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For any knot $K$ does there exist some diagram for $K$ such that Seifert's algorithm produces a Seifert surface for $K$ where the genus of said surface is $g(K)$ - the $3$-genus of $K$?

I have seen some knot blogs that claim it doesn't, but have yet to see an example where it will not produce one with minimal genus.

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The minimum genus of any Seifert surface coming from Seifert's algorithm is known as the canonical genus $g_c(K)$ of the knot. An example where the canonical genus is strictly greater than the $3$-genus is a Whitehead double of the trefoil (pictured below).

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This post gives an explanation of why the genus of a Whitehead double is one. Hugh Morton proved that the highest degree of one of the two variables of the HOMFLY polynomial gives a lower bound on twice the canonical genus $2g_c(K)$ (see the paper). This bound gives that the canonical genus of the Whitehead double of the trefoil is $3$.

See this paper of Brittenham and Jensen for more examples.

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