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Are braid links proper links? Or are the concepts involved unrelated?

flag	asked Jan 29 2011 at 18:34 tuppsphd 43●5

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Could you clarify what you mean by "braid link" and "proper link"? – Ryan Budney Jan 29 2011 at 18:40

Oops! Alexander's Thm says that all tame links are closed braids, so "braid links"="tame links." I'm studying a result of Murakami in "A recursive calculation of the Arf invariant of a link" (J. Math. Soc. Japan 38, #2 (1986). Murakami says "a link $L$ is proper if $lk(K,L−K)$ is even for every component of $K$ in $L$, where $lk$ means a linking number," and I don't have a very good visual picture for what that means. In that sense, are most tame links proper? Few? – tuppsphd Jan 30 2011 at 3:36

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For a picture of what "linking number" means, see: en.wikipedia.org/wiki/Linking_number As Paul mentions, Murakami's notion of "proper link" is fairly special and most links aren't of that sort. Just so you know "proper link" isn't a standard terminology. – Ryan Budney Jan 30 2011 at 6:40

Paul · Accepted Answer · 2011-01-30 05:01:09Z UTC

According to the definitions in your comment, the closure of the 2 stranded braid with braid word $\sigma_1^6$ is not proper, since the closure is a 2 component link with linking number 3.

It's hard to think of a more straightforward definition than what murakami says, but if you want examples, any link with all pairwise linking numbers even is proper. If you want odd linking number consider three fibers of the Hopf fibration.

Are braid links proper links?

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