Knot Theory Via Seifert Surfaces

Knot theory via Seifert surfaces.

Christopher William Davis Joint with Grant Roth (UWEC)

1/14 A link is an embedding of many disjoint circles into 3-dimensional space. It is often important to distinguish diﬀerent components of the same link.

The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other.

=? =?

Knots and Links

A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow.

2/14 It is often important to distinguish diﬀerent components of the same link.

The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other.

=? =?

Knots and Links

A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow. A link is an embedding of many disjoint circles into 3-dimensional space.

2/14 The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other.

=? =?

Knots and Links

A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow. A link is an embedding of many disjoint circles into 3-dimensional space. It is often important to distinguish diﬀerent components of the same link.

2/14 Knots and Links

The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other.

=? =?

2/14 Fact The linking number is not changed by isotopy.

lk(L1, L2) = +1 lk(L1, L2) = 0

No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve oﬀ of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 ∪ L2 the linking number lk(L1, L2) is given by counting how many times L1 passes “positively” over L2 and subtracting the number of “negative” times.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

Answer:

3/14 Fact The linking number is not changed by isotopy.

lk(L1, L2) = +1 lk(L1, L2) = 0

Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve oﬀ of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 ∪ L2 the linking number lk(L1, L2) is given by counting how many times L1 passes “positively” over L2 and subtracting the number of “negative” times.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

Answer: No.

3/14 Fact The linking number is not changed by isotopy.

lk(L1, L2) = +1 lk(L1, L2) = 0

The linking number is a formalization of this idea. For a 2-component link L1 ∪ L2 the linking number lk(L1, L2) is given by counting how many times L1 passes “positively” over L2 and subtracting the number of “negative” times.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve oﬀ of the red disk without crossing through the red curve.

3/14 Fact The linking number is not changed by isotopy.

lk(L1, L2) = +1 lk(L1, L2) = 0

For a 2-component link L1 ∪ L2 the linking number lk(L1, L2) is given by counting how many times L1 passes “positively” over L2 and subtracting the number of “negative” times.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

3/14 +

lk(L1, L2) = +1 lk(L1, L2) = 0

Fact The linking number is not changed by isotopy.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve oﬀ of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 ∪ L2 the linking number lk(L1, L2) is given by counting how many times L1 passes “positively” over L2 and subtracting the number of “negative” times.

3/14 lk(L1, L2) = +1 lk(L1, L2) = 0

Fact The linking number is not changed by isotopy.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

3/14 Fact The linking number is not changed by isotopy.

Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

lk(L1, L2) = +1 lk(L1, L2) = 0

3/14 Linking numbers

Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink?

lk(L1, L2) = +1 lk(L1, L2) = 0

A Seifert surface for a knot K is an oriented surface (2-dimensional shape) bounded by K.

lk(L1, L2) = 0

If F is ANY Seifert surface for L1 then lk(L1, L2) is given by counting up the number of times L2 passes through F in the positive direction and the subtracting the number of times is passes through in the negative direction.

Seifert surfaces

The deﬁnition previous is completely diagrammatic. Let’s reformulate the linking number in a more geometric way.

− − + +

lk(L1, L2) = 0

4/14 − − + +

lk(L1, L2) = 0

Seifert surfaces

The deﬁnition previous is completely diagrammatic. Let’s reformulate the linking number in a more geometric way. A Seifert surface for a knot K is an oriented surface (2-dimensional shape) bounded by K.

− − + +

lk(L1, L2) = 0

4/14 − − + +

lk(L1, L2) = 0

Seifert surfaces

− − + +

lk(L1, L2) = 0

4/14 − + +

lk(L1, L2) = 0

Seifert surfaces

− − + + −

lk(L1, L2) = 0

4/14 + +

lk(L1, L2) = 0

Seifert surfaces

− − + + − −

lk(L1, L2) = 0

4/14 +

lk(L1, L2) = 0

Seifert surfaces

− − + + − − +

lk(L1, L2) = 0

4/14 lk(L1, L2) = 0

Seifert surfaces

− − + + − − + +

lk(L1, L2) = 0

4/14 Seifert surfaces

− − + + − − + +

lk(L1, L2) = 0 lk(L1, L2) = 0

4/14 Many of these can be gathered together into one big invariant: The Seifert Matrix. Start with a Seifert surface F of genus g.(Informally: this means F consists of a disk with 2g bands added) I Find 2g curves on F , α1,α 2,α 3,β 4,...,α 2g−1,α 2g intersecting as to the right. + I αi = push αi oﬀ of the surface in the positive direction.

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

The Seifert matrix

A Seifert surface lets you compute linking number by counting up some points. There are many other invariants which a Seifert surface lets you compute in terms of “something easier.”

5/14 Start with a Seifert surface F of genus g.(Informally: this means F consists of a disk with 2g bands added) I Find 2g curves on F , α1,α 2,α 3,β 4,...,α 2g−1,α 2g intersecting as to the right. + I αi = push αi oﬀ of the surface in the positive direction.

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

The Seifert matrix

5/14 I Find 2g curves on F , α1,α 2,α 3,β 4,...,α 2g−1,α 2g intersecting as to the right. + I αi = push αi oﬀ of the surface in the positive direction.

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

The Seifert matrix

A Seifert surface lets you compute linking number by counting up some points. There are many other invariants which a Seifert surface lets you compute in terms of “something easier.” Many of these can be gathered together into one big invariant: The Seifert Matrix. Start with a Seifert surface F of genus g.(Informally: this means F consists of a disk with 2g bands added)

5/14 + I αi = push αi oﬀ of the surface in the positive direction.

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

The Seifert matrix

5/14 + I αi = push αi oﬀ of the surface in the positive direction.

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

The Seifert matrix

5/14 I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

The Seifert matrix

5/14 The Seifert matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ).

5/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + + lk(α1 ,α 1) lk(α1 ,α 2) V = + + lk(α2 ,α 1) lk(α2 ,α 2)

6/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + + lk(α1 ,α 1) lk(α1 ,α 2) V = + + lk(α2 ,α 1) lk(α2 ,α 2)

6/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + + 1 lk(α1 ,α 2) V = + + lk(α2 ,α 1) lk(α2 ,α 2)

6/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + 1 + 1 V = + + lk(α2 ,α 1) lk(α2 ,α 2)

6/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + 1 + 1 V = + + lk(α2 ,α 1) lk(α2 ,α 2)

6/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + 1 + 1 V = + 0 lk(α2 ,α 2)

6/14  0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + 1 + 1 V = 0 + 1

6/14 The Seifert Matrix

I Build a 2g × 2g matrix V whose + (i, j) entry lk(αi , αj ). + 1 + 1 V = 0 + 1

 0 0 −1 0   −1 0 0 −1  V =    −1 0 0 +2  0 −1 +1 0

6/14 1 −1 1 0 1 0 = 0 1 1 1 −1 1

1 0 V 0 = −1 1

NO Reason: There is more than one choice of curves α1,...,α 2g Changing the choice of αi changes the Seifert matrix by V 7→ V 0 = PT VP for some invertible matrix P. V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 V = 0 1

Is the Seifert matrix an invariant? Can you just look at this matrix and say that some knots are diﬀerent?

7/14 1 −1 1 0 1 0 = 0 1 1 1 −1 1

1 0 V 0 = −1 1

Reason: There is more than one choice of curves α1,...,α 2g Changing the choice of αi changes the Seifert matrix by V 7→ V 0 = PT VP for some invertible matrix P. V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 V = 0 1

Is the Seifert matrix an invariant? Can you just look at this matrix and say that some knots are diﬀerent? NO

7/14 1 −1 1 0 1 0 = 0 1 1 1 −1 1

1 0 V 0 = −1 1

Changing the choice of αi changes the Seifert matrix by V 7→ V 0 = PT VP for some invertible matrix P. V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 V = 0 1

Is the Seifert matrix an invariant? Can you just look at this matrix and say that some knots are diﬀerent? NO Reason: There is more than one choice of curves α1,...,α 2g

7/14 1 −1 1 0 1 0 = 0 1 1 1 −1 1

1 0 V 0 = −1 1

Changing the choice of αi changes the Seifert matrix by V 7→ V 0 = PT VP for some invertible matrix P. V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 V = 0 1

Is the Seifert matrix an invariant? Can you just look at this matrix and say that some knots are diﬀerent? NO Reason: There is more than one choice of curves α1,...,α 2g

7/14 1 −1 1 0 1 0 = 0 1 1 1 −1 1

Changing the choice of αi changes the Seifert matrix by V 7→ V 0 = PT VP for some invertible matrix P. V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 1 0 V = V 0 = 0 1 −1 1

Is the Seifert matrix an invariant? Can you just look at this matrix and say that some knots are diﬀerent? NO Reason: There is more than one choice of curves α1,...,α 2g

7/14 1 −1 1 0 1 0 = 0 1 1 1 −1 1

V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 1 0 V = V 0 = 0 1 −1 1

Is the Seifert matrix an invariant? Can you just look at this matrix and say that some knots are diﬀerent? NO Reason: There is more than one choice of curves α1,...,α 2g Changing the choice of αi changes the Seifert matrix by V 7→ V 0 = PT VP for some invertible matrix P.

7/14 V and PT VP are similar.

The Seifert matrix is not an invariant

1 1 1 0 1 −1 1 0 1 0 V = V 0 = = 0 1 −1 1 0 1 1 1 −1 1

7/14 The Seifert matrix is not an invariant

1 1 1 0 1 −1 1 0 1 0 V = V 0 = = 0 1 −1 1 0 1 1 1 −1 1

7/14 NO Reason: There is more than one Seifert surface for a single knot.

 1010   1100  V 0 =    1 0 0 1  0 0 0 0 An elementary enlargement changes the the Seifert matrix by:  V ∗ 0  V 7→  ∗ 0 1  0 0 0

Fact: Any two diﬀerent Seifert surfaces for a single knot are related by some elementary enlargements

The Seifert matrix is not an invariant II

Maybe any two Seifert matrices for a ﬁxed knot are similar?

1 0 V = 1 1

. 8/14  1010   1100  V 0 =    1 0 0 1  0 0 0 0 An elementary enlargement changes the the Seifert matrix by:  V ∗ 0  V 7→  ∗ 0 1  0 0 0

Fact: Any two diﬀerent Seifert surfaces for a single knot are related by some elementary enlargements

The Seifert matrix is not an invariant II

Maybe any two Seifert matrices for a ﬁxed knot are similar? NO Reason: There is more than one Seifert surface for a single knot.

1 0 V = 1 1

. 8/14  1010   1100  V 0 =    1 0 0 1  0 0 0 0 An elementary enlargement changes the the Seifert matrix by:  V ∗ 0  V 7→  ∗ 0 1  0 0 0

Fact: Any two diﬀerent Seifert surfaces for a single knot are related by some elementary enlargements

The Seifert matrix is not an invariant II

Maybe any two Seifert matrices for a ﬁxed knot are similar? NO Reason: There is more than one Seifert surface for a single knot.

1 0 V = 1 1

. 8/14 An elementary enlargement changes the the Seifert matrix by:  V ∗ 0  V 7→  ∗ 0 1  0 0 0

Fact: Any two diﬀerent Seifert surfaces for a single knot are related by some elementary enlargements

The Seifert matrix is not an invariant II

Maybe any two Seifert matrices for a ﬁxed knot are similar? NO Reason: There is more than one Seifert surface for a single knot.

 1010  1 0  1100  V = V 0 =   1 1  1 0 0 1  0 0 0 0

. 8/14 Fact: Any two diﬀerent Seifert surfaces for a single knot are related by some elementary enlargements

The Seifert matrix is not an invariant II

Maybe any two Seifert matrices for a ﬁxed knot are similar? NO Reason: There is more than one Seifert surface for a single knot.

 1010  1 0  1100  V = V 0 =   1 1  1 0 0 1  0 0 0 0 An elementary enlargement changes the the Seifert matrix by:  V ∗ 0  V 7→  ∗ 0 1  0 0 0

. 8/14 The Seifert matrix is not an invariant II

Maybe any two Seifert matrices for a ﬁxed knot are similar? NO Reason: There is more than one Seifert surface for a single knot.

 1010  1 0  1100  V = V 0 =   1 1  1 0 0 1  0 0 0 0 An elementary enlargement changes the the Seifert matrix by:  V ∗ 0  V 7→  ∗ 0 1  0 0 0

Fact: Any two diﬀerent Seifert surfaces for a single knot are

related by some elementary enlargements. 8/14 Theorem (Seifert, 1950’s) Any two Seifert matrices are related by a sequence of elementary enlargements (of matrices) and similarity. Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

The equivalence relation matrices generated by elementary enlargements and similarity is called S-equivalence.

The Seifert matrix is a knot invariant (up to S-equivalence)

I Any two Seifert surfaces for a single knot are related by some elementary enlargements

I Any two Seifert matrices over a single Seifert surface are similar

9/14 Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

The equivalence relation matrices generated by elementary enlargements and similarity is called S-equivalence.

The Seifert matrix is a knot invariant (up to S-equivalence)

I Any two Seifert surfaces for a single knot are related by some elementary enlargements

I Any two Seifert matrices over a single Seifert surface are similar

Theorem (Seifert, 1950’s) Any two Seifert matrices are related by a sequence of elementary enlargements (of matrices) and similarity. Resulting Philosophy:

9/14 The equivalence relation matrices generated by elementary enlargements and similarity is called S-equivalence.

The Seifert matrix is a knot invariant (up to S-equivalence)

I Any two Seifert surfaces for a single knot are related by some elementary enlargements

I Any two Seifert matrices over a single Seifert surface are similar

Theorem (Seifert, 1950’s) Any two Seifert matrices are related by a sequence of elementary enlargements (of matrices) and similarity. Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

9/14 The Seifert matrix is a knot invariant (up to S-equivalence)

I Any two Seifert surfaces for a single knot are related by some elementary enlargements

I Any two Seifert matrices over a single Seifert surface are similar

The equivalence relation matrices generated by elementary enlargements and similarity is called S-equivalence.

9/14 1 − t −t det = t2 + t + 1 1 1 − t

det V − tV T =

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.) 1 0 V = 1 1

Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

10/14 1 − t −t det = t2 + t + 1 1 1 − t

det V − tV T =

1 0 V = 1 1

Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.)

10/14 1 − t −t det = t2 + t + 1 1 1 − t

det V − tV T =

Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.) 1 0 V = 1 1

10/14 1 − t −t det = t2 + t + 1 1 1 − t

Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.) 1 0 V = 1 1 det V − tV T =

10/14 t2 + t + 1

Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.) 1 0 V = 1 1 1 − t −t det V − tV T = det = 1 1 − t

10/14 Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.) 1 0 V = 1 1 1 − t −t det V − tV T = det = t2 + t + 1 1 1 − t

10/14 Invariants of S-equivalence

Resulting Philosophy: Find a quantity you can compute out of a Seifert matrix which is not changed by similarity and elementary enlargement. Compute this quantity to distinguish knots.

Example: T The Alexander Polynomial: ∆K (t) = det(V − tV ), is an invariant (up to multiplication by ±t.) 1 0 V = 1 1 1 − t −t det V − tV T = det = t2 + t + 1 1 1 − t

Since The Alexander polynomial of the unknot (1) is diﬀerent from the Alexander polynomial of the trefoil, The trefoil is not the unknot.

10/14 So, the Seifert matrix doesn’t tell you everything. What does it tell you? Theorem (Naik-Stanford, ’99) Two knots K and J have S-equivalent Seifert matrices if and only if K and J are related by a sequence of isotopy and double delta moves.

Classiﬁcation of S-equivalence

Here are two fairly diﬀerent knots with identical Seifert matrices

11/14 So, the Seifert matrix doesn’t tell you everything. What does it tell you? Theorem (Naik-Stanford, ’99) Two knots K and J have S-equivalent Seifert matrices if and only if K and J are related by a sequence of isotopy and double delta moves.

Classiﬁcation of S-equivalence

Here are two fairly diﬀerent knots with identical Seifert matrices

Classiﬁcation of S-equivalence

Here are two fairly diﬀerent knots with identical Seifert matrices

11/14 What does it tell you? Theorem (Naik-Stanford, ’99) Two knots K and J have S-equivalent Seifert matrices if and only if K and J are related by a sequence of isotopy and double delta moves.

Classiﬁcation of S-equivalence

Here are two fairly diﬀerent knots with identical Seifert matrices

So, the Seifert matrix doesn’t tell you everything.

11/14 Theorem (Naik-Stanford, ’99) Two knots K and J have S-equivalent Seifert matrices if and only if K and J are related by a sequence of isotopy and double delta moves.

Classiﬁcation of S-equivalence

Here are two fairly diﬀerent knots with identical Seifert matrices

So, the Seifert matrix doesn’t tell you everything. What does it tell you?

11/14 Classiﬁcation of S-equivalence

Here are two fairly diﬀerent knots with identical Seifert matrices

So, the Seifert matrix doesn’t tell you everything. What does it tell you? Theorem (Naik-Stanford, ’99) Two knots K and J have S-equivalent Seifert matrices if and only if K and J are related by a sequence of isotopy and double delta moves.

11/14 Think of the curves α1, α2 as a link.

Double Delta moves in action.

Let’s illustrate the proof: Theorem (Naik-Stanford, ’99) Two knots K and J have S-equivalent Seifert matrices if and only if K and J are related by a sequence of isotopy and double delta moves. ! (Matveev ’87 and Murakami-Nakanishi ’89) Two links with the same linking numbers are ! related by “∆-moves”