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edited formatting, added 3-manifolds and branched-covers tags
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Marco Golla
Marco Golla
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In Chapter One of his notes (March 2002) Thurston says:

"If $K$ is the trivial knot the cyclic branched covers are $S^3$ . It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot."

If $K$ is the trivial knot the cyclic branched covers are $S^3$. It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot.

This sentence sounds a little bit enigmatic to me. Is he saying that (we expect that) "if $K \subset S^3$ is such that $\textit{all}$all its cyclic branched covers are $S^3$ then $K$ is the unknot", or that "if for some $k\geq 2$ the $k$-fold cyclic branched cover over $K$ is $S^3$ then $K$ is the unknot"?

I would like to know which is the state of the art this conjecture. In particular if $S^3$ $\textit{can}$can appear as a branched cover over a non-trivial knot I would like to see some (families of) examples.

In Chapter One of his notes (March 2002) Thurston says:

"If $K$ is the trivial knot the cyclic branched covers are $S^3$ . It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot."

This sentence sounds a little bit enigmatic to me. Is he saying that (we expect that) "if $K \subset S^3$ is such that $\textit{all}$ its cyclic branched covers are $S^3$ then $K$ is the unknot", or that "if for some $k\geq 2$ the $k$-fold cyclic branched cover over $K$ is $S^3$ then $K$ is the unknot"?

I would like to know which is the state of the art this conjecture. In particular if $S^3$ $\textit{can}$ appear as a branched cover over a non-trivial knot I would like to see some (families of) examples.

In Chapter One of his notes (March 2002) Thurston says:

If $K$ is the trivial knot the cyclic branched covers are $S^3$. It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot.

This sentence sounds a little bit enigmatic to me. Is he saying that (we expect that) "if $K \subset S^3$ is such that all its cyclic branched covers are $S^3$ then $K$ is the unknot", or that "if for some $k\geq 2$ the $k$-fold cyclic branched cover over $K$ is $S^3$ then $K$ is the unknot"?

I would like to know which is the state of the art this conjecture. In particular if $S^3$ can appear as a branched cover over a non-trivial knot I would like to see some (families of) examples.

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Antonio Alfieri
Antonio Alfieri
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$S^3$ as cyclic branched cover of itself

In Chapter One of his notes (March 2002) Thurston says:

"If $K$ is the trivial knot the cyclic branched covers are $S^3$ . It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot."

This sentence sounds a little bit enigmatic to me. Is he saying that (we expect that) "if $K \subset S^3$ is such that $\textit{all}$ its cyclic branched covers are $S^3$ then $K$ is the unknot", or that "if for some $k\geq 2$ the $k$-fold cyclic branched cover over $K$ is $S^3$ then $K$ is the unknot"?

I would like to know which is the state of the art this conjecture. In particular if $S^3$ $\textit{can}$ appear as a branched cover over a non-trivial knot I would like to see some (families of) examples.