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If $G$ is a graph and $G-v$ is linkless for some vertex $v$, is $G$ necessarily knotless?

Of course, one can assume that $v$ is adjacent to every vertex in $G-v$.

Here, a graph is linkless if it has an embedding in 3-space with no two linked cycles. And a graph is knotless if it has an embedding in 3-space such that every cycle is an unknot.

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No, this is false.

Your question was posed by Adams in 1994, and was disproved by Foisy in 2003.

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