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What is the definition of a genus in the context of knot theory?

I’m trying to understand what is meant by this question.
“define the genus of a knot, more formally, given an oriented knot K, its genus is the minimal integer r with the following property:
If S is a subsurface of a 3-manifold M such that K is isotopic to a knot in S, then K is homologous to zero in S.”
I’m trying to link this with the genus of a surface in the sense of topology. Then this condition is equivalent to saying that there exists a locally flat embedding of S in R^3 such that a cycle in that S represents K.
Could anyone clarify the relation between this and the topological genus of a surface?


The topological genus of an orientable surface $S$ is the smallest integer $g(S)$ such that any properly embedded arc in the plane defines an embedded surface of genus $g(S)$. A simple way to define the genus of a knot is then
Here $\partial
u$ denotes the boundary of a tubular neighborhood of the knot. This definition is intuitively obvious, it is equivalent to the number of double points of $\partial

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