Set of Jones polynomials as the knot varies












3














Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?






at.algebraic-topology knot-theory jones-polynomial







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  • 1




    "...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant.
    – David G. Stork
    3 hours ago






  • 2




    As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title.
    – pre-kidney
    3 hours ago


















3














Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?






at.algebraic-topology knot-theory jones-polynomial







share|cite|improve this question


















  • 1




    "...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant.
    – David G. Stork
    3 hours ago






  • 2




    As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title.
    – pre-kidney
    3 hours ago
















3












3








3







Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?






at.algebraic-topology knot-theory jones-polynomial







share|cite|improve this question













Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?






at.algebraic-topology knot-theory jones-polynomial




at.algebraic-topology knot-theory jones-polynomial






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









pre-kidney

510215




510215








  • 1




    "...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant.
    – David G. Stork
    3 hours ago






  • 2




    As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title.
    – pre-kidney
    3 hours ago
















  • 1




    "...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant.
    – David G. Stork
    3 hours ago






  • 2




    As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title.
    – pre-kidney
    3 hours ago










1




1




"...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant.
– David G. Stork
3 hours ago




"...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant.
– David G. Stork
3 hours ago




2




2




As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title.
– pre-kidney
3 hours ago






As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title.
– pre-kidney
3 hours ago












1 Answer
1






active

oldest

votes


















2














I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.



Theorem. Let $p(t)$ be any Laurent polynomial satisfying:





  1. $p(1) = pm 1$, and


  2. $p(t)=p(t^{-1})$.


There exists a knot $K$ whose Alexander polynomial $Delta_K(t)$ is $p(t)$.



It is well-known that the Alexander polynomial satisfies the above conditions. In the proof, Rolfsen shows how to explicitly construct a knot $K$ whose Alexander polynomial is a given Laurent polynomial $p(t)$ satisfying the conditions above.



Rolfsen gives the original reference of this result as




  • Seifert, H.; Über das Geschlecht von Knoten. Math. Ann. 110 (1935), no. 1, 571–592.






share|cite|improve this answer























  • Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
    – pre-kidney
    1 hour ago






  • 1




    I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
    – Adam Lowrance
    1 hour ago













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1 Answer
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active

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active

oldest

votes









2














I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.



Theorem. Let $p(t)$ be any Laurent polynomial satisfying:





  1. $p(1) = pm 1$, and


  2. $p(t)=p(t^{-1})$.


There exists a knot $K$ whose Alexander polynomial $Delta_K(t)$ is $p(t)$.



It is well-known that the Alexander polynomial satisfies the above conditions. In the proof, Rolfsen shows how to explicitly construct a knot $K$ whose Alexander polynomial is a given Laurent polynomial $p(t)$ satisfying the conditions above.



Rolfsen gives the original reference of this result as




  • Seifert, H.; Über das Geschlecht von Knoten. Math. Ann. 110 (1935), no. 1, 571–592.






share|cite|improve this answer























  • Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
    – pre-kidney
    1 hour ago






  • 1




    I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
    – Adam Lowrance
    1 hour ago


















2














I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.



Theorem. Let $p(t)$ be any Laurent polynomial satisfying:





  1. $p(1) = pm 1$, and


  2. $p(t)=p(t^{-1})$.


There exists a knot $K$ whose Alexander polynomial $Delta_K(t)$ is $p(t)$.



It is well-known that the Alexander polynomial satisfies the above conditions. In the proof, Rolfsen shows how to explicitly construct a knot $K$ whose Alexander polynomial is a given Laurent polynomial $p(t)$ satisfying the conditions above.



Rolfsen gives the original reference of this result as




  • Seifert, H.; Über das Geschlecht von Knoten. Math. Ann. 110 (1935), no. 1, 571–592.






share|cite|improve this answer























  • Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
    – pre-kidney
    1 hour ago






  • 1




    I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
    – Adam Lowrance
    1 hour ago
















2












2








2






I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.



Theorem. Let $p(t)$ be any Laurent polynomial satisfying:





  1. $p(1) = pm 1$, and


  2. $p(t)=p(t^{-1})$.


There exists a knot $K$ whose Alexander polynomial $Delta_K(t)$ is $p(t)$.



It is well-known that the Alexander polynomial satisfies the above conditions. In the proof, Rolfsen shows how to explicitly construct a knot $K$ whose Alexander polynomial is a given Laurent polynomial $p(t)$ satisfying the conditions above.



Rolfsen gives the original reference of this result as




  • Seifert, H.; Über das Geschlecht von Knoten. Math. Ann. 110 (1935), no. 1, 571–592.






share|cite|improve this answer














I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.



Theorem. Let $p(t)$ be any Laurent polynomial satisfying:





  1. $p(1) = pm 1$, and


  2. $p(t)=p(t^{-1})$.


There exists a knot $K$ whose Alexander polynomial $Delta_K(t)$ is $p(t)$.



It is well-known that the Alexander polynomial satisfies the above conditions. In the proof, Rolfsen shows how to explicitly construct a knot $K$ whose Alexander polynomial is a given Laurent polynomial $p(t)$ satisfying the conditions above.



Rolfsen gives the original reference of this result as




  • Seifert, H.; Über das Geschlecht von Knoten. Math. Ann. 110 (1935), no. 1, 571–592.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 2 hours ago









Martin Sleziak

2,92032028




2,92032028










answered 3 hours ago









Adam Lowrance

24122




24122












  • Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
    – pre-kidney
    1 hour ago






  • 1




    I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
    – Adam Lowrance
    1 hour ago




















  • Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
    – pre-kidney
    1 hour ago






  • 1




    I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
    – Adam Lowrance
    1 hour ago


















Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
– pre-kidney
1 hour ago




Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials?
– pre-kidney
1 hour ago




1




1




I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
– Adam Lowrance
1 hour ago






I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients.
– Adam Lowrance
1 hour ago




















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