Here you find some quick questions to test your understanding of the material presented in the lecture, grouped by chapter.

 

Chapter 1: Introduction

  1. Explain the difference between a geometrical and topological knot. Is any topological knot also a geometrical one?
  2. Explain the difference between a knot and a link. Is every link also a knot?
  3. Give an example of a link invariant.
  4. Draw diagrams of the unknot, the trefoil, the Whitehead link, the Hopf link, the 2-unlink, and the figure eight knot.
  5. How does the writhe of an oriented knot change when the orientation of the knot is reversed?
  6. State Reidemeister’s Theorem.
  7. Given two diagrams, how would you proceed to show that these diagrams are equivalent?
  8. Given two diagrams, how would you proceed to show that these diagrams are not equivalent?
  9. Explain the differences between planar, regular, and ambient isotopy of diagrams. Can one deduce some of these equivalences from others in the list? (For example, are two diagrams that are regular isotopic also automatically ambient or planar isotopic?)
  10. Show that the map $I$ from the set of all geometrical links to the set of all topological links, which maps each link $L$ to its equivalence class $[L]$, i.e. $I(L):=[L]$, is a complete link invariant. Discuss the usefulness of this invariant.

 

Chapter 2: Colourability, the Determinant, the Alexander Polynomial, and the Colouring Group

  1. Define what it means for a knot to be 3-colourable. Give an example of a knot that is 3-colourable, and an example of a knot that is not 3-colourable.
  2. Define the determinant of a link.
  3. Let $K$ be a knot with determinant $195$. For which integers $p$ is $K$ $p$-colourable?
  4. Does there exist a link that is $7$-colourable, but not $21$-colourable? Why?
  5. Show that the property of $3$-colourability does not change under Reidemeister moves of type II.
  6. A friend has been working on a puzzle for two weeks without solving it. The puzzle consists of two interlocked strings:
    puzzle
    The task is to disentangle the blue string from the yellow string, i.e. to “free” the blue part (without cutting the yellow string, of course). Having heard that you know about knot theory, your friend comes to you and asks for your help. How would you help him?
  7. If Col$(L)\cong{\mathbb Z}/3{\mathbb Z}+{\mathbb Z}/3{\mathbb Z}+{\mathbb Z}/15{\mathbb Z}$, for which primes $p$ is $L$ $p$-colourable?
  8. Let $K$ be a knot with crossing number $c(K)=5$ and determinant $\det K=105$. What are the possible forms of the colouring group ${\mathbb Z}/e_1{\mathbb Z}+…+{\mathbb Z}/e_n{\mathbb Z}$ of $K$?
  9. Explain why the Alexander polynomial is a stronger invariant than the determinant.
  10. Call a knot $K$ invertible if it has the property that once given an orientation, it is ambient isotopic to itself, but with the opposite orientation. Can $t^3+t+t^{-1}$ be the Alexander polynomial of an invertible knot?

 

Chapter 3: Polynomial Invariants

  1. Complete a table of the following form and learn it by heart. It is part of this exercise to understand the shorthand notations I have put into the table …
    Polynomial Symb. Variables Def. Inv. skein/rec. rel. ${\cal L}\sqcup{\cal U}$ comments
    Alexander $\Delta$ $t$ only def. up to factor $\pm t^m$, $m\in\mathbb Z$
    bracket pol. $\langle\cdot\rangle$ $A,B,d$ factor $d$
    Kauffman br. 0, II, III
    $X$
    Jones 0-III
    $P$
    $z$
  2. Explain why none of the following can be bracket polynomials of some diagram: $$A^3+BA^2d+2BABd^4+B^3,\quad 9A^3-B^3,\quad \frac{1}{2}A^2B+\frac{3}{2}AB^2,\quad 2A^{10}+4B^2A^8+B^3A^7+B^{10}$$
  3. How does one obtain the normalized bracket polynomial from the Kauffman bracket, and in which sense is the normalized bracket polynomial “better” than the Kauffman bracket?
  4. Compare the recursion relation of the Kauffman bracket to the skein relation of the HOMFLY polynomial, and explain the differences in the appearing local diagrams.
  5. Explain how one can use the Jones polynomial to test if a given knot is chiral or not. Also explain under which circumstances this test is inconclusive.
  6. How does the Jones polynomial of a knot change when the orientations of all its components are reversed?
  7. Let $3_1$ denote a topological oriented trefoil and $r(3_1)$ its reverse (not the mirror image, just the other orientation). Convince yourself of $3_1\# 3_1\cong 3_1\# r(3_1)$.
  8. What is a state of a diagram? Why is computing the bracket polynomial with its recursion relation often easier than computing it as a sum over states?
  9. Is the crossing number of the composition of two oriented knots $K_1,K_2$ always equal to the sum of their crossing numbers, $c(K_1\# K_2)=c(K_1)+c(K_2)$? Explain what the problem in showing this is. This question is still open, more than 100 years after it was first posed.
  10. If you want more practice in computing polynomial invariants (which is not really a quiz question): Take a knot catalogue and compute, say, the Jones, Conway, HOMFLY, etc polynomials of some randomly chosen knot $n_m$ and check your results with the Wolfram knot explorer or compare to the knot atlas by visiting the section “polynomial invariants” on http://katlas.org/wiki/n_m.

 

Chapter 4: Algebraic Invariants

  1. Explain the graphical notation for the matrix product and trace of matrices, and for the scalar product of (real) vectors.
  2. Let $T$ be a tensor with six upper and four lower indices, and $X$ a tensor with four upper and six lower indices, all from the same finite index set $I$. How many upper/lower indices do $2X$, $TX$, $XT$, $T\otimes X$, $X\otimes T$, $X+T$ have?
  3. Let $I$ be a finite index set, and $R\in M_{I^2}$ be defined by $R^{ab}_{cd}=\delta^{ab}\delta_{cd}$. Compute
    2tr
  4. Why does one need to consider oriented diagrams to compute partition functions?
  5. How does the partition function $Z_{I,R,\bar R}$ change when $R$ and $\bar R$ are scaled by some factor $\lambda\in\mathbb C$, i.e. what is $Z_{I,\lambda R,\lambda\bar R}$?
  6. How many components, arcs, and edges does the standard diagram of the Whitehead link have? How many colourings of edges does it have when there are $n$ colours?
  7. What are the relations of the Temperley-Lieb algebra?
  8. Show that for any two oriented diagrams $D,D’$, the partition function (with some $I,R,\bar R$) satisfies $Z(D\sqcup D’)=Z(D)\cdot Z(D’)$.
  9. Let $I$ be a finite index set and $R,\bar R\in M_{I^2}$. Come up with a condition on $R,\bar R$ that guarantees invariance of the partition function $Z_{I,R,\bar R}$ under type I moves.
  10. $\star$ Let $I$ be a finite index set, and $R\in M_{I^2}$ be defined by $R^{ab}_{cd}=\delta^a_d\delta^b_c$. Show that $R$ satisfies all conditions of Theorem 4.3, with $\bar R=R$, and even the “type I condition” from question 9, i.e. the corresponding partition function $Z$ is an invariant under ambient isotopy. But this partition function is a weak invariant: Show that $Z(L)=|I|^{N(L)}$, where $N(L)$ is the number of components of $L$.
    Hint: Don’t compute $Z(L)$ explicitly, but use $R=\bar R$ and knowledge about $Z$ on trivial links.

 

Chapter 5: Braids and Links

  1. Explain what a braid of $n$ strands is.
  2. Let $b:=\sigma_1\sigma_2\cdots\sigma_{n-1}\in B_n$ and $b’:=\sigma_{n-1}^{-1}\sigma_{n-2}^{-1}\cdots\sigma_1^{-1}\in B_n$. Show $bb’=e$ by calculation and also by drawing the corresponding braid diagrams.
  3. How many braids of $n=2$ strands are there?
  4. Represent the braidsomebraid
    on 5 strands as a suitable product of the elementary braids $\sigma_1^{\pm1}$, $\sigma_2^{\pm1}$, $\sigma_3^{\pm1}$, $\sigma_4^{\pm1}\in B_5$.
  5. Explain the Markov moves, and state Markov’s Theorem.
  6. Find a braid that closes to the Whitehead link.
  7. Explain how braids can give rise to link invariants.
  8. Realize a trefoil as the closure of a braid on five strands.
  9. List the relations in the braid group of $n$ strands.
  10. Describe the relation between the braid group and the Jones polynomial.