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 Incoming links: Integer surgery coefficient

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 141 / 1 / d / Solution Created 2026-10-03 Updated 2026-10-14

Take a Hopf link with its two unknot components labeled by integer surgery coefficients

2, 3

.

Figure 1.
Surgery diagrams and continued-fraction coefficients
. Details.

The slam-dunk move eliminates a meridional component with coefficient

b

and replaces the other coefficient

a

by

a - 1/ b

. Eliminating the component with coefficient three gives

[2, 3]^{-} := 2 - \frac{1}{3} = \frac{5}{3} .

Thus this Dehn surgery on a framed link becomes

5/3

Dehn filling on the unknot, with the required orientation. As a check, its surgery linking matrix is

(2113),

whose determinant is five.

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