I am aware that knots are characterized by polynomial invariants (Alexander , Jones, HOMFLY etc) but except for the Alexander polynomial, the other polynomials seem non-intuitive. In this context, could an arbitrary knot be represented as a sum of lissajous / fourier knots ( as in fourier series) ?? The difference between the former and the usual methods of knot decomposition would be the difference between representation as a sum of orthogonal knots and as a product of (prime / irreducible ??) knots . Would the invariant polynomials be the sum of the corresponding polynomials of the constituents ??
Each region would be bounded by edges between vertices with varying winding numbers (that are rational). Could 1 way of simplifying the above be to look at the dual graph of the above ??
Some more hypothetical leaps based on the above :
- Links to the roots of unity idea (vertices being roots of unity in each region) with possible links to hyperbolic triangles and polygons where the sizes of the arcs / segments at each edge do not matter (since we are looking at topology) with the winding number possibly representing the exponent. When we are looking at nested / interior regions, one would possibly add a constant winding number to all the regions bound by a particular curve (possibly corresponding to division / multiplication by an appropriate power of the independent variable).
I know the above is pretty speculative but anything formal has to be based on intuitive grasp of what is going on. If that makes sense, one can go ahead. Else, one has to continue the search for ideas
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