Celtic Knotwork: the ultimate tutorial

7. Extraction

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Suppose you find a nice knot like that one (okay, I could have done better... that’s only an example for God’s sake!):

You may want to reproduce it or make it fit to your needs. The best way to achieve that is by extracting its underlying graph. To do that, you have to color the regions of your knot with white and black, like a chequerboard. You notice that, forgetting the over/under pattern, your knot creates regions. There is a special region, the exterior of the knot, which is unbounded. Color it in black. Then, at each crossing, color the region opposite also in black:

You end up with this chequerboard pattern:

Now you build your graph, defining its vertices and the edges connecting them. Put a vertex in each white zone, and above each crossing, draw an edge between the two neighbouring white zones, connecting the two vertices associated to them, with one black zone on each side:

Once you’ve got your graph, you have to compare each crossing with the two guides, the plain and the dashed one, to decide what type of crossing it is:

or

You have to realize what are the effects of walls on the graph because the graph you extract may well be awfull at first but with the aid of a few walls here and there can be seen as snapped onto an easy grid. As you can see in the following example, closing a door amounts to identifying the two end points while slashing an edge is equivalent to simply discarding that edge:

Don’t hesitate to add "useless" new edges that you will slash or close
or

Now let’s build a bigger knot with the aid of encapsulation.

9 Forum messages

  • Extraction 22 May 2008 10:44, by Piffy

    Please forgive me for trying to run before I can walk here ;)

    I’m trying to learn this so that I can prepare a design for a piece of jewellry for a friend’s birthday. I’m taking a jewellry course next week in which I hope to make the piece.

    So I have a knot I am trying to extract so that I can replace it in my new design. I’m not quite sure I’m getting the gist though, especially as this knot has more sections than the one in your example.

    There’s a picture of my work so far.

    Grateful for any tips or hints
    With thanks

    Reply to this message

    • Extraction 22 May 2008 11:00, by Piffy

      That has given me a graph like this

      Reply to this message

      • One crossing for each edge 22 May 2008 21:05, by Christian Mercat

        There are way too many crossings. Remember: one crossing at the middle of each edge, no more, no less. It’s not a problem though because you just have to slash them with a wall (meaning there really is no edge there). It often more convenient to put an edge and slash it in order to have a well balanced graph like yours, rather than having a hollow graph that doesn’t stand firm on its legs.

        You missed a double crossing in the lower right part where there is the little wiggle.

        JPEG

        Don’t hesitate to split a large zone (like the red one at the upper left corner) into 2 vertices, connected by a thick wall (meaning they really are the same vertex).

        Reply to this message

        • With its dual graph 22 May 2008 21:17, by Christian Mercat

          Together with its dual (red) graph, here is the graph I get:

          JPEG - 77.1 kb

          Please send back the pictures of your jewel once you’ve done it, I am longing to see it done!

          Reply to this message

          • With its dual graph 6 June 2008 16:54, by Piffy

            Thanks for completing the extraction for me. I haven’t yet completed the design I was working on though every day I plod away a bit at trying to understand how to extract a graph from a knot. I’m able to do regular crossings but come apart with the split and bounced crossings and can’t seem to develop an ’eye’ to tell which are which from a completed knot. When I try to put vertices in they just don’t work out.

            I’m determined to get this but really struggling!

            Reply to this message

            • With its dual graph 6 June 2008 16:58, by Piffy

              Actually, I’ve gone back to trying to extract very basic knots such as the several stranded trefoils on your site and STILL can’t figure out how to do it. I seem to have encountered one of my blindspots in life.

              Reply to this message

              • Several stranded trefoil 10 June 2008 09:08, by Christian Mercat

                Well, beware that these are not natural knots, the slicing of a thread into several strands has an ugly effect on the graphs! Look at this article to understand what’s happening. My advice: Rely on the graph of such split knots as a last resort, for example when you want to open up some walls on the boundary to let them flaw away and mingle with another pattern, but not that much as a bulk pattern, splitting is a last touch adjustment.

                Keep up the good work and please post back when you stumble upon something.

                Reply to this message

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Monday 21 August 2006, by Christian Mercat

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