > andCP 1 ∼= S2 and such that γ1.γ1 = +1.


Twisting and sliceness in CP 2:

K

U ω,

B4

D = K2

Disk D2

K(−1 )

− int B4

CP2

+1

CP2

Kirby Calculus

[D] = ωγ ∈ H2(CP 2 − intB4, S3, Z).


Twisting and sliceness in CP 2:

K

U ω,

B4

D = K2

Disk D2

K(−1 )

− int B4

CP2

+1

CP2

Kirby Calculus

[D] = ωγ ∈ H2(CP 2 − intB4, S3, Z).

H2(CP 2 − intB4, S3, Z) =< γ > with γ.γ = +1.


Twisting and sliceness in CP2

K

U ω,

B4

D = K2

Disk D2

K)

− int B4

CP

CP2

Kirby Calculus

2−1

(+1

[D] = ωγ̄ ∈ H2(CP2− intB4, S3, Z).


Twisting and sliceness in CP2

K

U ω,

B4

D = K2

Disk D2

K)

− int B4

CP

CP2

Kirby Calculus

2−1

(+1

[D] = ωγ̄ ∈ H2(CP2− intB4, S3, Z).

H2(CP2− intB4, S3, Z) =< γ̄ > with γ̄.γ̄ = −1.


Twisting and sliceness in nCP 2:

K

U B4

D = K2

Disk D2

K

− int B4

CP2

CP2

Kirby Calculus

(−n

n

n

n

n CP2= CP

2# CP

2# ....... # CP

2

)ω, (−n < 0 )

n times

[D] = ωγ1 + ωγ2 + .... + ωγn ∈ H2(nCP 2 − intB4, S3, Z).


Twisting and sliceness in nCP 2:

K

U B4

D = K2

Disk D2

K

− int B4

CP2

CP2

Kirby Calculus

(−n

n

n

n

n CP2= CP

2# CP

2# ....... # CP

2

)ω, (−n < 0 )

n times

[D] = ωγ1 + ωγ2 + .... + ωγn ∈ H2(nCP 2 − intB4, S3, Z).

H2(nCP 2 − intB4, S3, Z) =< γ1, γ2, ..., γn > withγi.γi = +1 for i = 1, 2, ..., n.


Twisting and sliceness in nCP2:

K

U B4

D = K2

Disk D2

K

− int B4

CP2

CP2

Kirby Calculus

n

n

n CP2= CP

2# CP

2# ....... # CP

2

)ω, (n(n > 0 )

n−

n times

[D] = ωγ̄1 + ωγ̄2 + .... + ωγ̄n ∈ H2(nCP2− intB4, S3, Z).


Twisting and sliceness in nCP2:

K

U B4

D = K2

Disk D2

K

− int B4

CP2

CP2

Kirby Calculus

n

n

n CP2= CP

2# CP

2# ....... # CP

2

)ω, (n(n > 0 )

n−

n times

[D] = ωγ̄1 + ωγ̄2 + .... + ωγ̄n ∈ H2(nCP2− intB4, S3, Z).

H2(nCP 2 − intB4, S3, Z) =< γ̄1, γ̄2, ..., γ̄n > withγ̄i.γ̄i = −1 for i = 1, 2, ..., n.


Twisting and sliceness in S2 × S2

K

U B4

D = K2

Disk D2

K

− int B42

Sx2

(2n )ω,

S0

0

S2x S

2

Kirby Calculus

K(2n>0,ω)

→ K2 =⇒ [D2] = −ωα + nω2 β


Twisting and sliceness in S2 × S2

K

U B4

D = K2

Disk D2

K

− int B42

Sx2

(2n )ω,

S0

0

S2x S

2

Kirby Calculus

K(2n>0,ω)

→ K2 =⇒ [D2] = −ωα + nω2 β

K(2,2)→ K2 =⇒ [D2] = −2α+2β ∈ H2(S

2×S2−intB4, S3, Z)


Twisting and 4-manifolds: Main idea

K

U (n1

ω, 1 )U (n1 ω, 1

) .....(n2, ω2) (nm , ωm)

K

p CP2# qCP2

# xS2r S2

B4

− int B4

D = K2

Disk D2

X4=

K bounds a disk in a punctured standard 4-manifolds i.e. of

the form pCP 2#qCP 2#rS2 × S2 − intB4


Signature of (p, p + 6)-torus knots:

Letσ = σ(T (p, p + 6)).


Signature of (p, p + 6)-torus knots:

Letσ = σ(T (p, p + 6)).

σ =

−(p − 1)(p + 7)

2if p ≡ 5 (mod.12),

−(p − 1)(p + 7)

2− 6 if p ≡ 7 or 11(mod.12).


Signature of (p, p + 6)-torus knots: Proof

Ait Nouh-Yasuhara:

σ(T (p, p + r)) = −(p − 1)(p + r + 1)

2

+2

r/2∑

i=1

([

(2i − 1)p

2r

]

−

[

(2i − 1)p + r

2r

])

.



Ait Nouh-Yasuhara:

σ(T (p, p + r)) = −(p − 1)(p + r + 1)

2

+2

r/2∑

i=1

([

(2i − 1)p

2r

]

−

[

(2i − 1)p + r

2r

])

.

σ = −(p − 1)(p + 7)

2+2

3∑

i=1

([

(2i − 1)p

12

]

−

[

(2i − 1)p + 6

12

])

.



σ = −(p − 1)(p + 7)

2+ 2

(

[ p

12

]

−

[

p + 6

12

])

+

2

(

[p

4

]

−

[

p + 2

4

])

+ 2

([

5p

12

]

−

[

5p + 6

12

])

.



σ = −(p − 1)(p + 7)

2+ 2

(

[ p

12

]

−

[

p + 6

12

])

+

2

(

[p

4

]

−

[

p + 2

4

])

+ 2

([

5p

12

]

−

[

5p + 6

12

])

.

p = 12n + 5.


(p, p + 6)-torus knots is not twisted: p ≡ 5 (mod. 12)

Assume for a contradiction that T (p, p + 6) is(+1, ω)-twisted.


(p, p + 6)-torus knots is not twisted: p ≡ 5 (mod. 12)

Assume for a contradiction that T (p, p + 6) is(+1, ω)-twisted.

(∆, ∂∆) ⊂ (CP 2 − B4, S3) such that[∆] = ωγ̄ ∈ H2(CP 2 − B4, S3).

δ

B4

CP2 − B4

∆∆ = T(p,p+6)


ω is odd

T (−5, 1) ∼= U(−1,5)→ T (−5, 6) ∼= T (−6, 5)

(−2n,6)→

T (−6, 12n + 5) ∼= T (−p, 6)(−1,p)→ T (−p, p + 6).


ω is odd

T (−5, 1) ∼= U(−1,5)→ T (−5, 6) ∼= T (−6, 5)

(−2n,6)→

T (−6, 12n + 5) ∼= T (−p, 6)(−1,p)→ T (−p, p + 6).

(D, ∂D) ⊂ (CP 2#S2 × S2#CP 2 − B4, S3) such that:


ω is odd

T (−5, 1) ∼= U(−1,5)→ T (−5, 6) ∼= T (−6, 5)

(−2n,6)→

T (−6, 12n + 5) ∼= T (−p, 6)(−1,p)→ T (−p, p + 6).

(D, ∂D) ⊂ (CP 2#S2 × S2#CP 2 − B4, S3) such that:

[D] = 5γ1+6α+6nβ+pγ2 ∈ H2(CP 2#S2×S2#CP 2−B4, S3).


ω is odd

[ ] =

T(p,p+6)

T(−p,p+6)

4

4

− int B4

∆

D

∆ ω γ CP − int B2

[ ] =D 5 γ + 6 α + 6n pγβ + 1 2

M

Let M4 = CP 2#S2 × S2#CP 2 and X4 = CP 2#M4.


ω is odd

[ ] =

T(p,p+6)

T(−p,p+6)

4

4

− int B4

∆

D


[ ] =D 5 γ + 6 α + 6n pγβ + 1 2

M


X4 = CP 2#S2 × S2#CP 2#CP 2.


ω is odd

[ ] =

T(p,p+6)

T(−p,p+6)

4

4

− int B4

∆

D


[ ] =D 5 γ + 6 α + 6n pγβ + 1 2

M


X4 = CP 2#S2 × S2#CP 2#CP 2.

The sphere [S2] = [D ∪ ∆] ⊂ X4 satisfies:

[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).


Kikuchi’s Theorem

X4 be a closed, oriented and smooth 4-manifold such that:

H1(X4) has no 2-torsion; and


Kikuchi’s Theorem



b±12 ≤ 3


Kikuchi’s Theorem



b±12 ≤ 3

If [S2] ∈ H2(X4, Z) is a characteristic class then:

[S2].[S2] = σ(X4)


ω is odd

The sphere [S2] = [D ∪ ∆] satisfies:


ω is odd


[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).


ω is odd


[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).

X4 = CP 2#S2 × S2#CP 2#CP 2.


ω is odd


[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).

X4 = CP 2#S2 × S2#CP 2#CP 2.

Kikuchi’s theorem: [S2].[S2] = σ(X4)


ω is odd


[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).

X4 = CP 2#S2 × S2#CP 2#CP 2.

Kikuchi’s theorem: [S2].[S2] = σ(X4)

25 + 2 × 6 × 6n + p2 − ω2 = 1


ω is odd

25 + 2 × 6 × 6n + p2 − ω2 = 1.


ω is odd

25 + 2 × 6 × 6n + p2 − ω2 = 1.

p = 12n + 5.


ω is odd

25 + 2 × 6 × 6n + p2 − ω2 = 1.

p = 12n + 5.

p2 + 6p − 6 = ω2.


ω is odd

25 + 2 × 6 × 6n + p2 − ω2 = 1.

p = 12n + 5.

p2 + 6p − 6 = ω2.

p2 + 6p − 6 is not a perfect square (contradiction).


ω is even: (Gilmer-Viro’ theorem)

g (here g = 2)

K

K = δ gB4

S3= X 4

Σ

Σ

[ Σ ]g = ξ = Σi=1

i=nai γi H2 (X 4

, Z)

X 4compact and oriented.

2/ξ =⇒|ξ2

2− σ(X4) − σ(k) |≤ dimH2(X

4; Z2) + 2g



g (here g = 2)

K

K = δ gB4

S3= X 4

Σ

Σ

[ Σ ]g = ξ = Σi=1

i=nai γi H2 (X 4

, Z)


2/ξ =⇒|ξ2

2− σ(X4) − σ(k) |≤ dimH2(X

4; Z2) + 2g

d/ξ =⇒| d2−1

2.d2 .ξ2 − σ(X4) − σd(k) |≤ dimH2(X4; Zd) + 2g

(d ≥ 3 is a prime).



g (here g = 2)

K

K = δ gB4

S3= X 4

Σ

Σ

[ Σ ]g = ξ = Σi=1

i=nai γi H2 (X 4

, Z)


2/ξ =⇒|ξ2

2− σ(X4) − σ(k) |≤ dimH2(X

4; Z2) + 2g

d/ξ =⇒| d2−1

2.d2 .ξ2 − σ(X4) − σd(k) |≤ dimH2(X4; Zd) + 2g

(d ≥ 3 is a prime).

Tristram’s d-signature of a knot:

σd(k) = σ(ξdM + ξdMt), with ξd = e

2iπ(d−1)d .


ω is even

Gilmer-Viro’s theorem:

| −ω2

2− σ(T (p, p + 6)) − σ(CP 2) |≤ 2


ω is even


| −ω2

2− σ(T (p, p + 6)) − σ(CP 2) |≤ 2

ω2

2− 3 ≤ −σ ≤

ω2

2+ 1


ω is even


| −ω2

2− σ(T (p, p + 6)) − σ(CP 2) |≤ 2

ω2

2− 3 ≤ −σ ≤

ω2

2+ 1

−σ(T (p, p + 6)) =ω2

2

or

−σ(T (p, p + 6)) =ω2

2− 2


ω is even

By Proposition 2.2:

σ(T (p, p + 6)) = −(p − 1)(p + 7)

2if p ≡ 5 (mod. 12).


ω is even

By Proposition 2.2:

σ(T (p, p + 6)) = −(p − 1)(p + 7)

2if p ≡ 5 (mod. 12).

(p − 1)(p + 7) = ω2

or

(p − 1)(p + 7) − 4 = ω2


ω is even

By Proposition 2.2:

σ(T (p, p + 6)) = −(p − 1)(p + 7)

2if p ≡ 5 (mod. 12).

(p − 1)(p + 7) = ω2

or

(p − 1)(p + 7) − 4 = ω2

Neither (p − 1)(p + 7) nor (p − 1)(p + 7) − 4 is a perfectsquare, by a discriminant argument.


ω is even

By Proposition 2.2:

σ(T (p, p + 6)) = −(p − 1)(p + 7)

2if p ≡ 5 (mod. 12).

(p − 1)(p + 7) = ω2

or

(p − 1)(p + 7) − 4 = ω2

Neither (p − 1)(p + 7) nor (p − 1)(p + 7) − 4 is a perfectsquare, by a discriminant argument.

Contradiction.


(p, p + 6)-torus knots is not twisted: other cases

(p, p + 6)-torus knots is not twisted: p ≡ 7 (mod. 12)(similar argument).









p ≡ 9 and p ≡ 3 are thrown up because p and p + 6 wouldnot be coprime.


Thanks

Organizers of the conference:

Prof. Lew Ludwig and Collin Adams.


Thanks



Dr. Mohamed Ait Nouh (Supervisor and Coauthor).


Thanks



Dr. Mohamed Ait Nouh (Supervisor and Coauthor).

Audience !


Documents

Twisting of Torus Knots Speaker: Logan Godkin · Twisting of Torus Knots Speaker: Logan Godkin Supervisor: Dr. Mohamed Ait Nouh [email protected] Department of Mathematics University