Abstract
In classical knot theory, Markov’s theorem gives a way of describing all braids with isotopic closures as links in \(\mathbb {R}^3\). We present a version of Markov’s theorem for extended loop braids with closure in \(B^3 \times S^1\), as a first step towards a Markov’s theorem for extended loop braids and ribbon torus-links in \(\mathbb {R}^4\).
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- 1.
Notations are slightly changed: differences will be pointed out along this text.
- 2.
In [8] these groups are denote by \(PC_{n}^*\), while \(PC_{n}\) is used for the groups of automorphisms of the form \(\alpha :x_i\mapsto w_i^{-1}x_{\pi ({i})} w_i\).
- 3.
In the survey [8] the terminology ribbon braids refers to loop braids seen as braided objects in the 4-dimensional braid, while the terminology extended ribbon braids refers to extended loop braids. We chose to simplify.
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Acknowledgements
During the writing of this paper, the author was supported by a JSPS Postdoctral Fellowship For Foreign Researchers and by JSPS KAKENHI Grant Number 16F16793. The author thanks Emmanuel Wagner for valuable conversations.
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Damiani, C. (2019). Towards a Version of Markov’s Theorem for Ribbon Torus-Links in \(\mathbb {R}^4\). In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_15
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