2025/06/21 by Fiedler, Thomas
#57M25 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2506.17738
This paper contains the strongest and at the same time most calculable knot invariant ever. Let Θ be the topological moduli space of all ordered oriented tangles in 3-space. We construct a non-trivial combinatorial 1-cocycle \mathbbL for Θ that takes its values in H0(Θ;ℤ). The 1-cocycle \mathbbL has a very nice property, called the \em scan-property: if we slide a tangle T over or under a given crossing c of a fixed tangle T', then the value of \mathbbL on this arc scan(T) in Θ is already an isotopy invariant of T. In particular, let D be a framed long knot diagram. We take the product with a fixed long knot diagram K and we consider the 2-cable, with a fixed crossing c in 2K. \mathbbL(scan(2D)) gives an element in H0(Θ). To this element we associate the \em set of Alexander vectors, consisting of the corresponding integer multiples of the one-variable Alexander polynomials of (the standard closures) of all sub-tangles of each of the tangles. We can vary the knots (K,c) and moreover we can iterate our construction by starting now again the scan with the tangles in \mathbbL(scan(2D)) and so on. The result is the infinite \em Alexander tree, which is an isotopy invariant of the knot represented by D. \em As examples we show with just one edge of the Alexander tree that the knot 817 and the Conway knot are not invertible! This makes the Alexander tree a very promising candidate for a complete and "locally" calculable knot invariant, because the tangles in \mathbbL(scan(2D)) can be drawn with linear complexity and their Alexander polynomials can be calculated with quartic complexity with respect to the number of crossings of D.