Author:
Amanda Hager
Subject:
Mathematics
Material Type:
Lesson
Level:
Academic Lower Division
Provider:
University of Texas at Austin
Tags:
  • D2S2
  • Mindset
  • License:
    Creative Commons Attribution
    Language:
    English
    Media Formats:
    Text/HTML, Video

    Education Standards

    4.3 Knot Theory

    4.3 Knot Theory

    Overview

    TCCNS CourseMATH 1332: Contemporary Mathematics
    UT Austin CourseM 302: Introduction to Mathematics

    Introduction

    Suggested Resources and Preparation

    Materials and Technology

    • Objects that can be used as knots and links will be essential here. Twine can hold its shape a little bit and can be used. Pipe cleaners work very well here, as they hold their shape and can be held and rotated in space.

    Prerequisite Assumptions

    Students will need to be familiar with the concept of equivalence by distortion and will need to know how to prove that two objects are or are not equivalent by distortion. See Section 4.1.

    Overview and Student Objectives

    Lesson Length

    50 minutes

    Lesson Objectives

    Students will understand that:

    • Applying the math of topology to 1D shapes suspended in 3D space helps us make discoveries and insights in the areas of molecular biology and physics.
    • Knots have their own special class of topological invariants which can be used to discern which diagrams represent similar knots.

    Students will be able to:

    • Use knot diagrams to represent a knot that is modeled in 3D space.
    • Manipulate a knot (and alter a knot diagram) to switch a crossing, untangle a kink, move two strands past each other, or pass a strand under/over a crossing.
    • Compute two simple knot invariants on knot diagrams with up to six crossings.

    Knot Theory

    What is Knot Theory?

    Knot theory is a sub-discipline in topology that deals with 1-D strings rather than all kinds of surfaces and solids and 4-D things. There are special kinds of tests and theorems that apply to knots and links specifically that helps us run tests on computers and solve problems. We care about knots for a lot of reasons. For starters, our DNA is two meters long, and it's supposed to live inside the nucleus of a cell, which is very small. It achieves this through folding, and the shape it folds into helps us understand how it functions.

    image of dna strand at left. gradual zooming out shows how the dna packs and coils into a chromosome at right

    Richard Wheeler at en.wikipediaCC BY-SA 3.0, via Wikimedia Commons

    In the realm of digital security, folks are figuring out how to use the complicated structure of knots to encrypt information in new ways (instead of just using numbers).

    Terms

    • knot is a single 1-D loop in 3-D space. We don't let different parts of the loop touch each other.
    • link is one or more 1-D loops in 3-D space. This means that all knots also count as links.
    • knot invariant is a type of topological invariant specifically designed to be used on diagrams of knots.

    Examples

    trefoil knothopf linkoverhand knot with loose ends
    This is knot. It is also a link even though it has only one string.This is a two string link, but it does not count as a knot.This is a "knot" in a real world sense, but the loose ends mean that in math this doesn't count as a knot or a link.

    Topological invariants for knots

    To show that two knot diagrams are equivalent by distortion, the rules are the same as before: show motion. Either draw a sequence of diagrams or use a descriptive narrative or captions and arrows to communicate what you imagine happening to the strings.

    To show that two knot diagrams are not equivalent by distortion, you need a knot invariant. I'll ask you to learn about two different knot invariants in order to do homework problems in this class.

    Terms

    • The number of strings test is simply counting the number of strings in the diagram. If the number of strings is different, you know the diagrams are not equivalent. If the number of strings is the same, you get no information.
    • The unknotting number is the smallest number of crossings that you need to switch in order to untangle the knot diagram into a plain circle.

    Examples (Video)

    Quiz Questions

    Question 1

    2

    Question 2

    3

    Question 3

    A and B

    Question 4

    A, B, and D

    Question 5

    It is a link but not a knot.

    Question 6

    Yes it is alternating.

    Question 1

    What is the definition of a knot?

    1. It is a 1D loop in 3D space that has tangles in it.
    2. It is a single 1D loop in 3D space, possibly tangled.
    3. It is a tangled string.

    Question 2

    What is the definition of a link?

    1. One or more 1D loops in 3D space that are tangled up.
    2. Multiple 1D loops in 3D space that are tangled up.
    3. One or more 1D loops in 3D space, possibly tangled.
    4. Multiple 1D loops in 3D space, possibly tangled.

    Question 3

    Which of these diagrams represent knots? Select all that apply.

    diagram A is a one-string trefoil, diagram b is a single loop with two twists, diagram c is an overhand knot with two loose ends, diagram d is a two-strand hopf link

    Question 4

    Which of these diagrams represent links?

    diagram A is a one-string trefoil, diagram b is a single loop with two twists, diagram c is an overhand knot with two loose ends, diagram d is a two-strand hopf link

    Question 5

    Does this diagram represent a knot, a link, both, or neither?

    two string diagram with seven crossings

    Question 6

    A picture of a knot or link is alternating if as you follow along the string(s) you see an undercrossing, then an overcrossing, then an undercrossing, etc. until you return to where you started. 

    Does this diagram represent an alternating knot/link?

    two string diagram with seven crossings. following one string you see under over under over under over. following the second string you see over under over under over under over under.

    Homework Questions

    Question 4.3.1

    Which of the following pictures are mathematical knots? Which are mathematical links?

    A is the only knot. A, B, C are links.

    Question 4.3.2

    Grab the strand that goes over two times and pull it downward. You get a band that has two twists in it. Flop over the bottom part two times and you get a plain band. 

    Question 4.3.3

    0. This diagram can be fully untangled without altering any crossings. The piece of the string that goes "over-over" can be pushed up until the string is a figure 8. One twist and then it's a loop.

    Question 4.3.4

    1. It doesn't come untangled all the way as is. You have to switch a crossing. If you switch the upper left crossing then you have the object from Question 2, and we already showed that comes untangled.

    Question 4.3.5

    No. The left diagram has two strings, the right diagram has three.

    Question 4.3.6

    If the number of crossings is odd, it will have one string, and therefore it is a knot. If the number of crossings is even, it will have two strings, and therefore it is a link but not a knot.

    Question 4.3.7

    Because all knots can be unknotted in 4D space, therefore all knots are the unknot. Same with links.

    Question 4.3.8

    Reflective writing prompt. Responses vary.

    Question 4.3.1

    Which of the following pictures are mathematical knots? Which are mathematical links?

    Diagram ADiagram BDiagram CDiagram DDiagram E

    trefoil knot

     

    hopf link

     

    two string diagram with 8 crossings

     

     

    two strings twisted three times together, all four ends loose

     

     

    one string diagram with six crossings and two loose ends

     

     

    Question 4.3.2

    Show that this knot is equivalent by distortion to the unknot (a plain circle).

    one string diagram with crossings that go over over under over under under under over

    Question 4.3.3

    What is the unknotting number of this knot? Explain your response.

    one string diagram with three crossings that go over over over under under under

    Question 4.3.4

    What is the unknotting number of this knot? Is this knot equivalent by distortion to the unknot?
    one string diagram with four crossings, alternating

    Question 4.3.5

    Are these two links equivalent by distortion? Why or why not?
    Two string diagram with nine crossingsthree string diagram with eight crossings

    Question 4.3.6

    Suppose we take two strings that are twisted around each other a number of times and then we connect the top right end to the bottom right end with a "C" curve and the top left end to the bottom left end with a "C" curve. Here is an example of two strings twisted around each other three times and then connected on each side:
    two strings twisted three times together, all four ends loosetwo strings twisted three times together, left loose ends connected together, right loose ends connected together.
    What property of the twist determines if the resulting diagram represents a knot (which is also a link) or just a link? I recommend drawing a table showing 1 through 5 twists and then connecting the sides with "elephant ears."

    Question 4.3.7

    Why is the study of knots and links vacuous or pointless if you allow yourself to move in 4-dimensional space?

    Question 4.3.8

    Background

    The cool thing about knot theory is that it's surprisingly useful. It was invented about the same time we figured out that DNA existed (mid 1800s), but we didn't figure out that knot theory could be used to understand the shapes that DNA strands make until the 1980s. It's also now used by physicists who try to understand the universe though string theory.

    Assignment

    Think of something from a different class that you've taken that you didn't think was useful at the time, but later in your life it turned out to be useful. Describe the thing and how it turned out to be useful to you.

    If you can't possibly think of anything, that's okay. Instead, think of something that you really liked learning about, but you still have no idea how it's useful yet. Describe the thing and then describe something you could do to figure out how it's useful. Notice that you don't have to actually find the applications; just describe what work you could do to search for those applications.