Conjecture on even pretzel knots with maximal nonorientable concordance genus

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Let n≥3n\ge 3, and let an even nn-stranded pretzel knot be a pretzel knot K=P(a1,…,an)K=P(a_1,\dots,a_n) with even parameters in the sense used for pretzel knots. Write γc(K)\gamma_c(K) for the nonorientable concordance genus of KK. Pretzel-knot concordance genus conjecture. For every n≥3n\ge 3 there exist infinitely many even nn-stranded pretzel knots K=P(a1,…,an)K=P(a_1,\dots,a_n) with

γc(K)=n−1.\gamma_c(K)=n-1.

The preceding results establish the assertion for n=3n=3 and n=4n=4 using rational Witt spans; the conjecture asks for analogous infinite families for every number of strands at least three.

References

Primary source

Stanislav Jabuka, “The concordance crosscap number and rational Witt span of a knot”, arXiv:2012.14801 (2020).

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