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We develop a calculus for diagrams of knotted objects. We define arrow presentations, which encode the crossing information of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w–tree presentations, which can be seen as “higher-order Gauss diagrams”. This arrow calculus is used to develop an analogue of Habiro’s clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a “realization” of Polyak’s algebra of arrow diagrams at the welded level, and leads to a characterization of finite-type invariants of welded knots and long knots. As a corollary, we recover several topological results due to Habiro and Shima and to Watanabe on knotted surfaces in 4–space. We also classify welded string links up to homotopy, thus recovering a result of the first author with Audoux, Bellingeri and Wagner.
Keywords: knot diagrams, finite-type invariants, Gauss diagrams, claspers
Meilhan, Jean-Baptiste 1 ; Yasuhara, Akira 2
@article{10_2140_agt_2019_19_397,
author = {Meilhan, Jean-Baptiste and Yasuhara, Akira},
title = {Arrow calculus for welded and classical links},
journal = {Algebraic and Geometric Topology},
pages = {397--456},
publisher = {mathdoc},
volume = {19},
number = {1},
year = {2019},
doi = {10.2140/agt.2019.19.397},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.397/}
}
TY - JOUR AU - Meilhan, Jean-Baptiste AU - Yasuhara, Akira TI - Arrow calculus for welded and classical links JO - Algebraic and Geometric Topology PY - 2019 SP - 397 EP - 456 VL - 19 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.397/ DO - 10.2140/agt.2019.19.397 ID - 10_2140_agt_2019_19_397 ER -
%0 Journal Article %A Meilhan, Jean-Baptiste %A Yasuhara, Akira %T Arrow calculus for welded and classical links %J Algebraic and Geometric Topology %D 2019 %P 397-456 %V 19 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.397/ %R 10.2140/agt.2019.19.397 %F 10_2140_agt_2019_19_397
Meilhan, Jean-Baptiste; Yasuhara, Akira. Arrow calculus for welded and classical links. Algebraic and Geometric Topology, Tome 19 (2019) no. 1, pp. 397-456. doi: 10.2140/agt.2019.19.397
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