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).
The author was supported by JSPS KAKENHI Grant number JP18K13416.
The author would like to thank Tetsuya Abe for helpful discussion. The author thanks the referee for his/her careful reading and helpful comments.
"Notes on constructions of knots with the same trace." Hiroshima Math. J. 52 (1) 1 - 15, March 2022. https://doi.org/10.32917/h2021005