TY - JOUR

T1 - Arrow calculus for welded and classical links

AU - Meilhan, Jean Baptiste

AU - Yasuhara, Akira

N1 - Funding Information:
Acknowledgements The authors would like to thank Benjamin Audoux for stimulating conversations, and Haruko A Miyazawa for her useful comments. Thanks are also due to the referee for insightful comments and suggestions. This paper was completed during a visit of Meilhan at Tsuda University, Tokyo, whose hospitality and support is warmly acknowledged. Yasuhara is partially supported by a Grant-in-Aid for Scientific Research (C) (#17K05264) of the Japan Society for the Promotion of Science.

PY - 2019/2/6

Y1 - 2019/2/6

N2 - 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 finitetype 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.

AB - 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 finitetype 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.

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U2 - 10.2140/agt.2019.19.397

DO - 10.2140/agt.2019.19.397

M3 - Article

AN - SCOPUS:85062853989

VL - 19

SP - 397

EP - 456

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 1

ER -