Knot Polynomials The Alexander-Conway and the Jones polynomial
Adrian Dawid May 27, 2020
Institut f¨urMathematik, Humboldt Universit¨at zu Berlin Table of Contents
1. What are Knot Polynomials?
2. The Alexander-Conway Polynomial
3. The Jones Polynomial
1 What are Knot Polynomials? • polynomials are very easy to distinguish • form a ring • “intuitively” have more information than numbers or true-false properties
What are Knot Polynomials?
Idea: Associate a polynomial with a knot Advantages:
2 • form a ring • “intuitively” have more information than numbers or true-false properties
What are Knot Polynomials?
Idea: Associate a polynomial with a knot Advantages:
• polynomials are very easy to distinguish
2 • “intuitively” have more information than numbers or true-false properties
What are Knot Polynomials?
Idea: Associate a polynomial with a knot Advantages:
• polynomials are very easy to distinguish • form a ring
2 What are Knot Polynomials?
Idea: Associate a polynomial with a knot Advantages:
• polynomials are very easy to distinguish • form a ring • “intuitively” have more information than numbers or true-false properties
2 • only depends on knot type • easy to compute from diagram • gives us more information than current invariants
Goals
Must-have properties for a knot polynomial:
3 • easy to compute from diagram • gives us more information than current invariants
Goals
Must-have properties for a knot polynomial:
• only depends on knot type
3 • gives us more information than current invariants
Goals
Must-have properties for a knot polynomial:
• only depends on knot type • easy to compute from diagram
3 Goals
Must-have properties for a knot polynomial:
• only depends on knot type • easy to compute from diagram • gives us more information than current invariants
3 The Alexander-Conway Polynomial The Alexander-Conway Polynomial
Definition
A polynomial ∇L ∈ Z[z] is called the Alexander-Conway polynomial of an oriented link L if the following conditions are satisfied:
1. ∇K = 1 if K is the unknot
2. (Skein-Relation): For all L+, L−, L0 as defined on the next slide:
∇L+ − ∇L− = z∇L0
4 The Skein Relation
In an arbitrary oriented link diagram replace any crossing locally with:
(a) L+ (b) L− (c) L0
Skein-Relation: ∇L+ − ∇L− = z∇L0
5 The Alexander-Conway Polynomial
Theorem Let L be the (diagram) of a unlink with n ≥ 2 components, then
∇L = 0.
6 The Alexander-Conway Polynomial
Proof.
(Case n=2) View L as L0 in the skein relation. Then L+ and L−
are diagrams of the unknot. =⇒ ∇L− = ∇L+ = 1.
(a) Example of L+ or L− (b) Example of L
Thus we have
∇L+ − ∇L− = 0 = z∇L0 =⇒ ∇L0 = ∇L = 0.
7 The Alexander-Conway Polynomial
Proof.
(Case n ≥ 3) View L as L0 in the skein relation. Then L+ and L− are diagrams of unlinks with n − 1 components. Thus we get
∇L = 0 via the skein relation and induction from the case n = 2.
(b) Example of L (a) Example of L+ or L−
8 The Alexander-Conway Polynomial
Lemma Any knot diagram K can be transformed into an unknot with a finite number of crossing changes. Proof. Procedure: Go through the knot and see any crossing as an overcrossing on the first and as an undercrossing the second time.
−−−−→
9 The Alexander-Conway Polynomial
Lemma Let L be a link so that every component is the unknot then any diagram can be transformed into the unlink with a finite number of crossing changes. Proof. Pick any ordering of the components. Then in any crossing see the higher ranked strand as the “over” strand.
−−−−→
Figure 5: Example of unlinking. Order:1,2,3.
10 The Alexander-Conway Polynomial
Theorem
For all oriented links L the Alexander-Conway Polynomial ∇L is uniquely defined by the skein relation.
11 The Alexander-Conway Polynomial
Theorem
For all oriented links L the Alexander-Conway Polynomial ∇L is uniquely defined by the skein relation.
Note: We will assume without proof that ∇L is invariant under Reidemeister moves, i.e. that it is independent of the choice of diagram.
11 The Alexander-Conway Polynomial
Proof.
By the previous lemma ∃ links L2, ..., Lk s.t. Lk is an unlink and
L −−−−→ L2 −−−−→ L3 −−−−→ ... −−−−→ Lk
are related by single crossing changes. Then ∇Lk = 0. And ∀i = 1, ..., k the skein relation implies
∇ − ∇ = ±z∇ 0 . Li Li+1 Li
0 Where Li has one crossing less than Li . Because two or less crossings imply the unkot, the claim follows by induction.
12 Example: The Trefoil Knot
Let’s compute the Alexander-Conway Polynomial of the trefoil knot:
Figure 6: The well-known trefoil knot. 13 0 = T+ (b) T− (c) T0 = T+
(e) T 0 0 0 (d) T−
Example: The Trefoil Knot
Let’s compute the Alexander-Conway Polynomial of the trefoil knot:
(a) T
14 Example: The Trefoil Knot
Let’s compute the Alexander-Conway Polynomial of the trefoil knot:
0 (a) T = T+ (b) T− (c) T0 = T+
(e) T 0 0 0 (d) T− 14 The Trefoil Knot
By the skein relation we have
∇ − ∇ = z∇ = z∇ 0 . T+ T− T0 T+ |{z} =1 Also
∇ 0 − ∇ 0 = z ∇ . T+ T− T0 |{z} |{z} =0 =1 We then have in total
2 ∇T = z∇T0 + 1 = z(z + 0) + 1 = 1 + z
as the Alexander-Conway polynomial of the trefoil.
15 The Hopf Links
We can also compute the Alexander-Conway polynomial of a link.
(c) H0 (a) H = H+ (b) H−
Here we have
∇H − ∇H− = z∇H0 = z. |{z} =0
16 The Hopf Links
Now we change the orientations a bit:
(c) H0 (a) H = H− (b) H+
Suddenly we have
∇H = −z∇H0 + ∇H+ = −z. |{z} =0
17 The Alexander-Conway Polynomial
• The Alexander-Conway polynomial can distinguish links with different orientations • Can distingish many knots (e.g. trefoil and unknot)
Interesting Question: Can it detect the unknot?
18 The Conway-Knot
Sadly, it cannot:
Figure 10: The Conway knot has trivial Alexander-Conway polynomial but is not the unknot. 19 But not all hope is lost: Maybe we can define a more powerful knot polynomial.
20 The Jones Polynomial The Bracket-Polynomial
The definition of the Jones polynomial given by Jones cannot be stated using our methods. But Kauffman gave an easier definition we can use. Definition The Kauffman bracket h·i is a mapping from the unoriented link −1 diagrams into Z[A , A] given by the following three axioms: 1. h i = 1 2. h i = Ah i + A−1h i 3. h ∪ Li = (−A2 − A−2)hLi
The second axiom should be interpreted as a local change to a single crossing.
21 The Kauffman Bracket
Note: The Kauffman bracket of any link diagram is defined by these axioms because by resolving all crossing we always get the diagram of an unlink and h i = Ah i + A−1h i can then be applied recursively.
22 The Kauffman Bracket
Note: The Kauffman bracket of any link diagram is defined by these axioms because by resolving all crossing we always get the diagram of an unlink and h i = Ah i + A−1h i can then be applied recursively. But: We don’t know if the Kauffman bracket depends on the choice of link diagram.
22 The Kauffman Bracket
And it actually does: Theorem
The Kauffman bracket changes under a Ω1-move the following way:
h i = −A−3h i
h i = −A3h i
23 The Kauffman Bracket
Proof. By definition: h i = Ah i + A−1h i
It follows:
h i = Ah i + A−1h i = (A + (−A2 − A−2)A−1)h i = −A−3h i.
24 The Kauffman Bracket
However it looks better for Ω3, Ω2: Theorem
The Kauffman bracket does not change under Ω2 and Ω3 moves.
25 The Kauffman Bracket
Proof.
(For Ω2) By definition:
h i = Ah i + A−1h i = −A−2h i + h i + A−2h i = h i
26 The Kauffman Bracket
Proof.
(For Ω3) By definition:
h i = Ah i + A−1h i = Ah i + A−1h i = h i
27 • Goal: Define a correction term for Ω1-move.
The Kauffman Bracket
• Kauffman bracket is invariant under Ω2, Ω3. =⇒ There is hope.
• Change under Ω1 is known exactly.
28 The Kauffman Bracket
• Kauffman bracket is invariant under Ω2, Ω3. =⇒ There is hope.
• Change under Ω1 is known exactly.
• Goal: Define a correction term for Ω1-move.
28 The Writhe
Definition Let L be an oriented link diagram then the writhe w(L) is defined as follows: X w(L) = εi i
where for the ith crossing εi is
(a) ε = +1 (b) ε = −1
29 The Writhe
Lemma Let L be an oriented link diagram then w(L) = w(−L).
Proof.
(a) ε = +1 (b) ε = −1
30 The Writhe
Lemma The writhe behaves under Reidemeister moves as follows:
1. w( ) = w( ) − 1 2. w( ) = w( ) 3. w( ) = w( )
31 The Kaufmann Polynomial
Using the writhe we can now define a real link invariant. Definition Let L be an oriented link diagram then we define
χ(L) = (−A)−3w(L)hLi
as the Kaufmann polynomial of L.
32 The Kaufmann Polynomial
Theorem The Kaufmann polynomial χ(L) only depends on the link type.
33 The Kaufmann Polynomial
Proof.
We have Ω2, Ω3-invariance by definition. We also have
−3w( ) χ( ) = (−A) h i
= (−A)−3w( )+3h i = (−A)−3w( )+3(−A)−3h i = (−A)−3w( )h i
Thus the Kauffman polynomial is invariant under Reidemeister moves and thus a link invariant.
34 The Kaufmann Polynomial
Definition
Let L be an oriented link and DL any diagram of it. Then we call
χ(L) = χ(DL)
the Kaufmann polynomial of the link L.
35 The Jones Polynomial
Definition Let L be an oriented link. Then we call
1 −1 V (L) = χ(L) | 1 ∈ [t 2 , t 2 ] A−2=t 2 Z the Jones polynomial of L.
36 The Skein Relation
Theorem The Jones polynomial fulfills
1. V (K) = 1 if K is the unknot
2. (Skein-Relation): For all L+, L−, L0 as defined on the next slide: −1 1 −1 t V (L+) − tV (L−) = (t 2 − t 2 )V (L0)
37 The Skein Relation
In an arbitrary oriented link diagram replace any crossing locally with:
(a) L+ (b) L− (c) L0
−1 1 −1 Skein-Relation: t V (L+) − tV (L−) = (t 2 − t 2 )V (L0)
38 The Skein Relation
Proof. (1) Let K be the unknot. Then
V (K) = χ(K) | 1 = χ( ) | 1 A−2=t 2 A−2=t 2 −3w( ) = (−A) h i | 1 = 1 | 1 = 1 A−2=t 2 A−2=t 2
39 The Skein Relation
Proof. (2) By definition we have:
h i = Ah i + A−1h i h i = Ah i + A−1h i
40 The Skein Relation
Proof. (2) By definition we have:
h i = Ah i + A−1h i h i = Ah i + A−1h i
Thus we get
Ah i = A2h i + h i A−1h i = h i + A−2h i
40 The Skein Relation
Proof. (2) By definition we have:
h i = Ah i + A−1h i h i = Ah i + A−1h i
Thus we get
Ah i = A2h i + h i A−1h i = h i + A−2h i
Now we sum:
Ah i − A−1h i = A2h i − A−2h i = (A2 − A−2)h i
40 The Skein Relation
Proof.
(2 - continued) By definition w(L+) − 1 = w(L0) = w(L−) + 1.
−3w(L0) −3w(L+)+3 4 (−A) Ah i = −A(−A) h i = −A V (L+)
−3w(L0) −1 −1 −3w(L−)−3 4 (−A) A h i = −A (−A) h i = −A V (L+)
So we have
−3w(L0) −1 4 −4 (−A) (Ah i − A h i) = −A V (L+) + A V (L−) 2 −2 = (A − A )V (L0).
After a substution this is exactly the skein relation.
41 The Jones Polynomial
Note: As for the Alexander-Conway polynomial the Jones polynomial is determined by the skein relation and its value on the unknot.
42 The Jones Polynomial
Theorem Let L be a trivial n-component unlink. Then − 1 1 n−1 V (L) = (−t 2 − t 2 ) .
43 The Jones Polynomial
Theorem Let L be a trivial n-component unlink. Then − 1 1 n−1 V (L) = (−t 2 − t 2 ) .
Proof. 2 −2 We have h ∪ Ki = (−A − A )hKi Let DL be the trivial 2 −2 n−1 diagram. Then hDLi = (−A − A ) . Also w(DL) = 0. So we get 2 −2 n−1 χ(L) = hDLi = (−A − A ) . Thus − 1 1 n−1 V (L) = (−t 2 − t 2 ) .
43 Example: The Trefoil Knot
Let’s compute the Jones Polynomial of the trefoil knot:
0 (a) T+ (b) T− (c) T0 = T+
(e) T 0 0 0 (d) T−
44 The Trefoil Knot
By the skein relation we have
−1 1 −1 1 −1 0 t V (T+) − t V (T−) = (t 2 − t 2 )V (T0) = (t 2 − t 2 )V (T+). | {z } =1 Further we have
−1 0 0 1 − 1 t V (T+) − t V (T−) = t 2 − t 2 | {z } 1 1 =−t 2 −t− 2 −1 0 1 3 1 − 1 t V (T+) + t 2 + t 2 = t 2 − t 2 −1 0 3 − 1 t V (T+) + t 2 = −t 2 0 5 1 V (T+) = −t 2 − t 2
45 The Trefoil Knot
We then have in total
−1 1 − 1 0 t V (T+) − t = (t 2 − t 2 )V (T+) 3 1 5 1 2 V (T+) = (t 2 − t 2 )(−t 2 − t 2 ) − t 1 3 4 V (T+) = t + t − t
as the Jones polynomial of the trefoil.
46 0 = T− (b) T+ (c) T0 = T−
(e) T 0 0 0 (d) T+
Example: The Left-Handed Trefoil Knot
We can also consider the left-handed trefoil knot:
(a) Tl
47 Example: The Left-Handed Trefoil Knot
We can also consider the left-handed trefoil knot:
0 (a) Tl = T− (b) T+ (c) T0 = T−
(e) T 0 0 0 (d) T+
47 The Left-Handed Trefoil Knot
Similar as before, we get
−1 1 − 1 0 1 − 1 0 − 5 − 1 t (−t 2 − t 2 ) − tV (T−) = t 2 − t 2 =⇒ V (T−) = −t 2 − t 2
Thus we have
−1 1 − 1 0 t − tV (T−) = (t 2 − t 2 )V (T+) − 1 − 3 − 5 − 1 −2 V (T−) = (t 2 − t 2 )(−t 2 − t 2 ) − t −1 −3 −4 V (T−) = t + t − t .
So we get
−1 −3 −4 1 3 4 V (Tl ) = t + t − t 6= t + t − t .
48 Theorem Let K 0 be the mirror of an (oriented) knot K then
V (K)[t] = V (K 0)[t−1].
Proof. Omitted. See lecture notes by J. L. Gross.
Corollary If the Jones polynomial of a knot is not symmetric then it is a chiral knot.
The Jones Polynomial and Chiral Knots
The two polynomials look very similar. Maybe there is some additional structure there.
49 Corollary If the Jones polynomial of a knot is not symmetric then it is a chiral knot.
The Jones Polynomial and Chiral Knots
The two polynomials look very similar. Maybe there is some additional structure there. Theorem Let K 0 be the mirror of an (oriented) knot K then
V (K)[t] = V (K 0)[t−1].
Proof. Omitted. See lecture notes by J. L. Gross.
49 The Jones Polynomial and Chiral Knots
The two polynomials look very similar. Maybe there is some additional structure there. Theorem Let K 0 be the mirror of an (oriented) knot K then
V (K)[t] = V (K 0)[t−1].
Proof. Omitted. See lecture notes by J. L. Gross.
Corollary If the Jones polynomial of a knot is not symmetric then it is a chiral knot.
49 The Hopf Links
Corollary (Of the earlier computation)
(a) Positive Hopf (b) Negative
link H+ Hopf link H−
5 1 − 5 − 1 V (H+) = −t 2 − t 2 and V (H−) = −t 2 − t 2
50 The Conway Knot
What about the Conway knot?
Figure 17: The Conway knot again
51 The Conway Knot
What about the Conway knot?
Figure 17: The Conway knot again
V (ConwayKnot) = t−4(−1 + 2t − 2t2 + t6 − 2t7 + 2t8 − 2t9 + t10)
51 The Jones Polynomial
Conjecture: The Jones polynomial V (K) is trivial iff K is the unknot.
52 The Jones Polynomial
Conjecture: The Jones polynomial V (K) is trivial iff K is the unknot. This is unproven until now. But some progress has been made:
52 • Computer searches have not found a counterexample for knots (yet). • The minimal crossing number of a counterexample is ≥ 18.
The Jones Polynomial
Conjecture: The Jones polynomial V (K) is trivial iff K is the unknot. This is unproven until now. But some progress has been made:
• There are (a infinite number of) links with − 1 1 n−1 V (L) = (−t 2 − t 2 ) that are not trivial unlinks.
52 • The minimal crossing number of a counterexample is ≥ 18.
The Jones Polynomial
Conjecture: The Jones polynomial V (K) is trivial iff K is the unknot. This is unproven until now. But some progress has been made:
• There are (a infinite number of) links with − 1 1 n−1 V (L) = (−t 2 − t 2 ) that are not trivial unlinks. • Computer searches have not found a counterexample for knots (yet).
52 The Jones Polynomial
Conjecture: The Jones polynomial V (K) is trivial iff K is the unknot. This is unproven until now. But some progress has been made:
• There are (a infinite number of) links with − 1 1 n−1 V (L) = (−t 2 − t 2 ) that are not trivial unlinks. • Computer searches have not found a counterexample for knots (yet). • The minimal crossing number of a counterexample is ≥ 18.
52 Thank you for your attention!
52 References
Jonathan L. Gross. CS E6204 Lectures 9b and 10 Alexander-Conway and Jones Polynomials. url: http: //www.cs.columbia.edu/~cs6204/files/Lec9b,10.pdf (visited on 05/25/2020). W. B. Raymond Lickorish. An Introduction to Knot Theory by W. B. Raymond Lickorish. eng. Graduate Texts in Mathematics, 175. New York, NY, 1997, pp. 23 - 32. isbn: 9781461206910.
53 References
Knotilus. url: http://knotilus.math.uwo.ca/draw.php?knot=1%2C+- 4%2C+2%2C+-1%2C+3%2C+-7%2C+4%2C+-2%2C+- 5%2C+11%2C+-6%2C+-3%2C+7%2C+10%2C+-8%2C+5%2C+- 9%2C+6%2C+-10%2C+8%2C+-11%2C+9 (visited on 05/25/2020). Charles Livingston. Knotentheorie f¨urEinsteiger von Charles Livingston. ger. 1st ed. 1995. Wiesbaden, 1995, pp. 44 - 48. isbn: 9783322802873.
54 Further Reading
Oliver T. Dasbach and Stefan Hougardy. “Does the Jones polynomial detect unknottedness?”. In: Experiment. Math. 6.1 (1997), pp. 51–56. url: https://projecteuclid.org: 443/euclid.em/1047565283. Shalom Eliahou, Louis H. Kauffman, and Morwen B. Thistlethwaite. “Infinite families of links with trivial Jones polynomial”. In: Topology 42.1 (2003), pp. 155–169. issn: 0040-9383. doi: https://doi.org/10.1016/S0040-9383(02)00012-5. url: http://www.sciencedirect.com/science/article/ pii/S0040938302000125. Louis H. Kauffman. Remarks on Formal Knot Theory. 2006. arXiv: math/0605622 [math.GT].
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