Peter Cromwell's `Knots and Links' has a definition for a locally flat knot that I'm struggling to understand. It's more the terminology used rather than the concept (I hope) but I can't find a comparable formulation to help me. The part of the definition I'm struggling with is a follows:

"A point $p$ in a knot $K$ is locally flat if there exists some neighbourhood of $p$, call it $U$, such that the pair $(U, U\cap K)$ is homeomorphic to the unit ball $B_0(1)$, plus a diameter."

I simply don't understand the meaning of "plus a diameter", and I can't seem to find the phrase used in this context anywhere else. I (believe) I can visualise local flatness, and how it works in relation to knots, but I won't be confident as long as the crucial phrase remains a mystery to me!

Thanks in advance for any help.


1 Answer 1


Let $B$ be the unit ball in $R^3$ and let $X$ be the $x$-axis. What he means is that there be an homeomorphism of pairs between $(U,U\cap K)$ and $(B,B\cap X)$.

  • $\begingroup$ You could also ask that there be an homeo of pairs with $(I\times D^2,I\times\{(0,0)\})$ with $I=[0,1]$ and $D^2$ the open unit disc in the plane. $\endgroup$ May 15, 2014 at 19:31
  • $\begingroup$ Wonderful! It's exactly as I expected, but the terminology threw me. Thank you for your help! $\endgroup$ May 15, 2014 at 19:34

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