Number of (distinct) knots with a bounded number of crossings

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The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternating projection".






co.combinatorics graph-theory gt.geometric-topology







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    The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternating projection".






    co.combinatorics graph-theory gt.geometric-topology







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      The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternating projection".






      co.combinatorics graph-theory gt.geometric-topology







      share|cite|improve this question













      The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternating projection".






      co.combinatorics graph-theory gt.geometric-topology




      co.combinatorics graph-theory gt.geometric-topology






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      asked 2 hours ago









      Igor Rivin

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          Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered).



          Clearly from inclusion of sets $ak(n)leq a(n)leq l(n)$ and $ak(n)leq k(n)leq l(n)$.



          Then we have



          $$2.68 leq liminf_nto infty k(n)^frac1n leq liminf_nto infty l(n)^frac1n leq 10.398...,$$



          where the left is due to Welsh based on Ernst-Sumners (and only counts the growth of 2-bridge knots), and the right estimate is due to Stoimenow.



          For alternating knots, one has the same lower bound since 2-bridge knots are alternating. So one has



          $$2.68 leq liminf_nto infty ak(n)^frac1n leq lim_nto infty a(n)^frac1n = 6.14793...,$$
          the right upper bound coming from Sundberg-Thistlethwaite.



          From estimates on the growth of prime knots with $n$ crossings, one should be able to obtain the growth of all knots with $n$ crossings via the prime decomposition.






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            Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered).



            Clearly from inclusion of sets $ak(n)leq a(n)leq l(n)$ and $ak(n)leq k(n)leq l(n)$.



            Then we have



            $$2.68 leq liminf_nto infty k(n)^frac1n leq liminf_nto infty l(n)^frac1n leq 10.398...,$$



            where the left is due to Welsh based on Ernst-Sumners (and only counts the growth of 2-bridge knots), and the right estimate is due to Stoimenow.



            For alternating knots, one has the same lower bound since 2-bridge knots are alternating. So one has



            $$2.68 leq liminf_nto infty ak(n)^frac1n leq lim_nto infty a(n)^frac1n = 6.14793...,$$
            the right upper bound coming from Sundberg-Thistlethwaite.



            From estimates on the growth of prime knots with $n$ crossings, one should be able to obtain the growth of all knots with $n$ crossings via the prime decomposition.






            share|cite|improve this answer
























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              Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered).



              Clearly from inclusion of sets $ak(n)leq a(n)leq l(n)$ and $ak(n)leq k(n)leq l(n)$.



              Then we have



              $$2.68 leq liminf_nto infty k(n)^frac1n leq liminf_nto infty l(n)^frac1n leq 10.398...,$$



              where the left is due to Welsh based on Ernst-Sumners (and only counts the growth of 2-bridge knots), and the right estimate is due to Stoimenow.



              For alternating knots, one has the same lower bound since 2-bridge knots are alternating. So one has



              $$2.68 leq liminf_nto infty ak(n)^frac1n leq lim_nto infty a(n)^frac1n = 6.14793...,$$
              the right upper bound coming from Sundberg-Thistlethwaite.



              From estimates on the growth of prime knots with $n$ crossings, one should be able to obtain the growth of all knots with $n$ crossings via the prime decomposition.






              share|cite|improve this answer






















                up vote
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                up vote
                5
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                Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered).



                Clearly from inclusion of sets $ak(n)leq a(n)leq l(n)$ and $ak(n)leq k(n)leq l(n)$.



                Then we have



                $$2.68 leq liminf_nto infty k(n)^frac1n leq liminf_nto infty l(n)^frac1n leq 10.398...,$$



                where the left is due to Welsh based on Ernst-Sumners (and only counts the growth of 2-bridge knots), and the right estimate is due to Stoimenow.



                For alternating knots, one has the same lower bound since 2-bridge knots are alternating. So one has



                $$2.68 leq liminf_nto infty ak(n)^frac1n leq lim_nto infty a(n)^frac1n = 6.14793...,$$
                the right upper bound coming from Sundberg-Thistlethwaite.



                From estimates on the growth of prime knots with $n$ crossings, one should be able to obtain the growth of all knots with $n$ crossings via the prime decomposition.






                share|cite|improve this answer












                Let $k(n)$ denote the number of prime knots with $n$ crossings, $l(n)$ the number of prime links with $n$ crossings, $a(n)$ the number of alternating prime links with $n$ crossings, and $ak(n)$ the number of prime alternating knots (all unoriented and unordered).



                Clearly from inclusion of sets $ak(n)leq a(n)leq l(n)$ and $ak(n)leq k(n)leq l(n)$.



                Then we have



                $$2.68 leq liminf_nto infty k(n)^frac1n leq liminf_nto infty l(n)^frac1n leq 10.398...,$$



                where the left is due to Welsh based on Ernst-Sumners (and only counts the growth of 2-bridge knots), and the right estimate is due to Stoimenow.



                For alternating knots, one has the same lower bound since 2-bridge knots are alternating. So one has



                $$2.68 leq liminf_nto infty ak(n)^frac1n leq lim_nto infty a(n)^frac1n = 6.14793...,$$
                the right upper bound coming from Sundberg-Thistlethwaite.



                From estimates on the growth of prime knots with $n$ crossings, one should be able to obtain the growth of all knots with $n$ crossings via the prime decomposition.







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                answered 50 mins ago









                Ian Agol

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