I recently asked a question about the proof of theorem 1.8 in Miller and Piccirillo's paper:
I will continue to keep the same notation as in my previous question.
Following the discussion in the comment section of Lee Moscher's answer. I was wondering about the last part in the proof. The authors construct a map $S^2 \rightarrow S^4$ that is smooth away from 1 point. The singular point arises from taking the cone over the knot $K$. Then the authors claim that one can take a sufficiently small neighbourhood of the singular point and it will intersect the cone in exactly the knot $K$.
Does this follow from some other facts?
It seems that composing the cone with the embedding $X_K \rightarrow S^4$ would force the base of the cone to be some random knot and not $K$, why then can we by going up further in the cone assume that a section of the cone will be $K$. This seems like it should be false for an embedding $X_K \rightarrow S^4$ that sends $K \subset X_K$ to a knot that is not isotopic to it.