Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · Advanced undergraduate

Knot theory

Name: ______________________

Date: ______________________

You will learn to

Represent knots and links by diagrams, and state Reidemeister's theorem
Prove that Fox colourings and the linking number are invariants, and use them to distinguish knots and links
Compute the Kauffman bracket and the Jones polynomial of small knots, and show that the trefoil is not equivalent to its mirror image
Construct a Seifert surface by Seifert's algorithm and compute its genus
Separate what is known about knots from what is open, such as whether the Jones polynomial detects the unknot

1.
Which of these are knot invariants (the same for all diagrams of equivalent knots)? (Choose all that apply.)
- A.The crossing number: the fewest crossings in any diagram of the knot
- B.The number of crossings in a given diagram
- C.Tricolourability
- D.The number of Fox 5-colourings
- E.The writhe of a diagram
- F.The number of arcs in a diagram
2.
How many Fox 3-colourings (including the constant ones) does the standard 3-crossing diagram of the trefoil have?
3.
Neither the unknot nor the figure-eight knot is tricolourable. What can you conclude from this alone?
- A.Nothing about whether they are equivalent.
- B.They are equivalent knots.
- C.The figure-eight knot is unknotted.
- D.Tricolourability is not really an invariant.
4.
Solomon's knot, found in Roman mosaics, medieval manuscripts and textiles from many cultures, is a link of two loops whose standard diagram has $4$ crossings, all between the two components and all of the same sign $+1$ . What is its linking number?
5.
In the proof that Fox colourings survive move II, a strand of colour $x$ passes over a strand cut into arcs $a, m, b$ , with equations $2x = a + m$ and $2x = m + b$ . Why does this show that colourings before and after the move correspond one to one?
- A.The equations force $m = 2x - a$ and $b = a$ : the inside colour is determined, and the outgoing colour equals the incoming one, as it does with no crossings.
- B.Both diagrams have the same number of crossings.
- C.Because the operation $x \triangleright y = 2x - y$ is self-distributive.
- D.Because constant colourings always work.
6.
The standard diagram of the cinquefoil $5_1$ (the $(2, 5)$ torus knot) has $5$ crossings, and Seifert's algorithm produces $2$ Seifert circles. What is the genus of the resulting Seifert surface?
7.
The right-handed trefoil has Jones polynomial $V(t) = t + t^3 - t^4$ . What is the Jones polynomial of the left-handed trefoil, its mirror image?
- A. $t^{-1} + t^{-3} - t^{-4}$
- B. $-t - t^3 + t^4$
- C. $t + t^3 - t^4$ , the same
- D. $t^4 - t^3 - t$
8.
Which of these statements is actually proved in the lesson's Lean file?
- A.jonesMonomials trefoilPD ≠ jonesMonomials mirrorTrefoilPD
- B."The right-handed trefoil is not ambient isotopic to the left-handed trefoil"
- C.Reidemeister's theorem
- D.Every knot has a Seifert surface
9.
Write count_colourings(n_arcs, crossings, p) that counts the Fox $p$ -colourings of a diagram by brute force. crossings is a list of triples (over, under1, under2) of arc numbers in range(n_arcs); a colouring is a tuple of colours in range(p), and it is valid if (2 * c[over] - c[under1] - c[under2]) % p == 0 at every crossing.
Computational — Practice
10.
The figure-eight knot has $V(t) = t^{-2} - t^{-1} + 1 - t + t^2$ , which satisfies $V(t) = V(t^{-1})$ . What does this prove?
- A.Nothing about chirality by itself: it is consistent with the figure-eight knot being amphichiral, but does not prove it.
- B.The figure-eight knot is equivalent to its mirror image.
- C.The figure-eight knot is not equivalent to its mirror image.
- D.The figure-eight knot is the unknot.

Answer key — Knot theory

1.
Invariants: the crossing number, tricolourability and the number of Fox 5-colourings. The number of crossings, the number of arcs and the writhe of a particular diagram all change under move I (a kink), so they are properties of diagrams, not of knots.
2.
At each crossing the three arcs carry colours with $x + y + z \equiv 0 \pmod 3$ , and all three crossings give this same equation. Choosing two colours freely ( $3 \times 3 = 9$ ways) determines the third. So there are $9$ colourings: $3$ constant and $6$ using all three colours. The Lean file checks this by trying all $27$ assignments (trefoil_three_colourings).
3.
An invariant satisfies: equivalent knots have equal values. The converse fails in general, so equal values prove nothing. Here a stronger invariant does the job: the figure-eight knot has $25$ Fox 5-colourings and the unknot $5$ , so they are different.
4.
$\mathrm{lk} = \tfrac12 (1 + 1 + 1 + 1) = 2$ . Since the unlink has linking number $0$ , Solomon's knot cannot be pulled apart, and since the Hopf link has linking number $1$ , it is not the Hopf link either.
5.
From $2x = a + m$ we get $m = 2x - a$ , and subtracting the equations gives $b = a$ . So a colouring of the outside extends uniquely to the inside exactly when the two ends of the lower strand have the same colour — the same condition as before the move, where the lower strand is one arc. Hence the colourings correspond bijectively. In Lean this is reidemeister_two.
6.
$\chi = s - c = 2 - 5 = -3 = 1 - 2g$ , so $g = 2$ . In fact $g(5_1) = 2$ : for torus knots (more generally, for alternating and for positive diagrams) Seifert's algorithm gives a surface of minimal genus — a theorem we do not prove here.
7.
Reflecting switches every crossing: A- and B-smoothings swap, so $\langle \bar D\rangle(A) = \langle D\rangle(A^{-1})$ , and every sign flips, so $w(\bar D) = -w(D)$ . Hence $f(\bar D)(A) = f(D)(A^{-1})$ and $V_{\bar K}(t) = V_K(t^{-1}) = t^{-1} + t^{-3} - t^{-4}$ . Since this differs from $V_K$ , the two trefoils are not equivalent (Corollary 6).
8.
Only the computation jonesMonomials trefoilPD ≠ jonesMonomials mirrorTrefoilPD is proved in Lean. The topological statements — Reidemeister's theorem, the invariance of $V$ , chirality of the trefoil as knots, Seifert surfaces — are proved in the lesson on paper, not in Lean. Being precise about this boundary is part of honest formalisation.
9.
```
from itertools import product

def count_colourings(n_arcs, crossings, p):
    count = 0
    for c in product(range(p), repeat=n_arcs):
        if all((2 * c[o] - c[u] - c[v]) % p == 0 for o, u, v in crossings):
            count += 1
    return count
```
Brute force tries $p^{n}$ colourings, which is fine for small diagrams. Because the conditions are linear, the colourings form a vector space over $\mathbb{Z}/p$ (for $p$ prime), so their number is a power of $p$ and can be found quickly by Gaussian elimination — as the lesson's experiment does.
10.
$V_{\bar K}(t) = V_K(t^{-1})$ , so $V(t) = V(t^{-1})$ says only that $4_1$ and its mirror image have the same Jones polynomial — consistent with, but not a proof of, their equivalence. The figure-eight knot is in fact amphichiral (by an explicit isotopy), but the knot $9_{42}$ shows that a symmetric Jones polynomial can occur for a chiral knot. The computation does prove that $4_1$ is knotted, since $V \ne 1$ .