Knot Theory Senior Seminar by Tim Wylie December 3, 2002

Outline Introduction with brief history Lord Kelvins atom Defining a knot History and Progress Individual advances in the field Applications Conclusion

In 1867 Lord Kelvin proposed his theory of the vortex atom. He was inspired by a paper by Helmholtz on vortices, and by a paper by Riemann on Abelian functions. His theory stated that atoms are vortex rings, the movement of the vortex giving the illusion of matter. It also stated that chemical properties of elements were related to knotting that occurs between atoms.

Peter G. Tait In 1877 published the first paper addressing the enumeration of knots. Over the next few years he began working with C.N. Little and together they completed all the enumerations of knots past 10 crossings by Problems When his work began, the formal mathematics needed to address the subject was unavailable. He found the distinct knots but didnt have the math needed to prove they were distinct.

However, work at the beginning of the 20 th century placed the subject of topology on firm mathematical ground. Now it was possible to precisely define knots and to prove theorems about them. Algebraic methods introduced into the subject became especially important, providing the means to prove distinct knots. Topology The Tait conjectures-conjectures about knot projections that were unable to be proven at the time.

So, what is a knot? A simple definition is that a knot is a continuous simple closed curve in three-dimensional Euclidean space R3 This however, is not an entirely accurate definition because it is extremely difficult to deal with deformations, and it allows some infinite wild knots that are improbable. So, we define a knot as a simple closed polygonal curve in R3. For any two distinct points in 3-space, p and q, let [p,q] denote the line segment joining them. For an ordered set of distinct points (p1, p2,……, pn), the union of the segments [p1, p2], [p2,p3],…. [pn-1, pn] and [pn,p1] is called a closed polygonal curve. If each segment intersects exactly two other segments, intersecting each only at an endpoint, then the curve is said to be simple.

Right, so now some pictures. Knots are stick knots but are usually drawn and thought of as smooth. Intuitively, we realize that a smooth knot would be closely approximated with a very large number of segments in a polygonal curve.

This observation leads us to a useful definition of minimizing the points in a knot. If the ordered set (p1, p2, ….,pn) defines a knot, and no proper ordered subset defines the same knot, the elements of the set {pi} are called vertices of the knot. The two simplest knots are the right and left trefoil, which are each distinct.

In 1914 Max Dehn was the first to prove that two knots were distinct. He proved the right and left trefoils were distinct. Two knots are viewed as equivalent, or of the same type, if one can be deformed into the other knot. So to understand equivalence you have to understand deformations. Definition: A knot J is called an elementary deformation of the knot K if one of the two knots is determined by a sequence of points (p1, p2,…,pn) and the other is determined by the sequence (p0, p1, p2,…,pn), where 1.p0 is a point which is not collinear with p1 and pn, and 2.The triangle spanned by (p0, p1, pn) intersects the knot determined by (p1, p2,….,pn) only in the segment [p1, pn].

Definition: Knots K and J are called equivalent if there is a sequence of knots K=K0, K1,…,Kn=J with each K(i+1) an elementary deformation of Ki, for i greater than 0. Combinatorial Methods The techniques of knot theory based on the study of knot diagrams. 1. The Reidemeister Moves 2.Knot Colorings 3.The Alexander polynomial A knot diagram is the projection (or shadow) of a knot from 3-space to a plane. It can be proven that if two knots have the same projection they are equivalent regardless of their dimensions in R3.

The Reidemeister Moves In 1932 K. Reidemeister invented the Reidemeister moves. Theorem: If two knots are equivalent, their diagrams are related by a sequence of Reidemeister moves. In theory these tools are enough to distinguish any pair of distinct knots, however, for knots with complicated diagrams the calculations are often too lengthy to be of use.

Knot Colorings The method of distinguishing knots using the colorability of their diagrams was invented by Ralph Fox. A knot is called colorable if each arc can be drawn using one of three colors in such a way that: 1.At least two of the colors are used 2.At any crossing at which two colors appear, all three appear. Theorem: If a diagram of a knot, K, is colorable, then every diagram of K is colorable.

The Alexander Polynomial In 1928 James Alexander described a method of associating to each knot a polynomial such that if one knot can be deformed into another, both will have the same associated polynomial. Invariant: A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect intrinsic properties of the object of study. Calculating the Alexander Polynomial……. 1.Pick a diagram of the knot K 2.Number the arcs of the diagram 3.Separately, number the crossings 4.Define an N x N matrix where N is the number of crossings (and arcs)

5.Take a crossing numbered L. If it is right-handed with arc i passing over arcs j and k enter a (1-t) in column i of row L, enter a (-1) in column j of that row, and enter a (t) in column k of that row. If the crossing is left-handed, enter a (1-t) in column i of row L, enter a (t) in column j and enter a (-1) in column k of row L. All other entries in row L are 0. 6.Remove the last column and row of the N x N matrix. 7.Take the determinant of the (N-1) x (N-1) matrix AND WERE DONE!!!

The most simple knot to determine the alexander polynomial of is the trefoil. Applying the first several steps we end up with this matrix. (1-t)t-1 -1(1-t)t -1t (1-t) Deleting the bottom row and the last column gives a 2x2 matrix. Taking the determinant gives the Alexander polynomial A(t)=t^2 –t +1 Theorem: If the Alexander polynomial for a knot is computed using two different sets of choices for diagrams and labelings, the two polynomials will differ by a multiple of +-t^k for some integer k.

More History In the mid 1930s H. Seifert demonstrated that if a knot is the boundary of a surface in 3-space, then that surface can be used to study the knot. This discovery laid the foundation for the use of geometric methods in knot theory. In 1947 H. Schubert used geometric methods to prove that there are prime knots. A knot is called prime if it cannot be decomposed as a connected sum of nontrivial knots. Any knot can be decomposed uniquely as the connected sum of prime knots.

F. Waldhausen proved in 1968 that two knots are equivalent if and only if certain algebraic properties are the same. In 1957 C. Papakyriakopoulos succeeded in proving the Dehn Lemma, which says that if a knot were indistinguishable from the trivial knot using algebraic methods, then the knot is in fact trivial. William Thurston proved in 1978 that the complements of knots in 3-space have a complete hyberbolic structures. The Tait conjectures were finally proven in the late 80s based on a completely different polynomial invariant using the theory of operator algebras. The invariant was discovered in 1987 by Vaughan Jones.

Applications The first place knot theory was seriously used was in the study and manipulation of DNA. It was discovered in 1953 by Watson and Crick. They also discovered that DNA can become knotted which makes it difficult to carry out its function. Knot theory is also used in molecular chemistry and statistical mechanics. A recent use of knot theory is applying it to quantum computing.

The biggest contributions of knot theory have just recently developed thanks To the work of Vaughan Jones and his invariants. 1)all 3-manifolds can be describe in terms of knots and links via an operation called Dehn surgery; 2) there exists a set of moves, the Kirby calculus, that allow one to move between differing Dehn surgery descriptions of the same homeomorphic 3-manifold. Edward Witten has discovered that knots are connected to quantum field theory through generalized 3-manifold invariants.

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Acknowledgements God My Family My Wife, Rachel My Teachers My Friends

Sources Livingston, Charles. Knot theory. Washington, DC: Mathematical Association of America, Johnson, Scholz, et al. History of Topology. Amsterdam: Elsevier Science B.V., Crowell, Richard H., and Ralph H. Fox. Introduction to Knot Theory. New York, NY. Blaisdell Publishing Company, 1963 A Circular History of Knot Theory The Knotplot site Knot Theory: An Introduction Knot Theory Knot Theory Online

Q uestions ?