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Let $L$ be a framed knot in $S^3$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$.

How can I interpretate the following sentence?

"We identify $U$ with $S^1 \times B^2$ so that $L$ is identified $S^1 \times 0$, where $0$ is the center of $B^2$ and the given framing of $L$ is identified with a constant normal vector field on $S^1 \times 0 \subset S^1 \times B^2$."

I know $U$ is homeomorphic to $S^1 \times B^2$. What confuses me is that framing part.

For example, if framing number is $1$, is the identification the result of one Dehn twist along meridian on $S^1\times B^2$?

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Yes, that's exactly right. If you do $n$-framed surgery on a knot, you glue a solid torus into a regular neighborhood of the knot by Dehn twisting $n$ times around a meridian and then regluing. Rational surgery allows more elements of the mapping class group of the torus $S^1\times S^1$ than just these twists.

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