studying some knot theory and just had a question, wondering if anyone can clarify or shed some light:

I'm reading The Knot Book by Colin C. Adams, and it says that Composite knots are not completely determined by Dowker Notation.

Upon a little further research, I found that a composite knot 'may be reflected in the line between it's entry and exit points', when recovered by Dowker Notation.

I'm not sure exactly what this means , i.e. how can a Knot have an entry and exit point? And why is this the case? Is anyone able to clarify this a bit more for me?

Many thanks!

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    $\begingroup$ Think of knots as being contained in a box. When you connect sum two knots, the picture usually looks like two boxes, side by side, with two lines that do not cross connecting them. So, the line we are referring to is a line between those two connecting arcs. $\endgroup$ – N. Owad Mar 28 '14 at 15:36

The Wikipedia page on Dowser notation says

the recovered knot may differ from the original by being a reflection or (more generally) by having any connected sum component reflected in the line between its entry/exit points

which is similar to, but subtly different from what you quote.

If you take a knot and take its mirror image, then that mirror image is represented by the same Dowker notation, but is not, in general, an equivalent knot.

Now suppose you have a non-prime knot, i.e. one that you can write as the knot sum of two knots. Say we detached a small segment from $P_1$ to $P_2$ in one knot, a small segment from $Q_1$ to $Q_2$ in the other knot and connected $P_1$ to $Q_1$ and $P_2$ to $Q_2$. (There are of course restrictions on when this can be done.) Now you can take the mirror image of one of the two components. That doesn't change the Dowker notation, but can give a non-equivalent knot. The points $Q_1$ and $Q_2$ (or $P_1$ and $P_2$) are the entry and exit points of the component.

So it's not the entry and exit points of the whole knot that the Wikipedia page is talking about, but of the component that is being mirrored.

  • $\begingroup$ Thanks for your great answer. So, would it be correct to say that for a composite knot $J#K$, then the possible knots recovered by Dowker's Notation is $ J#K, *(J#K), *J#K, J#*K $? Where $*K$ is the mirror image of $J$? $\endgroup$ – JackReacher Mar 31 '14 at 6:12
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    $\begingroup$ @JackReacher Assuming $J$ and $K$ themselves are prime knots, yes, I think so. In a particular case, it might happen that, say, $J$ and $*J$ are equivalent. $\endgroup$ – Magdiragdag Mar 31 '14 at 6:54
  • $\begingroup$ Thanks. Sorry, I meant $*K$ is the mirror image of $K$ above, but I think you know what I meant. $\endgroup$ – JackReacher Mar 31 '14 at 6:58
  • $\begingroup$ @JackReacher I didn't even notice :-) $\endgroup$ – Magdiragdag Mar 31 '14 at 7:04

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