Lamm's equivariant crossing number conjecture for two-bridge knots

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Let KK be a two-bridge knot, and let c(K)c(K) denote its crossing number. A symmetric diagram for a strongly invertible knot is a transvergent diagram whose axis is a straight line in the plane of the diagram and which is unchanged under a half-rotation about that axis. The equivariant crossing number ct(K)c_t(K) is the minimum crossing number among all symmetric diagrams for KK. A continued fraction expansion [a1,…,an][a_1,\ldots,a_n] is of type (A) when all aia_i are positive, nn is even, and a2,a4,…,ana_2,a_4,\ldots,a_n are all even. It is of type (B) when all aia_i are positive, nn is odd, the integers are palindromic with a1=an,a2=an−1,…a_1=a_n,a_2=a_{n-1},\ldots, and the central integer a(n+1)/2a_{(n+1)/2} is odd.

Lamm's conjecture. ct(K)=c(K)c_t(K)=c(K) if and only if KK can be represented by a continued fraction expansion [a1,…,an][a_1,\ldots,a_n] of type (A) or type (B).

The conjecture characterizes precisely when a two-bridge knot admits a symmetric diagram realizing its ordinary crossing number. The paper studies this question through the restricted invariant c2(K)c_2(K) and reports computations up to 14 crossings, but the conjecture is presented as an unresolved claim.

References

Primary source

Jundai Nanasawa, “Equivariant crossing numbers for two-bridge knots”, arXiv:2304.00540 (2023).

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