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The Unknot, the Trefoil Knot, and the Figure-Eight Knot are Mutually Nonequivalent An Elementary Proof Jimmy Gillan Thursday, April 10, 2008

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Outline Introduction –A Brief History of Knot Theory –What is a knot? –Knot Terminology Defining Knots Equivalence and Knot Invariants The Proof

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A Brief History of Knot Theory Carl Friedrich Gauss (1777 – 1855) Lord Kelvin, atoms, and the “ether” P.G. Tait first to formally publish a paper on knots in 1877 –Enumerating and tabulating knots Early 20 th century classical knot theory becomes a formal branch of mathematics Knot theory now includes elements of algebra, combinatorics and geometry

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What is and what is not a knot? The “string analogy” Any tame mathematical knot can be physically represented by the “string analogy”

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What is and what is not a knot? Which of these two figures is a knot?

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What is and what is not a knot? A knot!Not a knot! Which of these two figures is a knot?

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Some Terminology A knot diagram is defined as a pictorial representation of a knot in R 2 Each diagram of a given knot K is defined as a projection of the knot K. Two projections of the same knot Two knot diagrams &

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Some Terminology A crossing is defined as a point in the projection of a knot where the knot intersects or crosses-over itself.

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Defining Knots Topological Definition A knot is an embedding of S 1 in R 3 or S 3. Simpler Definition A knot is a defined as a simple, closed curve in R 3 that is isotopic to a simple, closed polygonal curve with a finite vertex set.

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Defining Knots Let K be a curve in R 3 and let f: I → R 3 be a continuous function such that f (I) = K. closed – f (0) = f (1) simple – if f (x) = f (y), then either x = y or x,y ε {0,1}

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Defining Knots A simple, closed polygonal curve is defined as follows: Let (p 1,…,p n ) be an ordered set of points in R 3 such that no three points lie on a common line Let [p i, p j ] denote the line segment between points p i and p j

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Defining Knots Two curves are said to be isotopic if one can be deformed to form the other in R 3 without breaking the curve at any point The simple, closed polygonal curve determined by (a,b,c) and a knot to which it is isotopic.

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Equivalence and Invariants How do we know if two different knot diagrams represent different knots? Consider the two diagrams below, are these projections of the same knot? ?

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Equivalence and Invariants How do we know if two different knot diagrams represent different knots? Consider the two diagrams below, are these projections of the same knot? √ YES!

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Equivalence and Invariants Two projections are equivalent if you can deform one into the other without breaking the knot (“string analogy”) Planar isotopies –deformations that do not change the crossings of a projection

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Equivalence and Invariants Reidemeister moves OR TYPE I TYPE II TYPE III

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Equivalence and Invariants Alexander and Briggs Theorem If two knot projections are equivalent, then their diagrams are related by a series of Reidemeister moves A knot invariant is defined as a characteristic of a knot which is true for all of its projections Use knot invariants to determine whether two knots are not equivalent

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The Proof Consider the three knots with the fewest crossings in their simplest projections –The unknot, O (0 crossings) –The trefoil knot, T (3 crossings) –The figure-eight knot, F (4 crossings) INTRODUCTION

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The Proof Sketch of Proof First show T is not equivalent to O or F using the knot invariant Tricolorability Then show O and F are not equivalent using the knot invariant the Jones Polynomial O, T and F are mutually nonequivalent OUTLINE

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The Proof A knot is tricolorable if the pieces of a projection that are not intersected in its diagram can be colored with exactly 3 different colors such that at each crossing in the knot is the meeting of either 3 different colors or the same color. Is the trefoil knot T tricolorable? TRICOLORABILITY

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The Proof A knot is tricolorable if the pieces of a projection that are not intersected in its diagram can be colored with exactly 3 different colors such that at each crossing in the knot is the meeting of either 3 different colors or the same color. Is the trefoil knot T tricolorable? YES! TRICOLORABILITY

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The Proof Is the unknot O tricolorable? NO! Is the figure-eight knot F tricolorable? NO! TRICOLORABILITY

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The Proof Assign a Laurent polynomial to each knot If two projections have different polynomials, they are not equivalent J.W. Alexander [1928] - developed first knot polynomial using matrices & determinants John Conway [1969] - calculate the Alexander polynomial using skein relations Vaughn Jones [1984] developed a way to calculate the Alexander polynomial using the bracket polynomial and skein relations THE JONES POLYNOMIAL

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The Proof The Jones polynomial of a knot K is defined as: where: –X(K) is the Jones polynomial of K – denotes the bracket polynomial of K –w(K) denotes the writhe of K –A is the variable of bracket polynomial and A = t -1/4 THE JONES POLYNOMIAL

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The Proof There are three rules for computing the bracket polynomial of a knot K THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL

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The Proof By Rule 1, = 1 Computing the bracket polynomial of F is more involved Take the projection of F and enumerate the crossings THE JONES POLYNOMIAL – BRACKET POLYNOMIAL

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The Proof Consider crossing 1 By Rule 2, = A + A -1 where F V and F H are derivative knots created by changing crossing 1 from C to C V and C H respectively THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL

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The Proof We must use Rule 2 again with crossing 2 in order to compute both and, giving us four derivative knots,,,, and with: By substituting into the formula for we get: THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL

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The Proof Continue expansion with Rule 2 until all crossings have been eliminated THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL Clearly F VHHV is planar isotopic to the unknot and since = 1 by Rule 1, = 1

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The Proof Now Rules 1 and 3 can be used to compute the values of the resulting 16 derivative knots and have in terms of A After lots of drawing and simplification we ultimately get: THE JONES POLYNOMIAL – THE BRACKET POLYNOMIAL

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The Proof The writhe of F, w(F), is computed as follows: THE JONES POLYNOMIAL – WRITHE

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The Proof The writhe of F, w(F), is computed as follows: –Give F an orientation THE JONES POLYNOMIAL – WRITHE

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The Proof The writhe of F, w(F), is computed as follows: –Give F an orientation –Assign +1 or -1 to the crossings according to its type + 1 crossing– 1 crossing THE JONES POLYNOMIAL – WRITHE

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The Proof The writhe of F, w(F), is computed as follows: –Give F an orientation –Assign +1 or -1 to the crossings according to its type –Sum the assignments over all crossings THE JONES POLYNOMIAL – WRITHE

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The Proof Substitute the bracket polynomial and writhe of F and O into the original equation and replace A with t -1/4 Clearly X(F) ≠ X(O) THE JONES POLYNOMIAL – COMBINING THE TWO

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The Proof CONCLUSION T is not equivalent to O and T is not equivalent to F because T is tricolorable and O and F are not O and F are not equivalent because their Jones polynomials are different Thus O, T and F are mutually nonequivalent □

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Thank You I’d like to thank my advisors, Professor Ramin Naimi and Professor Ron Buckmire, and the Occidental Mathematics Department for all their help and support over the last four years. T hank you for coming!

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References Adams, Colin C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, (New York, NY: W.H. Freeman and Company, 1994) Kauffman, Louis H., On Knots, (Princeton, NJ: Princeton University Press, 1987) Livingston, Charles, Knot Theory, (Washington, D.C.: Mathematical Association of America, 1993)

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