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Knot Polynomials the Alexander-Conway and the Jones Polynomial

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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 rL 2 Z[z] is called the Alexander-Conway polynomial of an oriented link L if the following conditions are satisfied: 1. rK = 1 if K is the unknot 2. (Skein-Relation): For all L+; L−; L0 as defined on the next slide: rL+ − rL− = zrL0 4 The Skein Relation In an arbitrary oriented link diagram replace any crossing locally with: (a) L+ (b) L− (c) L0 Skein-Relation: rL+ − rL− = zrL0 5 The Alexander-Conway Polynomial Theorem Let L be the (diagram) of a unlink with n ≥ 2 components, then rL = 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. =) rL− = rL+ = 1. (a) Example of L+ or L− (b) Example of L Thus we have rL+ − rL− = 0 = zrL0 =) rL0 = rL = 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 rL = 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 rL is uniquely defined by the skein relation. 11 The Alexander-Conway Polynomial Theorem For all oriented links L the Alexander-Conway Polynomial rL is uniquely defined by the skein relation. Note: We will assume without proof that rL 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 9 links L2; :::; Lk s.t. Lk is an unlink and L −−−−! L2 −−−−! L3 −−−−! ::: −−−−! Lk are related by single crossing changes. Then rLk = 0. And 8i = 1; :::; k the skein relation implies r − r = ±zr 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 r − r = zr = zr 0 : T+ T− T0 T+ |{z} =1 Also r 0 − r 0 = z r : T+ T− T0 |{z} |{z} =0 =1 We then have in total 2 rT = zrT0 + 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 rH − rH− = zrH0 = z: |{z} =0 16 The Hopf Links Now we change the orientations a bit: (c) H0 (a) H = H− (b) H+ Suddenly we have rH = −zrH0 + rH+ = −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) = "i i where for the ith crossing "i is (a) " = +1 (b) " = −1 29 The Writhe Lemma Let L be an oriented link diagram then w(L) = w(−L). Proof. (a) " = +1 (b) " = −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.
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