 Solution

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

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

Insert a kink on any component of an oriented link diagram for

L

. Switching the kink's crossing gives another link diagram of

L

; its oriented smoothing separates off a small unknot in a ball disjoint from the rest of

L

. The skein relation therefore gives

(q - q^{- 1}) V (L) = (q^{1/2} - q^{- 1/2}) V (L ⨿ U_{1}) .

Cancel the nonzero factor

q^{1/2} - q^{- 1/2}

V (L ⨿ U_{1}) = (q^{1/2} + q^{- 1/2}) V (L) = V (U_{2}) V (L) .

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