Connection Between Knot Theory and Number Theory












5












$begingroup$


Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?










share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary
    $endgroup$
    – Jason Starr
    Mar 19 at 13:15






  • 6




    $begingroup$
    The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller).
    $endgroup$
    – Daniel Litt
    Mar 19 at 13:59










  • $begingroup$
    @DanielLitt thank you for promoting research outside the mainstream understanding.
    $endgroup$
    – Matt Cuffaro
    Mar 19 at 14:37
















5












$begingroup$


Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?










share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary
    $endgroup$
    – Jason Starr
    Mar 19 at 13:15






  • 6




    $begingroup$
    The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller).
    $endgroup$
    – Daniel Litt
    Mar 19 at 13:59










  • $begingroup$
    @DanielLitt thank you for promoting research outside the mainstream understanding.
    $endgroup$
    – Matt Cuffaro
    Mar 19 at 14:37














5












5








5


3



$begingroup$


Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?










share|cite|improve this question











$endgroup$




Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?







reference-request arithmetic-geometry algebraic-number-theory knot-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 19 at 15:05









Sean Lawton

3,97622248




3,97622248










asked Mar 19 at 13:12









fsociety_1729fsociety_1729

293




293








  • 5




    $begingroup$
    Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary
    $endgroup$
    – Jason Starr
    Mar 19 at 13:15






  • 6




    $begingroup$
    The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller).
    $endgroup$
    – Daniel Litt
    Mar 19 at 13:59










  • $begingroup$
    @DanielLitt thank you for promoting research outside the mainstream understanding.
    $endgroup$
    – Matt Cuffaro
    Mar 19 at 14:37














  • 5




    $begingroup$
    Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary
    $endgroup$
    – Jason Starr
    Mar 19 at 13:15






  • 6




    $begingroup$
    The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller).
    $endgroup$
    – Daniel Litt
    Mar 19 at 13:59










  • $begingroup$
    @DanielLitt thank you for promoting research outside the mainstream understanding.
    $endgroup$
    – Matt Cuffaro
    Mar 19 at 14:37








5




5




$begingroup$
Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary
$endgroup$
– Jason Starr
Mar 19 at 13:15




$begingroup$
Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary
$endgroup$
– Jason Starr
Mar 19 at 13:15




6




6




$begingroup$
The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller).
$endgroup$
– Daniel Litt
Mar 19 at 13:59




$begingroup$
The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller).
$endgroup$
– Daniel Litt
Mar 19 at 13:59












$begingroup$
@DanielLitt thank you for promoting research outside the mainstream understanding.
$endgroup$
– Matt Cuffaro
Mar 19 at 14:37




$begingroup$
@DanielLitt thank you for promoting research outside the mainstream understanding.
$endgroup$
– Matt Cuffaro
Mar 19 at 14:37










5 Answers
5






active

oldest

votes


















9












$begingroup$

The question seems very general, but the first book to come to mind is this:



The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)






share|cite|improve this answer









$endgroup$





















    5












    $begingroup$

    I recommend Chao Li/Charmaine Sia's notes for a brisk and illustrated overview of the ever-growing MKR dictionary detailed in Knots and Primes. The first three or so chapters of the source book is rather accessible.



    For the MKR dictionary, refer to the link in comments.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
      $endgroup$
      – Alex M.
      Mar 21 at 18:03



















    4












    $begingroup$

    I am not at all an expert in these matters, but there seems to be recent developments in this area around the work of Minhyong Kim and collaborators. See in particular:




    • https://arxiv.org/abs/1706.03336

    • https://arxiv.org/abs/1510.05818

    • https://arxiv.org/abs/1609.03012

    • https://arxiv.org/abs/1712.07602


    Since the classic article by Witten,
    Chern-Simons gauge theory from physics is well known to be related to knot theory. My understanding is this work by Kim and collaborators aims at developing an arithmetic analogue of this theory.






    share|cite|improve this answer









    $endgroup$





















      4












      $begingroup$

      Currently, the most complete book presenting the connections between knot theory and number theory is "Knots and Primes. An Introduction to Arithmetic Topology" by Masanori Morishita (Springer, 2012). (It also presents much of the needed prerequisites, so it should be the first choice for somebody serious about self-studying the subject.)



      If you feel confident enough that you already know the basics, it might be useful to take a look at "Primes and Knots" edited by Toshitake Kohno and Masanori Morishita (AMS, 2006). It contains the proceedings of two 2003 conferences on this topic and closely related ones (and, strangely, it is currently freely available, so grab it legally while you can!).






      share|cite|improve this answer









      $endgroup$





















        2












        $begingroup$

        I recommended look lower link.



        http://www.math.columbia.edu/~chaoli/tutorial2012/KnotsAndPrimes.html






        share|cite|improve this answer









        $endgroup$













        • $begingroup$
          It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
          $endgroup$
          – Alex M.
          Mar 21 at 17:59










        • $begingroup$
          your right. I forgot his answers.
          $endgroup$
          – hussein
          Mar 21 at 18:01












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        5 Answers
        5






        active

        oldest

        votes








        5 Answers
        5






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        9












        $begingroup$

        The question seems very general, but the first book to come to mind is this:



        The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)






        share|cite|improve this answer









        $endgroup$


















          9












          $begingroup$

          The question seems very general, but the first book to come to mind is this:



          The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)






          share|cite|improve this answer









          $endgroup$
















            9












            9








            9





            $begingroup$

            The question seems very general, but the first book to come to mind is this:



            The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)






            share|cite|improve this answer









            $endgroup$



            The question seems very general, but the first book to come to mind is this:



            The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Mar 19 at 13:20









            Sean LawtonSean Lawton

            3,97622248




            3,97622248























                5












                $begingroup$

                I recommend Chao Li/Charmaine Sia's notes for a brisk and illustrated overview of the ever-growing MKR dictionary detailed in Knots and Primes. The first three or so chapters of the source book is rather accessible.



                For the MKR dictionary, refer to the link in comments.






                share|cite|improve this answer











                $endgroup$









                • 1




                  $begingroup$
                  An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
                  $endgroup$
                  – Alex M.
                  Mar 21 at 18:03
















                5












                $begingroup$

                I recommend Chao Li/Charmaine Sia's notes for a brisk and illustrated overview of the ever-growing MKR dictionary detailed in Knots and Primes. The first three or so chapters of the source book is rather accessible.



                For the MKR dictionary, refer to the link in comments.






                share|cite|improve this answer











                $endgroup$









                • 1




                  $begingroup$
                  An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
                  $endgroup$
                  – Alex M.
                  Mar 21 at 18:03














                5












                5








                5





                $begingroup$

                I recommend Chao Li/Charmaine Sia's notes for a brisk and illustrated overview of the ever-growing MKR dictionary detailed in Knots and Primes. The first three or so chapters of the source book is rather accessible.



                For the MKR dictionary, refer to the link in comments.






                share|cite|improve this answer











                $endgroup$



                I recommend Chao Li/Charmaine Sia's notes for a brisk and illustrated overview of the ever-growing MKR dictionary detailed in Knots and Primes. The first three or so chapters of the source book is rather accessible.



                For the MKR dictionary, refer to the link in comments.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Mar 19 at 13:41

























                answered Mar 19 at 13:35









                Matt CuffaroMatt Cuffaro

                160116




                160116








                • 1




                  $begingroup$
                  An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
                  $endgroup$
                  – Alex M.
                  Mar 21 at 18:03














                • 1




                  $begingroup$
                  An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
                  $endgroup$
                  – Alex M.
                  Mar 21 at 18:03








                1




                1




                $begingroup$
                An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
                $endgroup$
                – Alex M.
                Mar 21 at 18:03




                $begingroup$
                An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University.
                $endgroup$
                – Alex M.
                Mar 21 at 18:03











                4












                $begingroup$

                I am not at all an expert in these matters, but there seems to be recent developments in this area around the work of Minhyong Kim and collaborators. See in particular:




                • https://arxiv.org/abs/1706.03336

                • https://arxiv.org/abs/1510.05818

                • https://arxiv.org/abs/1609.03012

                • https://arxiv.org/abs/1712.07602


                Since the classic article by Witten,
                Chern-Simons gauge theory from physics is well known to be related to knot theory. My understanding is this work by Kim and collaborators aims at developing an arithmetic analogue of this theory.






                share|cite|improve this answer









                $endgroup$


















                  4












                  $begingroup$

                  I am not at all an expert in these matters, but there seems to be recent developments in this area around the work of Minhyong Kim and collaborators. See in particular:




                  • https://arxiv.org/abs/1706.03336

                  • https://arxiv.org/abs/1510.05818

                  • https://arxiv.org/abs/1609.03012

                  • https://arxiv.org/abs/1712.07602


                  Since the classic article by Witten,
                  Chern-Simons gauge theory from physics is well known to be related to knot theory. My understanding is this work by Kim and collaborators aims at developing an arithmetic analogue of this theory.






                  share|cite|improve this answer









                  $endgroup$
















                    4












                    4








                    4





                    $begingroup$

                    I am not at all an expert in these matters, but there seems to be recent developments in this area around the work of Minhyong Kim and collaborators. See in particular:




                    • https://arxiv.org/abs/1706.03336

                    • https://arxiv.org/abs/1510.05818

                    • https://arxiv.org/abs/1609.03012

                    • https://arxiv.org/abs/1712.07602


                    Since the classic article by Witten,
                    Chern-Simons gauge theory from physics is well known to be related to knot theory. My understanding is this work by Kim and collaborators aims at developing an arithmetic analogue of this theory.






                    share|cite|improve this answer









                    $endgroup$



                    I am not at all an expert in these matters, but there seems to be recent developments in this area around the work of Minhyong Kim and collaborators. See in particular:




                    • https://arxiv.org/abs/1706.03336

                    • https://arxiv.org/abs/1510.05818

                    • https://arxiv.org/abs/1609.03012

                    • https://arxiv.org/abs/1712.07602


                    Since the classic article by Witten,
                    Chern-Simons gauge theory from physics is well known to be related to knot theory. My understanding is this work by Kim and collaborators aims at developing an arithmetic analogue of this theory.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Mar 19 at 14:44









                    Abdelmalek AbdesselamAbdelmalek Abdesselam

                    11.1k12871




                    11.1k12871























                        4












                        $begingroup$

                        Currently, the most complete book presenting the connections between knot theory and number theory is "Knots and Primes. An Introduction to Arithmetic Topology" by Masanori Morishita (Springer, 2012). (It also presents much of the needed prerequisites, so it should be the first choice for somebody serious about self-studying the subject.)



                        If you feel confident enough that you already know the basics, it might be useful to take a look at "Primes and Knots" edited by Toshitake Kohno and Masanori Morishita (AMS, 2006). It contains the proceedings of two 2003 conferences on this topic and closely related ones (and, strangely, it is currently freely available, so grab it legally while you can!).






                        share|cite|improve this answer









                        $endgroup$


















                          4












                          $begingroup$

                          Currently, the most complete book presenting the connections between knot theory and number theory is "Knots and Primes. An Introduction to Arithmetic Topology" by Masanori Morishita (Springer, 2012). (It also presents much of the needed prerequisites, so it should be the first choice for somebody serious about self-studying the subject.)



                          If you feel confident enough that you already know the basics, it might be useful to take a look at "Primes and Knots" edited by Toshitake Kohno and Masanori Morishita (AMS, 2006). It contains the proceedings of two 2003 conferences on this topic and closely related ones (and, strangely, it is currently freely available, so grab it legally while you can!).






                          share|cite|improve this answer









                          $endgroup$
















                            4












                            4








                            4





                            $begingroup$

                            Currently, the most complete book presenting the connections between knot theory and number theory is "Knots and Primes. An Introduction to Arithmetic Topology" by Masanori Morishita (Springer, 2012). (It also presents much of the needed prerequisites, so it should be the first choice for somebody serious about self-studying the subject.)



                            If you feel confident enough that you already know the basics, it might be useful to take a look at "Primes and Knots" edited by Toshitake Kohno and Masanori Morishita (AMS, 2006). It contains the proceedings of two 2003 conferences on this topic and closely related ones (and, strangely, it is currently freely available, so grab it legally while you can!).






                            share|cite|improve this answer









                            $endgroup$



                            Currently, the most complete book presenting the connections between knot theory and number theory is "Knots and Primes. An Introduction to Arithmetic Topology" by Masanori Morishita (Springer, 2012). (It also presents much of the needed prerequisites, so it should be the first choice for somebody serious about self-studying the subject.)



                            If you feel confident enough that you already know the basics, it might be useful to take a look at "Primes and Knots" edited by Toshitake Kohno and Masanori Morishita (AMS, 2006). It contains the proceedings of two 2003 conferences on this topic and closely related ones (and, strangely, it is currently freely available, so grab it legally while you can!).







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Mar 20 at 18:03









                            Alex M.Alex M.

                            2,49441633




                            2,49441633























                                2












                                $begingroup$

                                I recommended look lower link.



                                http://www.math.columbia.edu/~chaoli/tutorial2012/KnotsAndPrimes.html






                                share|cite|improve this answer









                                $endgroup$













                                • $begingroup$
                                  It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
                                  $endgroup$
                                  – Alex M.
                                  Mar 21 at 17:59










                                • $begingroup$
                                  your right. I forgot his answers.
                                  $endgroup$
                                  – hussein
                                  Mar 21 at 18:01
















                                2












                                $begingroup$

                                I recommended look lower link.



                                http://www.math.columbia.edu/~chaoli/tutorial2012/KnotsAndPrimes.html






                                share|cite|improve this answer









                                $endgroup$













                                • $begingroup$
                                  It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
                                  $endgroup$
                                  – Alex M.
                                  Mar 21 at 17:59










                                • $begingroup$
                                  your right. I forgot his answers.
                                  $endgroup$
                                  – hussein
                                  Mar 21 at 18:01














                                2












                                2








                                2





                                $begingroup$

                                I recommended look lower link.



                                http://www.math.columbia.edu/~chaoli/tutorial2012/KnotsAndPrimes.html






                                share|cite|improve this answer









                                $endgroup$



                                I recommended look lower link.



                                http://www.math.columbia.edu/~chaoli/tutorial2012/KnotsAndPrimes.html







                                share|cite|improve this answer












                                share|cite|improve this answer



                                share|cite|improve this answer










                                answered Mar 21 at 12:38









                                husseinhussein

                                191




                                191












                                • $begingroup$
                                  It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
                                  $endgroup$
                                  – Alex M.
                                  Mar 21 at 17:59










                                • $begingroup$
                                  your right. I forgot his answers.
                                  $endgroup$
                                  – hussein
                                  Mar 21 at 18:01


















                                • $begingroup$
                                  It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
                                  $endgroup$
                                  – Alex M.
                                  Mar 21 at 17:59










                                • $begingroup$
                                  your right. I forgot his answers.
                                  $endgroup$
                                  – hussein
                                  Mar 21 at 18:01
















                                $begingroup$
                                It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
                                $endgroup$
                                – Alex M.
                                Mar 21 at 17:59




                                $begingroup$
                                It has already been mentioned by Matt Cufaro in his answer, just with a different URL.
                                $endgroup$
                                – Alex M.
                                Mar 21 at 17:59












                                $begingroup$
                                your right. I forgot his answers.
                                $endgroup$
                                – hussein
                                Mar 21 at 18:01




                                $begingroup$
                                your right. I forgot his answers.
                                $endgroup$
                                – hussein
                                Mar 21 at 18:01


















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