Murakami and Yasuhara's sharpness conjecture for the non-orientable slice genus

Let KK be a knot, let g4(K)g_4(K) denote its smooth orientable slice genus, and let b34(K)b3_4(K) denote its smooth non-orientable slice genus. A knot is non-slice if g4(K)>0g_4(K)>0. Murakami and Yasuhara's conjecture. There exists a non-slice knot KK such that

γ4(K)=2g4(K)+1.\gamma_4(K)=2g_4(K)+1.

Murakami and Yasuhara proved the general inequality γ4(K)2g4(K)+1\gamma_4(K)\leq 2g_4(K)+1; the conjecture asks whether this bound is attained by some non-slice knot.

Sources & referencesView supporting material

Primary source

Stanislav Jabuka and Tynan Kelly, “The non-orientable 4-genus for knots with 8 or 9 crossings”, arXiv:1708.03000 (2020).

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