Conjecture on orientation reversal for even-winding satellite patterns

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Let PP be a pattern with winding number, the algebraic intersection number of the pattern with a meridional disk of the solid torus, a nonzero even integer. For a knot KK, write P(K)P(K) for the satellite knot obtained from PP and KK, and write −K-K for the inverse of KK in the smooth knot concordance group. Even-winding reversal conjecture. There exists a knot KK such that P(−K)P(-K) is not concordant to −P(K)-P(K). No such example is known for even winding number unless the pattern induces the zero map, so the conjecture remains open.

References

Primary source

Randall Johanningsmeier, Hillary Kim and Allison N. Miller, “A partial resolution of Hedden's conjecture on satellite homomorphisms”, arXiv:2308.06890 (2023).

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