Exchange complexes and contractibility of the complex of incompressible Seifert surfaces

Abstract

Let $K\subset S^3$ be a non-trivial knot and let $\mathrm{IS}(K)$ be the simplicial complex whose vertices are the ambient isotopy classes of incompressible Seifert surfaces in the exterior $E(K)$, a finite set of distinct vertices spanning a simplex exactly when its classes admit simultaneously pairwise disjoint representatives. Kakimizu proved that $\mathrm{IS}(K)$ is connected; whether it is contractible was asked by Przytycki and Schultens, who also identified the obstruction, namely that the projection used in the connectedness proof is not known to be well defined on isotopy classes. We prove that $\mathrm{IS}(K)$ is contractible. We also prove contractibility of every genus truncation ${\mathrm{IS}}_{\ell }(K)$ with $\ell \ge g(K)$ and every non-empty lexicographic complexity sublevel. The argument factors through an unconditional combinatorial theorem: every non-empty connected flag exchange complex is contractible, where an exchange complex carries a complexity function subject to two axioms which require only that an exchanging vertex exist, never that a coherent selection rule be supplied. The combinatorial proof uses hereditary descending links and transfinite induction. We compare its attachment step with existing Morse criteria, including a formulation applicable to every countable exchange complex, and distinguish the given exchange order from a dismantling order. The geometric argument also applies to links satisfying a linking condition that forces every spanning surface to be connected. For the geometric input we prove fixed-boundary area attainment among smooth neat embeddings, using smooth convex replacement domains, area-controlled disc cleaning and boundary-preserving smoothing. All area complexities use this smooth competing class. No interpretation of an undefined piecewise smooth existence class is needed.

69 pages. Revised exchange-complex proof; expanded fixed-boundary area attainment, uniform area gain, and isotopy arguments