Twisting of Torus Knots
Speaker: Logan Godkin
Supervisor: Dr. Mohamed Ait Nouh
Department of Mathematics
University of California, Riverside
Riverside, CA 92521 USA
Twisting of Torus Knots – p.1/30
Plan
twisted knots: Definition.
Twisting of Torus Knots – p.2/30
Plan
twisted knots: Definition.
Connection between twisting of knots and 4-manifolds.
Twisting of Torus Knots – p.2/30
Plan
twisted knots: Definition.
Connection between twisting of knots and 4-manifolds.
Signature of (p, p + 6)-torus knots.
Twisting of Torus Knots – p.2/30
Plan
twisted knots: Definition.
Connection between twisting of knots and 4-manifolds.
Signature of (p, p + 6)-torus knots.
(p, p + 6)-torus knots is not twisted for p ≥ 11.
Twisting of Torus Knots – p.2/30
What is a Twisting operation?
If ωg = 2:
C
(+1)−twisting
(−1)−twisting
C
Twisting of Torus Knots – p.3/30
What is a Twisting operation?
If ωg = 2:
C
(+1)−twisting
(−1)−twisting
C
If ωg = 3:
(+1)−twisting
(−1)−twisting
C
C
Twisting of Torus Knots – p.3/30
What is a Twisted Knot?
Trefoil Knot T(3, 2)
Unknot
U(+1)−twist
+ 1
C
Unknotting
(+1)−twisting
twist knotC
Twisting of Torus Knots – p.4/30
Yves Mathieu 1990: Is any knot in S3 twisted ?
Y. Ohyama: Any knot can be untied by at most twotwisting operations.
Twisting of Torus Knots – p.5/30
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 – p.5/30
Twisting of torus knots
Miyazaki-Motegi(1999):
If T (p, q) (0 < p < q and q 6= kp ± 1) is n-twisted, thenn = ±1.
Twisting of Torus Knots – p.6/30
Twisting of torus knots
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.
Twisting of Torus Knots – p.6/30
Twisting of torus knots
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)
Twisting of Torus Knots – p.7/30
Twisting of torus knots
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 !
Twisting of Torus Knots – p.8/30
Twisting of torus knots
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.
Twisting of Torus Knots – p.9/30
4-manifolds
X4 is a simply-connected (π1(X4, Z) = {1}), orientable
and closed 4-manifold.
Twisting of Torus Knots – p.10/30
4-manifolds
X4 is a simply-connected (π1(X4, Z) = {1}), orientable
and closed 4-manifold.
H2(X4, Z) =< γ1, γ2, ..., γn > free abelian group of n
generators.
Twisting of Torus Knots – p.10/30
4-manifolds
X4 is a simply-connected (π1(X4, Z) = {1}), orientable
and closed 4-manifold.
H2(X4, Z) =< γ1, γ2, ..., γn > free abelian group of n
generators.
The intersection matrix (form) is: M = (γi.γj)1≤i,j≤n
Twisting of Torus Knots – p.10/30
4-manifolds
X4 is a simply-connected (π1(X4, Z) = {1}), orientable
and closed 4-manifold.
H2(X4, Z) =< γ1, γ2, ..., γn > free abelian group of n
generators.
The intersection matrix (form) is: M = (γi.γj)1≤i,j≤n
The signature of M is called the signature of X4, and isgiven by the formula: σ(X4) = b+
2 − b−2
Twisting of Torus Knots – p.10/30
4-manifolds
X4 is a simply-connected (π1(X4, Z) = {1}), orientable
and closed 4-manifold.
H2(X4, Z) =< γ1, γ2, ..., γn > free abelian group of n
generators.
The intersection matrix (form) is: M = (γi.γj)1≤i,j≤n
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 of Torus Knots – p.10/30
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 of Torus Knots – p.11/30
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 of Torus Knots – p.11/30
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 of Torus Knots – p.12/30
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 of Torus Knots – p.12/30
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 of Torus Knots – p.13/30
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 of Torus Knots – p.13/30
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 of Torus Knots – p.14/30
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 of Torus Knots – p.14/30
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 of Torus Knots – p.15/30
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 of Torus Knots – p.15/30
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
Twisting of Torus Knots – p.16/30
Signature of (p, p + 6)-torus knots:
Letσ = σ(T (p, p + 6)).
Twisting of Torus Knots – p.17/30
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).
Twisting of Torus Knots – p.17/30
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
])
.
Twisting of Torus Knots – p.18/30
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
])
.
σ = −(p − 1)(p + 7)
2+2
3∑
i=1
([
(2i − 1)p
12
]
−
[
(2i − 1)p + 6
12
])
.
Twisting of Torus Knots – p.18/30
Signature of (p, p + 6)-torus knots: Proof
σ = −(p − 1)(p + 7)
2+ 2
(
[ p
12
]
−
[
p + 6
12
])
+
2
(
[p
4
]
−
[
p + 2
4
])
+ 2
([
5p
12
]
−
[
5p + 6
12
])
.
Twisting of Torus Knots – p.19/30
Signature of (p, p + 6)-torus knots: Proof
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.
Twisting of Torus Knots – p.19/30
(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.
Twisting of Torus Knots – p.20/30
(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)
Twisting of Torus Knots – p.20/30
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).
Twisting of Torus Knots – p.21/30
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:
Twisting of Torus Knots – p.21/30
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).
Twisting of Torus Knots – p.21/30
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.
Twisting of Torus Knots – p.22/30
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.
X4 = CP 2#S2 × S2#CP 2#CP 2.
Twisting of Torus Knots – p.22/30
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.
X4 = CP 2#S2 × S2#CP 2#CP 2.
The sphere [S2] = [D ∪ ∆] ⊂ X4 satisfies:
[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).
Twisting of Torus Knots – p.22/30
Kikuchi’s Theorem
X4 be a closed, oriented and smooth 4-manifold such that:
H1(X4) has no 2-torsion; and
Twisting of Torus Knots – p.23/30
Kikuchi’s Theorem
X4 be a closed, oriented and smooth 4-manifold such that:
H1(X4) has no 2-torsion; and
b±12 ≤ 3
Twisting of Torus Knots – p.23/30
Kikuchi’s Theorem
X4 be a closed, oriented and smooth 4-manifold such that:
H1(X4) has no 2-torsion; and
b±12 ≤ 3
If [S2] ∈ H2(X4, Z) is a characteristic class then:
[S2].[S2] = σ(X4)
Twisting of Torus Knots – p.23/30
ω is odd
The sphere [S2] = [D ∪ ∆] satisfies:
Twisting of Torus Knots – p.24/30
ω is odd
The sphere [S2] = [D ∪ ∆] satisfies:
[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).
Twisting of Torus Knots – p.24/30
ω is odd
The sphere [S2] = [D ∪ ∆] satisfies:
[S2] = 5γ1 + 6α + 6nβ + pγ2 + ωγ̄ ∈ H2(X4, Z).
X4 = CP 2#S2 × S2#CP 2#CP 2.
Twisting of Torus Knots – p.24/30
ω is odd
The sphere [S2] = [D ∪ ∆] satisfies:
[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)
Twisting of Torus Knots – p.24/30
ω is odd
The sphere [S2] = [D ∪ ∆] satisfies:
[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
Twisting of Torus Knots – p.24/30
ω is odd
25 + 2 × 6 × 6n + p2 − ω2 = 1.
Twisting of Torus Knots – p.25/30
ω is odd
25 + 2 × 6 × 6n + p2 − ω2 = 1.
p = 12n + 5.
Twisting of Torus Knots – p.25/30
ω is odd
25 + 2 × 6 × 6n + p2 − ω2 = 1.
p = 12n + 5.
p2 + 6p − 6 = ω2.
Twisting of Torus Knots – p.25/30
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).
Twisting of Torus Knots – p.25/30
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
Twisting of Torus Knots – p.26/30
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
d/ξ =⇒| d2−1
2.d2 .ξ2 − σ(X4) − σd(k) |≤ dimH2(X4; Zd) + 2g
(d ≥ 3 is a prime).
Twisting of Torus Knots – p.26/30
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
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 .
Twisting of Torus Knots – p.26/30
ω is even
Gilmer-Viro’s theorem:
| −ω2
2− σ(T (p, p + 6)) − σ(CP 2) |≤ 2
Twisting of Torus Knots – p.27/30
ω is even
Gilmer-Viro’s theorem:
| −ω2
2− σ(T (p, p + 6)) − σ(CP 2) |≤ 2
ω2
2− 3 ≤ −σ ≤
ω2
2+ 1
Twisting of Torus Knots – p.27/30
ω is even
Gilmer-Viro’s theorem:
| −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
Twisting of Torus Knots – p.27/30
ω is even
By Proposition 2.2:
σ(T (p, p + 6)) = −(p − 1)(p + 7)
2if p ≡ 5 (mod. 12).
Twisting of Torus Knots – p.28/30
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
Twisting of Torus Knots – p.28/30
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.
Twisting of Torus Knots – p.28/30
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.
Twisting of Torus Knots – p.28/30
(p, p + 6)-torus knots is not twisted: other cases
(p, p + 6)-torus knots is not twisted: p ≡ 7 (mod. 12)(similar argument).
Twisting of Torus Knots – p.29/30
(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, p + 6)-torus knots is not twisted: p ≡ 11 (mod. 12)(similar argument).
Twisting of Torus Knots – p.29/30
(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, p + 6)-torus knots is not twisted: p ≡ 11 (mod. 12)(similar argument).
p ≡ 9 and p ≡ 3 are thrown up because p and p + 6 wouldnot be coprime.
Twisting of Torus Knots – p.29/30
Thanks
Organizers of the conference:
Prof. Lew Ludwig and Collin Adams.
Twisting of Torus Knots – p.30/30
Thanks
Organizers of the conference:
Prof. Lew Ludwig and Collin Adams.
Dr. Mohamed Ait Nouh (Supervisor and Coauthor).
Twisting of Torus Knots – p.30/30
Thanks
Organizers of the conference:
Prof. Lew Ludwig and Collin Adams.
Dr. Mohamed Ait Nouh (Supervisor and Coauthor).
Audience !
Twisting of Torus Knots – p.30/30