Parameterized complexity of untangling knots

TL;DR

This paper investigates the parameterized complexity of knot untangling, showing it is W[P]-complete when parameterized by defect, a measure of move efficiency, thus providing new insights into the computational complexity of knot theory problems.

Contribution

It establishes the W[P]-completeness of the knot untangling problem with respect to defect, a natural parameter, and demonstrates that certain moves can be performed greedily in shortest sequences.

Findings

Untangling with Reidemeister II-moves can be done greedily.

The problem is W[P]-hard and belongs to W[P] when parameterized by defect.

This characterizes the problem's complexity class in parameterized complexity theory.

Abstract

Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that the II- moves in a shortest untangling sequence can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.