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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-141/2/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 141 / 2 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Reflection through the projection plane reverses the orientation of

S^{3}

and exchanges the overpassing and underpassing strands at every crossing. Thus switching all crossings gives the mirror of a link, with the component orientations transported by the reflection.

Define

W (L) (q) = V (\overline{L}) (q^{- 1})

. Reflection exchanges

D_{-}

with

D_{+}

and preserves the oriented smoothing. Apply the printed skein relation to the reflected triple at parameter

q^{- 1}

q^{- 1} W (D_{+}) - q W (D_{-}) = (q^{- 1/2} - q^{1/2}) W (D_{0}) .

(1)

Multiplication by

- 1

gives the original skein relation for

W

. Also

W (U_{1}) = 1

. The stated uniqueness of the Jones polynomial now gives

W = V

, hence

V (\overline{L}) (q) = V (L) (q^{- 1}) .

(2)

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