Set of Jones polynomials as the knot varies












7














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
    yesterday






  • 4




    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
    yesterday


















7














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
    yesterday






  • 4




    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
    yesterday
















7












7








7







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 yesterday









pre-kidney

530215




530215








  • 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
    yesterday






  • 4




    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
    yesterday
















  • 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
    yesterday






  • 4




    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
    yesterday










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
yesterday




"...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
yesterday




4




4




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
yesterday






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
yesterday












1 Answer
1






active

oldest

votes


















6














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
    yesterday






  • 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
    yesterday













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1 Answer
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active

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6














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
    yesterday






  • 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
    yesterday


















6














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
    yesterday






  • 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
    yesterday
















6












6








6






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 yesterday









Martin Sleziak

2,92032028




2,92032028










answered yesterday









Adam Lowrance

28122




28122












  • 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
    yesterday






  • 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
    yesterday




















  • 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
    yesterday






  • 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
    yesterday


















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
yesterday




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
yesterday




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
yesterday






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
yesterday




















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