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Question: is it possible to define the Jones polynomial for knotted surfaces (or $S^2$ for simplicity) in $R^4$?

Jones polynomial has several definitions (see How many definitions are there of the Jones polynomial?). Is there any one of them (such as the skein relation along double curves or representations of 2-braids) can be generalized to define the Jones polynomial of 2-knots? One of the motivation of this question is, if we can define the Jones polynomial for 2-knots, maybe we can use it to determine the minimal number of triple points of the 'alternating' 2-knot diagram.

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    $\begingroup$ As far as I know there's no natural and direct way to extend a Jones polynomial definition from 1-knots to 2-knots. But as you've stated it, this is a little poorly-defined as far as questions go. Would you want the Jones polynomial of a 2-knot to relate to the Jones polynomial of a 1-knot in any way? $\endgroup$
    – Ryan Budney
    Commented Feb 14, 2014 at 2:49
  • $\begingroup$ Thanks for Budney's reply. I do not want the Jones polynomial of a 2-knot relate to the Jones polynomial of a 1-knot. In fact I want to know whether there exists an invariant of 2-knot which can determine the minimal number of triple points. Since Jones polynomial tells us that reduced alternating knot diagram has minimal crossing number. I do not know whether this can be generalized to 2-knots. The only approach I know to the minimal triple points of a knotted surface diagram is the quandle 3-cocycle invariants. $\endgroup$
    – Zhiyun Cheng
    Commented Feb 16, 2014 at 15:55
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    $\begingroup$ It's not clear to me polynomial invariants are of much use to 2-knot theory. Relatively little is known about the connect-sum operation for 2-knots at this point. $\endgroup$
    – Ryan Budney
    Commented May 19, 2014 at 20:29

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