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**Extensions of some classical local moves on knot diagrams.**
*(English)*
Zbl 1406.57003

This paper consists in a study of a series of local moves on classical and welded string link diagrams. The authors distinguish classical local moves, involving only classical crossings (crossing change move \(CC\), self-crossing change \(SC\), Delta move \(\Delta\), band-pass move \(BP\)) and non-classical local moves involving also virtual/welded crossings (virtualization move \(V\), self-virtualization move \(SV\), virtual conjugation move \(VC\), fused move \(F\), welded band-pass move \(wBP\)).

They translate these moves to the language of Gauss diagrams, which appear to be significantly more efficient than usual knotted objects’ diagrams for handling global manipulations. Using Gauss diagrams as a tool, they describe the interrelation of the local moves. This leads to their main result: a complete classification of welded string links up to each one of the considered moves in terms of virtual linking numbers. They remark that, considering ribbon tubes [B. Audoux et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 2, 713–761 (2017; Zbl 1375.57032)] as a topological incarnation of welded diagrams through the tube map [S. Satoh, J. Knot Theory Ramifications 9, No. 4, 531–542 (2000; Zbl 0997.57037)], the virtual linking numbers correspond to the evaluation of any longitude of the components of the ribbon tube in \(\mathbb Z\).

As a consequence of the main result, they obtain that \(VC\), \(\Delta\), \(F\), \(BP\) and \(wBP\) are unknotting operations for welded knots.

The main result and its consequence fill two literature gaps: the first one has been partially explored by A. Fish and E. Keyman [J. Knot Theory Ramifications 25, No. 7, Article ID 1650042, 8 p. (2016; Zbl 1353.57008)], who proved that welded links with only classical crossings up to \(F\) are classified by their linking numbers; the second one by S. Satoh [Rocky Mt. J. Math. 48, No. 3, 967–879 (2018; Zbl 1397.57018)] who proved that some of these local moves are unknotting operations for welded knots.

Finally the authors transport the results obtained for classical and welded string links to the case of welded links and welded braids.

Throughout the paper the authors give a careful literature review and plenty of well addressed references, and clearly highlight the topological implications of their work.

They translate these moves to the language of Gauss diagrams, which appear to be significantly more efficient than usual knotted objects’ diagrams for handling global manipulations. Using Gauss diagrams as a tool, they describe the interrelation of the local moves. This leads to their main result: a complete classification of welded string links up to each one of the considered moves in terms of virtual linking numbers. They remark that, considering ribbon tubes [B. Audoux et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 2, 713–761 (2017; Zbl 1375.57032)] as a topological incarnation of welded diagrams through the tube map [S. Satoh, J. Knot Theory Ramifications 9, No. 4, 531–542 (2000; Zbl 0997.57037)], the virtual linking numbers correspond to the evaluation of any longitude of the components of the ribbon tube in \(\mathbb Z\).

As a consequence of the main result, they obtain that \(VC\), \(\Delta\), \(F\), \(BP\) and \(wBP\) are unknotting operations for welded knots.

The main result and its consequence fill two literature gaps: the first one has been partially explored by A. Fish and E. Keyman [J. Knot Theory Ramifications 25, No. 7, Article ID 1650042, 8 p. (2016; Zbl 1353.57008)], who proved that welded links with only classical crossings up to \(F\) are classified by their linking numbers; the second one by S. Satoh [Rocky Mt. J. Math. 48, No. 3, 967–879 (2018; Zbl 1397.57018)] who proved that some of these local moves are unknotting operations for welded knots.

Finally the authors transport the results obtained for classical and welded string links to the case of welded links and welded braids.

Throughout the paper the authors give a careful literature review and plenty of well addressed references, and clearly highlight the topological implications of their work.

Reviewer: Celeste Damiani (Leeds)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

20F36 | Braid groups; Artin groups |

### Keywords:

knot; link; string-link; welded knot; welded link; welded string-link; welded braid; virtual knot; virtual link; virtual string link; invariant; linking number; virtual linking number; unknotting operation; local move; crossing change; Delta move; band-pass move; Gauss diagram; fused link
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\textit{B. Audoux} et al., Mich. Math. J. 67, No. 3, 647--672 (2018; Zbl 1406.57003)

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