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Knots and (A Little Of) Braids

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Knots and (A Little Of) Braids Knots and (a little of) Braids Camila A. Ramirez University of Iowa February 28, 2016 C. Ramirez (UI) Knots and Braids February 28, 2016 1 / 38 Questions Given a tangled closed piece of string, is it really knotted or can it be untangled without having to cut it? Given two knotted pieces of closed string, when can we change one into the other? Does there exist an algorithm that can answer these questions? C. Ramirez (UI) Knots and Braids February 28, 2016 2 / 38 1 Introduction History 2 Knots Definition Projections Reidemeister Moves Knot Invariants Links More Knots 3 Braids Definition Braid Equivalence Braid Multiplication Closure of Braids 4 Applications C. Ramirez (UI) Knots and Braids February 28, 2016 3 / 38 Introduction History History In the late 19th century, physicists postulated that a substance called ether permeated all throughout space. In the 1860s, Lord Kelvin proposed that elements should have unique signatures based on how the atoms knotted up the ether around them. This theory led Scottish mathematical-physicist Peter Guthrie Tait to the creation of the first knot tables. In 1887 the Michelson-Morley experiment provided evidence against the existence of the ether, but it was too late for mathematicians, they were already tangled up! C. Ramirez (UI) Knots and Braids February 28, 2016 4 / 38 Knots Definition What is a knot? Take a piece of string and tie a knot on it. Glue the ends together. You have a "mathematical" knot! 3 Definition (A tame knot is a simple closed polygonal curve in R ) A knot is the union of the segments [p1; p2]; [p2; p3]; :::; [pn − 1; pn] of an ordered set of distinct points fp1; p2; :::; p3g in which each segment intersect exactly two others. C. Ramirez (UI) Knots and Braids February 28, 2016 5 / 38 Knots Projections Knot Projection Definition A knot projection is called a regular projection if no three points on the knot project to the same point and no vertex projects to the same point as any other point on the knot. Projections of a knot to the plane allow the representation of a knot as a knot diagram. C. Ramirez (UI) Knots and Braids February 28, 2016 6 / 38 Knots Projections Types of Knots Definition An oriented knot consists of a knot and an ordering of its vertices. Two orderings are called equivalent if they differ by a cyclic permutation. Definition An alternating knot is a knot with a diagram that has crossings alternating between over and under, for a fixed orientation. C. Ramirez (UI) Knots and Braids February 28, 2016 7 / 38 Knots Projections Knot Symmetry Definition A chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is amphicheiral. C. Ramirez (UI) Knots and Braids February 28, 2016 8 / 38 Knots Projections Knot Addition: K = K1#K2 Definition A composite knot is a knot which can be formed by the connected sum of two or more nontrivial knots. The knots that make up a composite knot are called factor knots. The connected sum of two knots, K1 and K2 is formed by removing a small arc from each knot and then connecting the four endpoints by two new arcs in such way that no new crossings are introduced. C. Ramirez (UI) Knots and Braids February 28, 2016 9 / 38 Knots Projections Prime knots Definition A knot is called prime if for any decomposition as a connected sum, one of the factors is the unknot. Theorem (Prime Decomposition Theorem) Every knot can be decomposed as the connected sum of nontrivial prime knots. If K = K1#K2#:::#Kn and K = J1#:::J2#:::#Jm with each Ki and Ji nontrivial prime knots, then m = n and, after reordering each Ki is equivalent to Ji . C. Ramirez (UI) Knots and Braids February 28, 2016 10 / 38 Knots Projections Are two knots the same? Given a knot, is it the unknot? C. Ramirez (UI) Knots and Braids February 28, 2016 11 / 38 Knots Projections Equivalence of knots Definition Two knots K and J are called equivalent if there is a sequence of knots K = K0; K1; :::; Kn = J with each Ki + 1 an elementary deformation of Ki . Theorem If two knots K and J have identical diagrams, then they are equivalent. C. Ramirez (UI) Knots and Braids February 28, 2016 12 / 38 Knots Reidemeister Moves Reidemeister Moves Kurt Reidemeister (1893-1971) Reidemeister was a German mathematician, whose interests were mainly in combinatorial group theory, combinatorial topology, geometric group theory, and the foundations of geometry. C. Ramirez (UI) Knots and Braids February 28, 2016 13 / 38 Knots Reidemeister Moves Reidemeister Moves In 1926, Reidemeister (1927, J. W. Alexander and G. B. Briggs) demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves. Theorem Two knots (or links) are equivalent if and only if their diagrams are related by a sequence of Reidemeister moves. C. Ramirez (UI) Knots and Braids February 28, 2016 14 / 38 Knots Reidemeister Moves Example C. Ramirez (UI) Knots and Braids February 28, 2016 15 / 38 Knots Reidemeister Moves Example? Can you find a sequence of Reidemeister Moves deforming the right-handed trefoil into the left-handed trefoil? C. Ramirez (UI) Knots and Braids February 28, 2016 16 / 38 Knots Knot Invariants Ways of determining knot equivalence Definition Suppose that to each knot, K, we assign a specific quantity q(K). If for two equivalent knots the assigned quantities are always equal, then we call such a quantity a knot invariant. crossing number chirality unknotting number knot genus linking number Alexander polynomial tricolorability Jones polynomial stick number Homfly polynomial Question Does there exist a knot polynomial which distinguishes all knots from each other, or even simpler which distinguishes the unknot from all other knots? C. Ramirez (UI) Knots and Braids February 28, 2016 17 / 38 Knots Knot Invariants Crossing number The crossing number of a knot K, denoted c(K), is the least number of crossings that occur, ranging over all possible diagrams of a knot. Question: Is c(Ki #K2) = c(K1) + c(K2) for a composite knot? True for alternating knots (Kauffman, 1988) C. Ramirez (UI) Knots and Braids February 28, 2016 18 / 38 Knots Knot Invariants Unknotting number The unkotting number of a knot K, denoted u(K), is the least number of crossing changes that are required for the knot to become unknotted, ranging over all possible diagrams. The unknotting number of a knot is always less than half of its crossing number. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. (Scharlemann 1985) u(K) is not necessarily realized in a projection with minimal crossing number. (S. Bleiler, Y. Nalsanishi 1983) C. Ramirez (UI) Knots and Braids February 28, 2016 19 / 38 Knots Knot Invariants Tricolorability A knot or link is called tricolorable if it is possible to do the following: 1 Color every strand in the diagram of the knot. 2 Use a total of three colors. 3 At each crossing, either make all the strands one color or make each strand a different color. Can you tricolor the figure eight knot? Theorem If a diagram of a knot K is tricorolable, then every diagram of K is tricolorable. C. Ramirez (UI) Knots and Braids February 28, 2016 20 / 38 Knots Links Links Definition A link is a collection of knots; the individual knots which make up a link are called the components of the link. C. Ramirez (UI) Knots and Braids February 28, 2016 21 / 38 Knots Links Linking number Given an oriented link we calculated its linking number in order to measure how linked up the components are. In order to do this, we first assign signs to the crossings as follows: Definition The linking number is a link invariant defined for a two-component oriented link as the sum of +1 crossings and -1 crossing over all crossings between the two links divided by 2. C. Ramirez (UI) Knots and Braids February 28, 2016 22 / 38 Knots Links Linking number C. Ramirez (UI) Knots and Braids February 28, 2016 23 / 38 Knots More Knots Other Types of Knots Stick Knot A stick knot is a knot formed out of straight sticks instead of flexible rope. The stick number of any knot is the least number of straight sticks necessary to form the knot. Example: The trefoil knot has a stick number of 6. Fact: No nontrivial knot can be formed by less than 6 sticks. C. Ramirez (UI) Knots and Braids February 28, 2016 24 / 38 Knots More Knots Other Types of Knots Wild Knots A wild knot is any knot that is not tame, meaning it cannot be 3 represented as a polygonal path in R . C. Ramirez (UI) Knots and Braids February 28, 2016 25 / 38 Braids Braids The idea behind braid theory is that braids can be organized into algebraic groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second braid on the twisted strings'. Even though braids are closely related to knots, the theory has developed semi-independently of knot theory. Braid groups were introduced explicitly by Emil Artin in 1925, although they were already implicit in Adolf Hurwitz's work as the fundamental group of a configuration space (1891). C. Ramirez (UI) Knots and Braids February 28, 2016 26 / 38 Braids Definition What is a braid? Take a unit cube, and place in it n strands of string, subject to the following conditions: 1 No part of any strand lies outside the cube.
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