Practice: Knot theory · The Lumenary

Knot theory

Warm-upHuman

Which of these are knot invariants (the same for all diagrams of equivalent knots)? (Choose all that apply.)

Choose all that apply

The crossing number: the fewest crossings in any diagram of the knotThe number of crossings in a given diagramTricolourabilityThe number of Fox 5-colouringsThe writhe of a diagramThe number of arcs in a diagram

Knot theory

Warm-upHuman

How many Fox 3-colourings (including the constant ones) does the standard 3-crossing diagram of the trefoil have?

Knot theory

CoreHuman

Neither the unknot nor the figure-eight knot is tricolourable. What can you conclude from this alone?

Choose one

Nothing about whether they are equivalent.They are equivalent knots.The figure-eight knot is unknotted.Tricolourability is not really an invariant.

Knot theory

CoreHuman

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?

Knot theory

CoreHuman

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?

Choose one

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.Both diagrams have the same number of crossings.Because the operation

x \triangleright y = 2x - y

is self-distributive.Because constant colourings always work.

Knot theory

CoreHuman

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?

Knot theory

CoreHuman

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?

Choose one

t^{-1} + t^{-3} - t^{-4}

-t - t^3 + t^4

t + t^3 - t^4

, the same

t^4 - t^3 - t

Knot theory

CoreFormal

Which of these statements is actually proved in the lesson's Lean file?

Choose one

jonesMonomials trefoilPD ≠ jonesMonomials mirrorTrefoilPD"The right-handed trefoil is not ambient isotopic to the left-handed trefoil"Reidemeister's theoremEvery knot has a Seifert surface

Knot theory

CoreComputational

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.

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Knot theory

StretchHuman

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?

Choose one

Nothing about chirality by itself: it is consistent with the figure-eight knot being amphichiral, but does not prove it.The figure-eight knot is equivalent to its mirror image.The figure-eight knot is not equivalent to its mirror image.The figure-eight knot is the unknot.