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Let $Y$ be a closed, oriented 3-manifold equipped with an oriented null homologous knot $K$. I want to understand the relative $spin^c$ structures for $(Y,K)$. There is canonical zero surgery $Y_0(K)$. For a $spin^c$ structure $s\in Spin^c(Y_0(K))$ consider restriction of $s$ to $Y-K$. My question is how can we uniquely extend it to $Y$ ?

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