Kanenobu knots crossing-number upper-bound conjecture

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Let K(p,q)K(p,q) be the Kanenobu knot determined by integers pp and qq, and let c(K(p,q))c(K(p,q)) denote its crossing number. Assume

pq<0,∣p∣≠1,∣q∣≠1.pq<0,\qquad |p|\ne1,\qquad |q|\ne1.

Kanenobu knots crossing-number upper-bound conjecture. The crossing number satisfies

c(K(p,q))≤∣p∣+∣q∣+8.c(K(p,q))\leq |p|+|q|+8.

The preceding proposition already states this same upper bound, and the paper gives a diagram realizing it; the conjectural content is therefore redundant with the established upper bound and leaves the corresponding equality as the unresolved issue.

References

Primary source

Khaled Qazaqzeh and Isra Mansour, “Further Study of Kanenobu Knots”, arXiv:1405.0683 (2014).

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