2
$\begingroup$

In Khovanov homology the Jones polynomial of a given link is computed as the graded Euler characteristic of a certain complex associated with the link. From other side the Atiyah-Singer theorem provides a representation of the Euler characteristic as the index of a certain elliptic operator. Then my question is : It is possible to obtain the Jones polynomial as the equivariant index of a certain Dirac operator associated with the given link?

$\endgroup$
  • 3
    $\begingroup$ Euler characteristic is one of the most heavily interpreted concepts in topology, with a variant for essentially every interest. Is there something motivating this particular question? $\endgroup$ – Ryan Budney Dec 2 '14 at 22:00
  • 1
    $\begingroup$ Note that the complex whose Euler characteristic gives rise to the Jones polynomial is not the same as the complex whose Euler characteristic is computed by the index of the de Rham operator (namely, the de Rham complex). I don't have a reason why the Jones polynomial is not related to the index theorem somehow, but "both involve Euler characteristics of complexes" is not convincing evidence to me. $\endgroup$ – Paul Siegel Dec 3 '14 at 7:53
  • 1
    $\begingroup$ Simon Willerton has written about a rough analogue of your question for the Kontsevich Invariant: arxiv.org/abs/math/0105190 $\endgroup$ – Daniel Moskovich Jan 14 '15 at 13:33
  • $\begingroup$ Professor @DanielMoskovich many thanks for the paper which is very interesting. From other side your papers about tangle machines are impressive. All the best. $\endgroup$ – Juan Ospina Jan 14 '15 at 15:12
  • $\begingroup$ A recent paper very close to the answer of my question is xxx.lanl.gov/pdf/1502.03116.pdf $\endgroup$ – Juan Ospina Feb 12 '15 at 11:48

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.