7
$\begingroup$

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through Knot Theory." Therein he discusses, among other things, constructions of knotted surfaces by means of slice disks for certain rational knots. In particular, example 12 combines two slice disks of the Stevedore's knot ($6_1$) to give a knotted sphere that has interesting homological properties: the Alexander ideal is not principal.

In 1965, Sir Christopher published his Transactions AMS paper that describes twist-spinning. Later in a letter to Cameron Gordon, Rick Litherland demonstrated that Fox's example 12 was indeed the $2$-twist-spin of the trefoil. Other proofs followed including a version by Kaneobu that discussed Fox's example 15 --- the general case o Fox's construction. Kamada also gives a proof.

My question is for those who might have been present or who heard from either Fox or Zeeman. Was Fox aware of the connection to the trefoil? He must have been since $6_1$ is so closely related to the trefoil. Did Zeeman actually reinterpret Fox's construction for these examples and subsequently through out the motivation once he saw the general idea of twist-spinning? How much communication about Fox's examples occurred during the conference? Sir Christopher wrote that he addresses a problem from Fox's quick trip. In short, what did these men know and when did they know it?

$\endgroup$
  • $\begingroup$ I wasn't alive at the time so I have no first-hand information. I would guess Fox figured it out (or was told) eventually of the connection, as the Alexander module gives you a big hint. Gordon, or Milnor or Przytycki might know. None of them frequent this forum as far as I know. $\endgroup$ – Ryan Budney Sep 26 '12 at 2:34
  • $\begingroup$ Can you ask Zeeman? $\endgroup$ – Bruce Westbury Sep 26 '12 at 6:56
  • $\begingroup$ I have heard that he is ill. But I thought I would try here first. $\endgroup$ – Scott Carter Sep 26 '12 at 11:51
  • $\begingroup$ Have you asked Gordon? The conference was a little before his time, but he may know. $\endgroup$ – Autumn Kent Feb 5 '13 at 17:37
  • $\begingroup$ I think I spoke to him about this. I will ask him again. $\endgroup$ – Scott Carter Feb 7 '13 at 0:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.