The Jones polynomial of
ribbon knots and symmetric unions
Michael Eisermann (Institut Fourier,Grenoble)
Christoph Lamm (Dresdner Bank,Frankfurt)
August 3,2007
AMSPTM International Meeting
Warsaw,July 31 – August 3,2007
Keywords
Symmetric unions of knots
(KinoshitaTerasaka 1957)
Ribbon knots & slice knots
(FoxMilnor 1957,1961,1965)
A twovariable reﬁnement of the Jones polynomial
(preliminary report)
Keywords
Symmetric unions of knots
(KinoshitaTerasaka 1957)
Ribbon knots & slice knots
(FoxMilnor 1957,1961,1965)
A twovariable reﬁnement of the Jones polynomial
(preliminary report)
Keywords
Symmetric unions of knots
(KinoshitaTerasaka 1957)
Ribbon knots & slice knots
(FoxMilnor 1957,1961,1965)
A twovariable reﬁnement of the Jones polynomial
(preliminary report)
Symmetric unions (KinoshitaTerasaka 1957)
(a) symmetric
sum,3
1
]3
1
(b) symmetric
union,6
1
(c) symmetric
union,8
20
(d) 8
8
(e) 9
27
(f) 9
27
Symmetric unions (KinoshitaTerasaka 1957)
(a) symmetric
sum,3
1
]3
1
(b) symmetric
union,6
1
(c) symmetric
union,8
20
(d) 8
8
(e) 9
27
(f) 9
27
Ribbon & slice (FoxMilnor 1957,1961,1965)
General setting
fsymmetric unionsg
¿ ?
fribbon knotsg
¿ ?
fslice knotsg
Every symmetric union is a ribbon knot:
(a) ribbon singularity
(b) 8
20
(c) 10
87
Ribbon & slice (FoxMilnor 1957,1961,1965)
General setting
fsymmetric unionsg
¿ ?
fribbon knotsg
¿ ?
fslice knotsg
Every symmetric union is a ribbon knot:
(a) ribbon singularity
(b) 8
20
(c) 10
87
Which ribbon knots are symmetric unions?
Empirical evidence
All “small” ribbon knots are symmetric unions.
Prime knots up to 10 crossings:21 ribbon knots.
Can be presented as symmetric unions.(Lamm 2000)
Twobridge knots:three inﬁnite families of ribbon knots.
(CassonGordon 1975;Lisca 2007)
Can be presented as symmetric unions.(Lamm 2005)
Question
Is every ribbon knot a symmetric union?
Which ribbon knots are symmetric unions?
Empirical evidence
All “small” ribbon knots are symmetric unions.
Prime knots up to 10 crossings:21 ribbon knots.
Can be presented as symmetric unions.(Lamm 2000)
Twobridge knots:three inﬁnite families of ribbon knots.
(CassonGordon 1975;Lisca 2007)
Can be presented as symmetric unions.(Lamm 2005)
Question
Is every ribbon knot a symmetric union?
Which ribbon knots are symmetric unions?
Empirical evidence
All “small” ribbon knots are symmetric unions.
Prime knots up to 10 crossings:21 ribbon knots.
Can be presented as symmetric unions.(Lamm 2000)
Twobridge knots:three inﬁnite families of ribbon knots.
(CassonGordon 1975;Lisca 2007)
Can be presented as symmetric unions.(Lamm 2005)
Question
Is every ribbon knot a symmetric union?
Which ribbon knots are symmetric unions?
Empirical evidence
All “small” ribbon knots are symmetric unions.
Prime knots up to 10 crossings:21 ribbon knots.
Can be presented as symmetric unions.(Lamm 2000)
Twobridge knots:three inﬁnite families of ribbon knots.
(CassonGordon 1975;Lisca 2007)
Can be presented as symmetric unions.(Lamm 2005)
Question
Is every ribbon knot a symmetric union?
Paradigm:“knots with extra structure”
Familiar theme:fknots + extra structureg
forget extra
!
structure
fknotsg
fdiagramsg !fknotsg:Reidemeister’s theorem.
fbraidsg !fknotsg:Theorems of Alexander and Markov.
Reduced alternating diagrams:MenascoThistlethwaite.
Legendrian knots,transverse knots,etc...
Here:
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
!
slice knots +
speciﬁc slice
?
?
y
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg !fslice knotsg
Paradigm:“knots with extra structure”
Familiar theme:fknots + extra structureg
forget extra
!
structure
fknotsg
fdiagramsg !fknotsg:Reidemeister’s theorem.
fbraidsg !fknotsg:Theorems of Alexander and Markov.
Reduced alternating diagrams:MenascoThistlethwaite.
Legendrian knots,transverse knots,etc...
Here:
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
!
slice knots +
speciﬁc slice
?
?
y
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg !fslice knotsg
Paradigm:“knots with extra structure”
Familiar theme:fknots + extra structureg
forget extra
!
structure
fknotsg
fdiagramsg !fknotsg:Reidemeister’s theorem.
fbraidsg !fknotsg:Theorems of Alexander and Markov.
Reduced alternating diagrams:MenascoThistlethwaite.
Legendrian knots,transverse knots,etc...
Here:
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
!
slice knots +
speciﬁc slice
?
?
y
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg !fslice knotsg
Paradigm:“knots with extra structure”
Familiar theme:fknots + extra structureg
forget extra
!
structure
fknotsg
fdiagramsg !fknotsg:Reidemeister’s theorem.
fbraidsg !fknotsg:Theorems of Alexander and Markov.
Reduced alternating diagrams:MenascoThistlethwaite.
Legendrian knots,transverse knots,etc...
Here:
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
!
slice knots +
speciﬁc slice
?
?
y
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg !fslice knotsg
Paradigm:“knots with extra structure”
Familiar theme:fknots + extra structureg
forget extra
!
structure
fknotsg
fdiagramsg !fknotsg:Reidemeister’s theorem.
fbraidsg !fknotsg:Theorems of Alexander and Markov.
Reduced alternating diagrams:MenascoThistlethwaite.
Legendrian knots,transverse knots,etc...
Here:
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
!
slice knots +
speciﬁc slice
?
?
y
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg !fslice knotsg
Paradigm:“knots with extra structure”
Familiar theme:fknots + extra structureg
forget extra
!
structure
fknotsg
fdiagramsg !fknotsg:Reidemeister’s theorem.
fbraidsg !fknotsg:Theorems of Alexander and Markov.
Reduced alternating diagrams:MenascoThistlethwaite.
Legendrian knots,transverse knots,etc...
Here:
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
!
slice knots +
speciﬁc slice
?
?
y
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg !fslice knotsg
Symmetric Reidemeister moves
Remark:the associated ribbon changes only up to isotopy.
Partial knots
Resolve
7!
and
7!
.Finally split
7!
.
D
7!
D
]D
+
7!
D
;D
+
Partial knots are invariant under Smoves.
det K = det K
det K
+
= a square
K = K
K
+
if only full twists on the axis
Question
Is there a similar geometric construction for ribbon knots?
Algebraically we have K = f(t) f(t
1
) for some f 2 Z[t
].
Partial knots
Resolve
7!
and
7!
.Finally split
7!
.
D
7!
D
]D
+
7!
D
;D
+
Partial knots are invariant under Smoves.
det K = det K
det K
+
= a square
K = K
K
+
if only full twists on the axis
Question
Is there a similar geometric construction for ribbon knots?
Algebraically we have K = f(t) f(t
1
) for some f 2 Z[t
].
Partial knots
Resolve
7!
and
7!
.Finally split
7!
.
D
7!
D
]D
+
7!
D
;D
+
Partial knots are invariant under Smoves.
det K = det K
det K
+
= a square
K = K
K
+
if only full twists on the axis
Question
Is there a similar geometric construction for ribbon knots?
Algebraically we have K = f(t) f(t
1
) for some f 2 Z[t
].
Symmetric union representations are not unique
The following diagrams are asymmetrically equivalent:
(a) 8
8
(b) 9
27
(a) Not symmetrically equivalent
because partial knots differ (4
1
6= 5
1
).
(b) Not symmetrically equivalent
although partial knots coincide (5
2
).
Symmetric union representations are not unique
The following diagrams are asymmetrically equivalent:
(a) 8
8
(b) 9
27
(a) Not symmetrically equivalent
because partial knots differ (4
1
6= 5
1
).
(b) Not symmetrically equivalent
although partial knots coincide (5
2
).
Symmetric union representations are not unique
The following diagrams are asymmetrically equivalent:
(a) 8
8
(b) 9
27
(a) Not symmetrically equivalent
because partial knots differ (4
1
6= 5
1
).
(b) Not symmetrically equivalent
although partial knots coincide (5
2
).
A variation of Kauffman’s ansatz
Consider arbitrary diagrams (not necessarily symmetric).
Crossings off the axis:
(A)
D
E
= A
+1
D
E
+A
1
D
E
:
Crossings on the axis:
(B)
D
E
= B
+1
D
E
+B
1
D
E
;
D
E
= B
1
D
E
+B
+1
D
E
:
Circle evaluation?
A variation of Kauffman’s ansatz
Consider arbitrary diagrams (not necessarily symmetric).
Crossings off the axis:
(A)
D
E
= A
+1
D
E
+A
1
D
E
:
Crossings on the axis:
(B)
D
E
= B
+1
D
E
+B
1
D
E
;
D
E
= B
1
D
E
+B
+1
D
E
:
Circle evaluation?
A variation of Kauffman’s ansatz
Consider arbitrary diagrams (not necessarily symmetric).
Crossings off the axis:
(A)
D
E
= A
+1
D
E
+A
1
D
E
:
Crossings on the axis:
(B)
D
E
= B
+1
D
E
+B
1
D
E
;
D
E
= B
1
D
E
+B
+1
D
E
:
Circle evaluation?
Circle evaluation
If C is a collection of n circles
having 2mintersections with the axis,then
hCi = (A
2
A
2
)
n1
B
2
B
2
A
2
A
2
m1
(C)
= (A
2
A
2
)
nm
(B
2
B
2
)
m1
:
Examples:
= 1;
=
1
B
2
B
2
;
*
+
= B
2
B
2
;
=
A
2
A
2
B
2
B
2
;
*
+
=
B
2
B
2
A
2
A
2
;
=
(A
2
A
2
)
2
B
2
B
2
:
Circle evaluation
If C is a collection of n circles
having 2mintersections with the axis,then
hCi = (A
2
A
2
)
n1
B
2
B
2
A
2
A
2
m1
(C)
= (A
2
A
2
)
nm
(B
2
B
2
)
m1
:
Examples:
= 1;
=
1
B
2
B
2
;
*
+
= B
2
B
2
;
=
A
2
A
2
B
2
B
2
;
*
+
=
B
2
B
2
A
2
A
2
;
=
(A
2
A
2
)
2
B
2
B
2
:
A twovariable reﬁnement of the Jones polynomial
Theorem (invariance)
The normalized map W:D!Z(A;B) deﬁned by
W(D):= hDi (A
3
)
Awrithe(D)
(B
3
)
Bwrithe(D)
is invariant under all Reidemeister moves respecting the axis.
Convention:A
2
= t
1
=
2
,B
2
= s
1
=
2
.
t
1
W
t
+1
W
= (t
1
=
2
t
1
=
2
)W
s
1
W
s
+1
W
= (s
1
=
2
s
1
=
2
)W
W
=
s
1
=
2
+s
1
=
2
t
1
=
2
+t
1
=
2
W
W
= 1
A twovariable reﬁnement of the Jones polynomial
Theorem (invariance)
The normalized map W:D!Z(A;B) deﬁned by
W(D):= hDi (A
3
)
Awrithe(D)
(B
3
)
Bwrithe(D)
is invariant under all Reidemeister moves respecting the axis.
Convention:A
2
= t
1
=
2
,B
2
= s
1
=
2
.
t
1
W
t
+1
W
= (t
1
=
2
t
1
=
2
)W
s
1
W
s
+1
W
= (s
1
=
2
s
1
=
2
)W
W
=
s
1
=
2
+s
1
=
2
t
1
=
2
+t
1
=
2
W
W
= 1
Properties of the Wpolynomial
Usual properties
W
D]D
0 = W
D
W
D
0.
mirror image:W
D
(s;t) = W
D
(s
1
;t
1
).
W
D
is insensitive to orientation reversal,mutation,ﬂypes.
Integrality
If D is a symmetric union,then W(D) 2 Z[s
1
;t
1
].
*
Theorem (Eisermann 2007)
For every ncomponent ribbon link L the Jones polynomial
V (L) is divisible by V (
n
) = (t
1=2
+t
1=2
)
n1
.
Properties of the Wpolynomial
Usual properties
W
D]D
0 = W
D
W
D
0.
mirror image:W
D
(s;t) = W
D
(s
1
;t
1
).
W
D
is insensitive to orientation reversal,mutation,ﬂypes.
Integrality
If D is a symmetric union,then W(D) 2 Z[s
1
;t
1
].
*
Theorem (Eisermann 2007)
For every ncomponent ribbon link L the Jones polynomial
V (L) is divisible by V (
n
) = (t
1=2
+t
1=2
)
n1
.
Properties of the Wpolynomial
Usual properties
W
D]D
0 = W
D
W
D
0.
mirror image:W
D
(s;t) = W
D
(s
1
;t
1
).
W
D
is insensitive to orientation reversal,mutation,ﬂypes.
Integrality
If D is a symmetric union,then W(D) 2 Z[s
1
;t
1
].
*
Theorem (Eisermann 2007)
For every ncomponent ribbon link L the Jones polynomial
V (L) is divisible by V (
n
) = (t
1=2
+t
1=2
)
n1
.
Special values of W(s;t)
Specializations in t
W
D
(s;) = 1 for 2 f1;i;e
2=3
g
@W
D
@t
(s;1) = 0
) W
D
1 is divisible by (t 1)
2
(t
2
+1)(t
2
+t +1).
Specializations in s
W
D
(t;t) = V
K
(t)
W
D
(1;t) = V
K
(t) V
K
+
(t)
) W
D
(1;1) = det(K) = det(K
) det(K
+
)
Special values of W(s;t)
Specializations in t
W
D
(s;) = 1 for 2 f1;i;e
2=3
g
@W
D
@t
(s;1) = 0
) W
D
1 is divisible by (t 1)
2
(t
2
+1)(t
2
+t +1).
Specializations in s
W
D
(t;t) = V
K
(t)
W
D
(1;t) = V
K
(t) V
K
+
(t)
) W
D
(1;1) = det(K) = det(K
) det(K
+
)
Two symmetric unions for 9
27
(a) D
1
9
27
(b) D
2
9
27
W
D
1
(s;t) = 1 +s
1
f(t) s
2
g(t)
W
D
2
(s;t) = 1 s
0
f(t) +s
1
g(t)
f(t) = t
5
3t
4
+6t
3
9t
2
+11t
1
12 +11t 9t
2
+6t
3
3t
4
+t
5
g(t) = t
4
2t
3
+3t
2
4t
1
+4 4t +3t
2
2t
3
+t
4
W(t;t) = t
5
+3t
4
5t
3
+7t
2
8t
1
+9 7t +5t
2
3t
3
+t
4
W(1;t) = (t t
2
+2t
3
t
4
+t
5
t
6
)(t
1
t
2
+2t
3
t
4
+t
5
t
6
)
Summary
Algebraically,a ribbon knot K looks like a connected sum
of a knot K
+
with its mirror image K
.
Geometrically,this is modelled by symmetric unions.
This allows us to deﬁne partial knots and a reﬁned
polynomial W(s;t) that keep track of the symmetry.
Summary
Algebraically,a ribbon knot K looks like a connected sum
of a knot K
+
with its mirror image K
.
Geometrically,this is modelled by symmetric unions.
This allows us to deﬁne partial knots and a reﬁned
polynomial W(s;t) that keep track of the symmetry.
Summary
Algebraically,a ribbon knot K looks like a connected sum
of a knot K
+
with its mirror image K
.
Geometrically,this is modelled by symmetric unions.
This allows us to deﬁne partial knots and a reﬁned
polynomial W(s;t) that keep track of the symmetry.
Perspectives
Extension to ribbon knots?
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg
Obstructions to being a symmetric union/ribbon knot?
Properties of the Jones polynomial of ribbon knots & links?
Perspectives
Extension to ribbon knots?
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg
Obstructions to being a symmetric union/ribbon knot?
Properties of the Jones polynomial of ribbon knots & links?
Perspectives
Extension to ribbon knots?
symmetric
unions
!
ribbon knots +
speciﬁc ribbon
?
?
y
?
?
y
symmetrizable
ribbon knots
!fribbon knotsg
Obstructions to being a symmetric union/ribbon knot?
Properties of the Jones polynomial of ribbon knots & links?
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