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March 2022 Notes on constructions of knots with the same trace
Keiji Tagami
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Hiroshima Math. J. 52(1): 1-15 (March 2022). DOI: 10.32917/h2021005

Abstract

The m-trace of a knot is the 4-manifold obtained from $B^4$ by attaching a 2-handle along the knot with m-framing. In 2015, Abe, Jong, Luecke and Osoinach introduced a technique to construct infinitely many knots with the same (diffeomorphic) m-trace, which is called the operation (∗m). In 2018, Miller and Piccirillo gave pairs of knots with diffeomorphic m-traces by utilizing Gompf and Miyazaki’s dualizable pattern. In this paper, we clarify the relation between the two techniques. In particular, we prove that the “twistings” appearing in both techniques are corresponding. In addition, we show that the family of knots admitting the same 4-surgery given by Teragaito can be explained by the operation (∗m).

Funding Statement

The author was supported by JSPS KAKENHI Grant number JP18K13416.

Acknowledgment

The author would like to thank Tetsuya Abe for helpful discussion. The author thanks the referee for his/her careful reading and helpful comments.

Citation

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Keiji Tagami. "Notes on constructions of knots with the same trace." Hiroshima Math. J. 52 (1) 1 - 15, March 2022. https://doi.org/10.32917/h2021005

Information

Received: 26 January 2021; Revised: 15 September 2021; Published: March 2022
First available in Project Euclid: 25 March 2022

Digital Object Identifier: 10.32917/h2021005

Subjects:
Primary: 57K10

Keywords: annulus presentation , annulus twist , dualizable pattern , knot , m-trace

Rights: Copyright © 2022 Hiroshima University, Mathematics Program

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Vol.52 • No. 1 • March 2022
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