SL(2, C)-representation of a knot












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When studying knot theory I often encounter $SL(2, mathbb{C})$-representation of knots (of the knot group) or the $SL(2, mathbb{C})$ character variety of a knot group. But I just don't seem to understand what this is all about and when the special linear group comes into play.
Can anyone recommend me literature that covers the basics to this topic and where to start? Perhaps a nice gentle introduction preferably with examples?






rt.representation-theory gt.geometric-topology knot-theory







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  • $begingroup$
    Are you familiar with the theory of hyperbolic structures on knot complements? Thurston's notes is a good place to start, especially Ch 4 library.msri.org/books/gt3m
    $endgroup$
    – Neal
    13 hours ago
















7












$begingroup$


When studying knot theory I often encounter $SL(2, mathbb{C})$-representation of knots (of the knot group) or the $SL(2, mathbb{C})$ character variety of a knot group. But I just don't seem to understand what this is all about and when the special linear group comes into play.
Can anyone recommend me literature that covers the basics to this topic and where to start? Perhaps a nice gentle introduction preferably with examples?






rt.representation-theory gt.geometric-topology knot-theory







share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you familiar with the theory of hyperbolic structures on knot complements? Thurston's notes is a good place to start, especially Ch 4 library.msri.org/books/gt3m
    $endgroup$
    – Neal
    13 hours ago














7












7








7


1



$begingroup$


When studying knot theory I often encounter $SL(2, mathbb{C})$-representation of knots (of the knot group) or the $SL(2, mathbb{C})$ character variety of a knot group. But I just don't seem to understand what this is all about and when the special linear group comes into play.
Can anyone recommend me literature that covers the basics to this topic and where to start? Perhaps a nice gentle introduction preferably with examples?






rt.representation-theory gt.geometric-topology knot-theory







share|cite|improve this question











$endgroup$




When studying knot theory I often encounter $SL(2, mathbb{C})$-representation of knots (of the knot group) or the $SL(2, mathbb{C})$ character variety of a knot group. But I just don't seem to understand what this is all about and when the special linear group comes into play.
Can anyone recommend me literature that covers the basics to this topic and where to start? Perhaps a nice gentle introduction preferably with examples?






rt.representation-theory gt.geometric-topology knot-theory




rt.representation-theory gt.geometric-topology knot-theory






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edited 13 hours ago









YCor

27.5k481134




27.5k481134










asked 13 hours ago









Jake B.Jake B.

545211




545211












  • $begingroup$
    Are you familiar with the theory of hyperbolic structures on knot complements? Thurston's notes is a good place to start, especially Ch 4 library.msri.org/books/gt3m
    $endgroup$
    – Neal
    13 hours ago


















  • $begingroup$
    Are you familiar with the theory of hyperbolic structures on knot complements? Thurston's notes is a good place to start, especially Ch 4 library.msri.org/books/gt3m
    $endgroup$
    – Neal
    13 hours ago
















$begingroup$
Are you familiar with the theory of hyperbolic structures on knot complements? Thurston's notes is a good place to start, especially Ch 4 library.msri.org/books/gt3m
$endgroup$
– Neal
13 hours ago




$begingroup$
Are you familiar with the theory of hyperbolic structures on knot complements? Thurston's notes is a good place to start, especially Ch 4 library.msri.org/books/gt3m
$endgroup$
– Neal
13 hours ago










1 Answer
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$(P)SL(2, mathbb{C})$ is the isometry group of $mathbb{H}^3,$ so $SL(2, mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast literature on the subject, but you might want to look at some of the foundational work:



Morgan, John W.; Shalen, Peter B., Valuations, trees, and degenerations of hyperbolic structures. I, Ann. Math. (2) 120, 401-476 (1984). ZBL0583.57005.






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    1 Answer
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    active

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    7












    $begingroup$

    $(P)SL(2, mathbb{C})$ is the isometry group of $mathbb{H}^3,$ so $SL(2, mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast literature on the subject, but you might want to look at some of the foundational work:



    Morgan, John W.; Shalen, Peter B., Valuations, trees, and degenerations of hyperbolic structures. I, Ann. Math. (2) 120, 401-476 (1984). ZBL0583.57005.






    share|cite|improve this answer









    $endgroup$


















      7












      $begingroup$

      $(P)SL(2, mathbb{C})$ is the isometry group of $mathbb{H}^3,$ so $SL(2, mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast literature on the subject, but you might want to look at some of the foundational work:



      Morgan, John W.; Shalen, Peter B., Valuations, trees, and degenerations of hyperbolic structures. I, Ann. Math. (2) 120, 401-476 (1984). ZBL0583.57005.






      share|cite|improve this answer









      $endgroup$
















        7












        7








        7





        $begingroup$

        $(P)SL(2, mathbb{C})$ is the isometry group of $mathbb{H}^3,$ so $SL(2, mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast literature on the subject, but you might want to look at some of the foundational work:



        Morgan, John W.; Shalen, Peter B., Valuations, trees, and degenerations of hyperbolic structures. I, Ann. Math. (2) 120, 401-476 (1984). ZBL0583.57005.






        share|cite|improve this answer









        $endgroup$



        $(P)SL(2, mathbb{C})$ is the isometry group of $mathbb{H}^3,$ so $SL(2, mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast literature on the subject, but you might want to look at some of the foundational work:



        Morgan, John W.; Shalen, Peter B., Valuations, trees, and degenerations of hyperbolic structures. I, Ann. Math. (2) 120, 401-476 (1984). ZBL0583.57005.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 13 hours ago









        Igor RivinIgor Rivin

        79.2k8113308




        79.2k8113308






























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