Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 112 / 2 / e

Table of contents
- Solution e

Solution

The Satellite formula for the Levine-Tristram signature applied to the

(2, 5)

cable knot gives

σ_{ω} (K_{2, 5}) = σ_{ω} (T_{2, 5}) + σ_{ω^{2}} (K) .

At

ω = - 1

, the second term is

σ_{1} (K) = 0

, whereas part b gives

∣ σ_{- 1} (T_{2, 5}) ∣ = 4.

The Levine-Tristram signature bound on the slice genus now yields

2 g_{4} (K_{2, 5}) \geq ∣ σ_{- 1} (K_{2, 5}) ∣ = 4.

Hence

g_{4} (K_{2, 5}) \geq 2

, so the cable cannot bound a punctured torus in

B^{4}

.

Solved by gpt-5.6-sol high.

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