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Thread: How to get mirror image of knot

  1. #1
    Feb 2011

    How to get mirror image of knot

    Dear mathematicians

    Kindly I have a small question:

    The projection of a knot in R^2 can be obtained by choosing a vector v in R^3 and 2-dimensional hyperplane in R^3 that is disjoint from the knot such that we project the knot in the direction of v onto the hyperplane.

    Let -K denote the reflection of a knot K in R^3 given by f(x,y,z)=(x,y,-z). Then how to project -K? Should we use same vector or the opposite of it. If we use same vector then in this case the upper arcs and lower arcs of K are exchanged in -K, but we reverse the orientation. In order to be confident I took the left-handed trefoil as an example and then I reflected it in R^3. The projection of the trefoil and its mirror in R^2 are actually same (both give the left handed trefoil). So surly this is not correct.

    On the other hand if we reverse the direction of v so that the plane for example can be the xy plane. But here also I found a problem. For trefoil case this definition is fine but I found a non-alternating knot (see the attached picture) such that the diagram of the image of reflection can not be obtained by exchanging upper sheets and lower sheets of the original diagram which conflicts with the definition.

    So what is the correct definition.

    Please guide me and every help is highly highly appreciated
    Attached Thumbnails Attached Thumbnails How to get mirror image of knot-nonaltknot.jpeg  
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