Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\R^2$

Algorithmic simplification of knot diagrams: new moves and experiments

This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diagram D' with the minimal possible number of crossings for the isotopy class of the knot represented by D. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain Z1, Z2, Z3 generalizing the classical Reidemeister moves R1, R2, R3, and another one C (together with a variant) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.Comment: 38 pages, 33 figure

Quantum money from knots

Quantum money is a cryptographic protocol in which a mint can produce a quantum state, no one else can copy the state, and anyone (with a quantum computer) can verify that the state came from the mint. We present a concrete quantum money scheme based on superpositions of diagrams that encode oriented links with the same Alexander polynomial. We expect our scheme to be secure against computationally bounded adversaries.Comment: 22 pages, 5 figure

The Unbearable Hardness of Unknotting

We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number)

Computational aspects of knots and knot transformation

In this thesis we study the computational aspects of knots and knot trans- formations. Most of the problems of recognising knot properties (such as planarity, unknottedness, equivalence) are known to be decidable, however for many problems their precise time or space complexity is still unknown. On the other hand, their complexity in terms of computational power of devices needed to recognise the knot properties was not studied yet. In this thesis we address this problem and provide first known bounds for some knot problems within this context. In order to estimate and characterise complexity of knot problems represented by Gauss words, we consider vari- ous tools and mathematical models including automata models over infinite alphabets, standard computational models and definability in logic. In particular we show that the planarity problem of signed and unsigned Gauss words can be recognised by a two-way deterministic register au- tomata. Then we translate this result in terms of classical computational models to show that these problems belong to the log-space complexity class L, Further we consider definability questions in terms of first order logic and its extensions and show that planarity of both signed and unsigned Gauss words cannot be expressed by a formula of first-order predicate logic, while extensions of first-order logic with deterministic transitive closure operator allow to define planarity of both signed unsigned Gauss words. Follow- ing the same line of research we provide lower and upper bounds for the planarity problem of Gauss paragraphs and unknottedness. In addition we consider knot transformations in terms of string rewriting systems and provide a refined classification of Reidemeister moves formu- lated as string rewriting rules for Gauss words. Then we analyse the reach- ability properties for each type and present some bounds on the complexity of the paths between two knot diagrams reachable by a sequence of moves of the same type. Further we consider a class of non-isomorphic knot diagrams generated by type I moves from the unknot and discover that the sequence corresponding to the number of diagrams with respect to the number of crossings is equal to a sequence related to a class of Eulerian maps with respect to the number of edges. We then investigate the bijective mapping between the two classes of objects and as a result we present two algo- rithms to demonstrate the transformations from one object to the other. It is known that unknotting a knot may lead to a significant increase in number of crossings during the transformations. We consider the question of designing a set of rules that would not lead to the increase in the number of crossings during knot transformations. In particular we introduce a new set moves in this regard which can be used to substitute one of the rules of type II that increases the number of crossings. We show that such new moves coupled with Reidemeister moves can unknot all known examples of complex trivial knot diagrams without increasing number of crossings