2025/05/25 by Kwon, Bo-hyun
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2505.19082
In this paper, we define the normal form and normal coordinate of a rational 3-tangle T with respect to ∂ E1, where E1 is the fixed two punctured disk in Σ0,6. Among all normal coordinates of T with respect to ∂ E1, we investigate the collection of minimal normal coordinates of T. We show that the simplicial complex constructed with normal forms of the rational 3-tangle is contractible. As an effectiveness of the contractibility of the simplicial complex by normal forms of T, we would choose a minimal normal coordinate of T with a certain rule for the representative for the rational 3-tangle T. This classifies rational 3-tangles up to isotopy.