Exercises

Problem 1: Crossing Number

Show that there exists no knot with crossing number two. What if you allow for links?

Problem 2: Writhe and Linking Number

Compute the writhe and linking number of both the following oriented link diagrams. Are these diagrams equivalent?

Problem 3: Mirror Images of Links and Chirality

For a two-dimensional plane $E$ in ${\mathbb R}^3$, we denote by $R_E:{\mathbb R}^3\to{\mathbb R}^3$ the reflection about $E$.

1. Given a geometrical link $L$ and any two planes $E,E’\subset{\mathbb R}^3$, show that the reflected links $R_E(L)$ and $R_{E’}(L)$ are equivalent.
Hint: This requires a bit of linear algebra/geometry. What kind of transformation is $R_E\cdot R_{E’}$?
2. We define the mirror image ${\cal L}^*$ of a topological link $\cal L$ as follows. If ${\cal L}=[L]$ for some geometrical link $L$, we set ${\cal L}^*:=\{L’\,:\,L’\cong R_E(L)\}=\{\text{all links equivalent to } R_E(L)\}$, where $E$ is some two-dimensional plane in ${\mathbb R}^3$. Show that this definition does not depend on the choice of the plane $E$, and also not on the choice of the representative $L$ within the class $\cal L$.
3. Denoting the just defined mirror image of ${\cal L}$ by ${\cal L}^*$, show that the diagrams of ${\cal L}^*$ are precisely the diagrams of $\cal L$, but with all overcrossings replaced by undercrossings and vice versa.
4. A link $\cal L$ is called chiral if ${\cal L}\neq{\cal L}^*$ (such links occur in different “left” and “right” versions) and achiral (or amphichiral) if ${\cal L}={\cal L}^*$. Show that the figure eight knot is achiral by working with a model of string/cable — memorise the necessary moves so that you can demonstrate achirality. Be sure not to knot a trefoil instead of a figure eight or you will have a hard time!

Problem 4: Invariance properties of writhe

Give a proof of Lemma 1.16 by using Reidemeister moves.
Lemma 1.16:  Writhe is an invariant of oriented links under regular isotopy, but not under ambient isotopy.

is non-trivial, i.e., not equivalent to the 2-unlink.

Problem 6: Colourability, Determinants, and the Alexander Polynomial

Consider the following five knot diagrams

a) Working with colourings, show that $0_1$, $4_1$ and $5_1$ are not $3$-colourable, but $3_1$ and $6_1$ are.

b) Working with determinants, show that no two of the shown diagrams belong to equivalent knots except maybe the pair $4_1$, $5_1$.

c) Working with the Alexander polynomial, show that also the knots represented by the diagrams $4_1$ and $5_1$ are not equivalent.

Problem 7$\star$: The Alexander Polynomial and Mirror Images

Show that given a topological knot $\cal K$ and its mirror image ${\cal K}^*$ (defined in problem 3), there exist orientations on $K,K^*$, diagrams $D$ of $\cal K$, $D^*$ of ${\cal K}^*$, and labelings of (the crossings and arcs of) $D,D^*$ such that the Alexander polynomials $\Delta_D$ and $\Delta_{D^*}$ coincide. Thus the Alexander polynomial can not distinguish between a knot and its mirror image.

Hint: Use the freedom of choosing the reflection plane to find a simple connection between the matrices underlying $\Delta_D$ and $\Delta_{D^*}$.

Problem 8: Colouring groups

Compute the colouring group of the knot $K$ with the diagram

That is, find integers $d_1,…,d_n\in{\mathbb N}_0$ such that Col$(K)\cong{\mathbb Z}/d_1{\mathbb Z}+…+{\mathbb Z}/d_n{\mathbb Z}$.

Problem 9: Bracket polynomials

a) Compute the bracket polynomial $\langle D\rangle$ as a polynomial in $A,B,d$, from its definition as a sum over states, for the standard diagram $D$ of the Hopf link, namely

b) Compute the bracket polynomial $\langle D\rangle$ as a polynomial in $A,B,d$ by using its recursion relation for the diagram

Problem 10: Brackets and bracelets

a) Prove that the following relation holds for the Kauffman bracket (i.e., the bracket polynomial with $B=A^{-1}$ and $d=-A^2-A^{-2}$)

b) $\star$ Calculate the Kauffman bracket of the “bracelet” link diagram $D_n$ with $n\in\mathbb N$ components, depicted below.

Problem 11: Jones and HOMFLY Polynomials

Compute the Jones and HOMFLY polynomials of the following five links (you need to pick an orientation)

Problem 12: Conway and Alexander polynomials

Compute the Conway polynomials $\nabla_{3_1}(z), \nabla_{4_1}(z)$ of the (oriented) trefoil $3_1$ and figure of eight knots $4_1$ (choose orientations). You may use the Conway polynomials of the Hopf links,

Then replace the variable $z$ by $t^{-1/2}-t^{1/2}$ and compare with the Alexander polynomials of the trefoil (computed in lecture 8) and the figure of eight (computed in problem 6).

Problem 13: The Jones polynomial and composite knots

Show that for any two oriented knots $K,K’$, there holds the equation $$V_{K\# K’}(t)=V_K(t)\cdot V_{K’}(t)$$ between the three Jones polynomials $V_{K\# K’},V_K,V_{K’}$.

To do so, you can proceed as follows:

1. Use the skein relation of the Jones polynomial to express $V_{K\# K’}$ in terms of the Jones polynomial $V_{K\sqcup K’}$ of the 2-link $K\sqcup K’$, which consists of the two components $K,K’$, with no crossings between $K$ and $K’$ (see the schematic picture below).
2. Consider the bracket polynomial, and prove that $\langle K\sqcup K’\rangle=d\,\langle K\rangle\cdot\langle K’\rangle$ as polynomials in $A,B,d$, by working with sums over states. Deduce an analogous equation for the Kauffman bracket.
3. Show that the writhe satisfies $w(K\sqcup K’)=w(K)+w(K’)$, and use this and part b) to find a relation expressing $V_{K\sqcup K’}$ in terms of $V_K$ and $V_{K’}$.
4. Use part (1) and (3) to show the claimed equation $V_{K\# K’}=V_K\cdot V_{K’}$.

You can also test the validity of $V_{K\# K’}=V_K\cdot V_{K’}$ by considering examples for $K,K’$.

Problem 14: Partition functions

Consider the diagram $D$ of the oriented figure of eight knot
an arbitrary index set $I$, and tensors $R,\bar R\in M_{I^2}$.

• Write down the partition function of $D$.
• Now assume that $R=G\otimes H$ and ${\bar R}=G^{-1}\otimes H^{-1}$ for invertible matrices $G,H\in M_I$. Show that in this case, $$Z(D)={\rm Tr}(G^2)\cdot {\rm Tr}(H^{-2})\cdot {\rm Tr}(HG^{-1}HG^{-1})\,.$$ What do you get when $H=G=1\in M_I$?

Problem 15: Adjoints and tensor products

Let $I_1,I_2,J_1,J_2$ be finite index sets, and $U\in M_{I_1,J_1}$, $V\in M_{I_2,J_2}$ be arbitrary matrices. Using the graphical notation for matrices, verify that
$$(U\otimes V)^*=U^*\otimes V^*\,.$$

Problem 16: Temperley-Lieb algebra

Let $I$ be a finite set of $n=|I|$ elements, and $E_{a,b}\in M_I$ the canonical matrix units (that is, $(E_{a,b})^i_j=\delta^i_a\delta^b_j$ for all $a,b,i,j\in I$). Define
$$U_1:=n^{-1/2}\sum_{i,j\in I}E_{ij}\otimes 1\,,\qquad U_2:=n^{1/2}\sum_{i\in I}E_{ii}\otimes E_{ii}\,,$$ and show that the following equations hold $$U_1^2=n^{1/2}\,U_1\,,\qquad U_2^2=n^{1/2}U_2\,,\qquad U_2U_1U_2=U_2\,,\qquad U_1U_2U_1=U_1\,.$$

Problem 17 $\star$ : Yang-Baxter equation

Let $I$ be a finite index set and $R\in M_{I^2}$ a solution of the Yang-Baxter equation, i.e. $$(R\otimes 1)(1\otimes R)(R\otimes 1)=(1\otimes R)(R\otimes 1)(1\otimes R)\,.$$
Show that for invertible $A\in M_I$, also $S:=(A\otimes A)R(A^{-1}\otimes A^{-1})$ is a solution of the Yang-Baxter equation. Show furthermore that if $S=R$, then also $T=(1\otimes A)R(1\otimes A^{-1})$ is a solution of the Yang-Baxter equation.