An elementary proof that linking problems are hard

Abstract

We give a new, elementary proof of what we believe is the simplest known example of a ``natural'' problem in computational 3-dimensional topology that is $\mathsf{NP}$-hard -- namely, the \emph{Trivial Sublink Problem}: given a diagram $L$ of a link in $S^3$ and a positive integer $k$, decide if $L$ contains a $k$ component sublink that is trivial. This problem was previously shown to be $\mathsf{NP}$-hard in independent works of Koenig-Tsvietkova and de Mesmay-Rieck-Sedgwick-Tancer, both of which used reductions from $\mathsf{3SAT}$. The reduction we describe instead starts with the Independent Set Problem, and allows us to avoid the use of Brunnian links such as the Borromean rings. On the technical level, this entails a new conceptual insight: the Trivial Sublink Problem is hard entirely due to mod 2 pairwise linking, with no need for integral or higher order linking. On the pedagogical level, the reduction we describe is entirely elementary, and thus suitable for introducing undergraduates and non-experts to complexity-theoretic low-dimensional topology. To drive this point home, in this work we assume no familiarity with low-dimensional topology, and -- other than Reidemeister's Theorem and Karp's result that the Clique Problem is $\mathsf{NP}$-hard -- we provide more-or-less complete definitions and proofs. We have also constructed a web app that accompanies this work and allows a user to visualize the new reduction interactively.