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December 13, 2022

The Knottedness and Chirality of the Trefoil

Filed under: knots,Mathematics,the knottedness and chirality of the trefoil — Rob burchett @ 3:15 am


Here is an outline in rough:

1. Start with an oriented trefoil in S^3.

2. Define two new spaces. A negative space and a mixed space.

3. Let the trefoil be in the mixed space.

4. If we wish, we can move the trefoil into positive space from the mixed space or move it into negative space from the mixed space. To move from positive to negative or from negative to positive we need to go through a transition space. (mixed space)

5. Make moving points (labels) a, b type to represent where the trefoil crosses over and under itself. These exist in mixed space and negative space but not in positive space. These moving points move as the diagram is moved, so that they are always above or below in a crossing. They can come out of the crossing and combine to form joinings (a(r) or (b(s) due to elimination via moves 1 and 2 type.

6. New labels s and r only exist in negative space and are counterparts to a and b. s and t may switch as the knot passes through itself as this is allowed in negative space.

7.Suppose I have a diagram with the same shadow as the usual representation of the trefoil. Then this is either an unknotted case with a trefoil shadow or a positive or negatively oriented trefoil.

8. Start with a positively oriented trefoil .Label the crossings in the transition space. Move the trefoil to the special format where I remove a, b and form ordered joining pairs (a(r) and (b(s) type.

9. After this, the trefoil may make any 1,2,3 moves recorded in r,s crossings. Eventually returning to have a trefoil shadow. These moves can all be shown to be mirrored as if we were in positive space. Then this represents the full knot.

10. At any time I can undo the diagram to see that the circle with (a(r), and (b(s) joining pairs is an invariant in the negative space.

11. I need to be able to reform a and b crossings by recombining the (a(r) and (b(s) joinings to get back into transition space before I can move to positive space. This must be either possible or not. The restriction on what type of crossings I can reform is given by the (a(r), (b(s) sequence. I can rotate the sequence to match with a copy of itself or overturn it. Since I am in the new space there are also other moves I can make.

12. Since I can’t form the unknot, the trefoil is knotted, since I can’t form the negative orientation, the trefoil is chiral.

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