As far as I understood, the knot equivalence (ambient isotopy) is a $3$-dimensional phenomenon for knots while the knot concordance is a $4$-dimensional phenomenon.

The knot concordance does not imply the knot equivalence. For example, we consider the unknot and the stevedore knot. They are concordant but not equivalent.

How about the converse direction? Can we produce counter examples or can we prove the implication?


1 Answer 1


Knot equivalence implies knot concordance.

Specifically, let $K_0$ and $K_1$ be equivalent, smoothly embedded knots in $S^3$. Then there is a smooth ambient isotopy $f: [0,1] \times S^3 \to S^3$ where each map $f(t, -)$ is a smooth diffeomorphism $S^3 \to S^3$, and $f(1, -)$ sends $K_0$ to $K_1$.

To define a smooth concordance between $K_0$ and $K_1$, define a smooth embedding $$g: [0,1] \times K_0 \to [0,1] \times S^3$$ by $$g(t, x) = (t, f(t,x)).$$


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