### Abstract

A knot diagram looks like a two-dimensional drawing of aknotted rubberband. Proving that a given knot diagram can be untangled(that is, is a trivial knot, called an unknot) is one of the most famousproblems of knot theory. For a small knot diagram, one can try to finda sequence of untangling moves explicitly, but for a larger knot diagramproducing such a proof is difficult, and the produced proofs are hardto inspect and understand. Advanced approaches use algebra, with anadvantage that since the proofs are algebraic, a computer can be usedto produce the proofs, and, therefore, a proof can be produced evenfor large knot diagrams. However, such produced proofs are not easy toread and, for larger diagrams, not likely to be human readable at all.We propose a new approach combining advantages of these: the proofsare algebraic and can be produced by a computer, whilst each part ofthe proof can be represented as a reasonably small knot-like diagram(a new representation as a labeled tangle diagram), which can be easilyinspected by a human for the purposes of checking the proof and findingout interesting facts about the knot diagram.

Original language | English |
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Title of host publication | 10th International Conference on Theory and Applications of Diagrams |

Editors | P. Chapman, G. Stapleton, A. Moktefi , S. Perez-Kriz, F. Bellucci |

Place of Publication | Edinburgh |

Publisher | Springer |

Pages | 89-104 |

Volume | 10871 |

ISBN (Electronic) | 9783319913766 |

ISBN (Print) | 9783319913759 |

DOIs | |

Publication status | Published - 17 May 2018 |

Event | 10th International Conference on Theory and Applications of Diagrams - Edinburgh, UK, 18-22 June 2018 Duration: 1 Jan 2018 → … |

### Publication series

Name | Lecture Notes in Computer Science |
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ISSN (Print) | 0302-9743 |

### Conference

Conference | 10th International Conference on Theory and Applications of Diagrams |
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Period | 1/01/18 → … |

### Bibliographical note

The final authenticated version is available online at https://doi.org/10.1007/978-3-319-91376-6_12## Fingerprint Dive into the research topics of 'Visual algebraic proofs for unknot detection'. Together they form a unique fingerprint.

## Cite this

Fish, A., Lisitsa, A., & Vernitski, A. (2018). Visual algebraic proofs for unknot detection. In P. Chapman, G. Stapleton, A. Moktefi , S. Perez-Kriz, & F. Bellucci (Eds.),

*10th International Conference on Theory and Applications of Diagrams*(Vol. 10871, pp. 89-104). (Lecture Notes in Computer Science). Edinburgh: Springer. https://doi.org/10.1007/978-3-319-91376-6_12