Number of minima in a ribbon disk? The 2019 Stack Overflow Developer Survey Results Are In ...

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Number of minima in a ribbon disk?



The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)What is a 2-surgery on a disk?fundamental group of ribbon knotadditivity of crossing number of composite knotsgenus of knot and crossing number invariantsWhat is the relationship between the Euler number of Seifert surface of a link and its linking numberPotential proof for the Slice-Ribbon conjecture (may be wrong).Is there a nontrivial oriented link with two components with linking number 0 which is related to the unknot by a single skein relation?Bounds on the number of knots with a fixed number of crossingsAlternating and Non-Altenating Knot projections with same crossing number?Framed Links are Ribbon Graphs












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$begingroup$


I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ribbon disks of $K$ ("I" for invariant - I'm guessing that it has a name and a literature).



Is there a family of ribbon knots that show that $I(K)$ can be arbitrarily large? In other words, how do I get a hold on lower bounds for $I(K)$ - hopefully without assuming slice = ribbon.










share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ribbon disks of $K$ ("I" for invariant - I'm guessing that it has a name and a literature).



    Is there a family of ribbon knots that show that $I(K)$ can be arbitrarily large? In other words, how do I get a hold on lower bounds for $I(K)$ - hopefully without assuming slice = ribbon.










    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ribbon disks of $K$ ("I" for invariant - I'm guessing that it has a name and a literature).



      Is there a family of ribbon knots that show that $I(K)$ can be arbitrarily large? In other words, how do I get a hold on lower bounds for $I(K)$ - hopefully without assuming slice = ribbon.










      share|cite|improve this question









      $endgroup$




      I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ribbon disks of $K$ ("I" for invariant - I'm guessing that it has a name and a literature).



      Is there a family of ribbon knots that show that $I(K)$ can be arbitrarily large? In other words, how do I get a hold on lower bounds for $I(K)$ - hopefully without assuming slice = ribbon.







      geometric-topology knot-theory knot-invariants






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 27 at 23:28









      user101010user101010

      1,972516




      1,972516






















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          A handle decomposition of a ribbon disk gives a band presentation of the knot. If I understand how this goes correctly, the band presentation is an unlink along with attached bands between them such that




          1. performing embedded arc surgery along these bands (which carry framing information for the surgery) gives the knot, and


          2. if the unknots are thought of as vertices and the bands as edges, the graph forms a tree (since loops in the graph would correspond to a maxima, which don't exist).



          The number of components in the unlink is the number of minima, and so your $I(K)$ is the minimal number of such components over all band presentations of the ribbon knot $K$.



          For example, here is a band presentation of a square knot:



          square knot band decomposition



          By the way, there appear to be some moves that can go between any two band presentations for the same ribbon disk of a ribbon knot:



          band presentation moves



          (See at least Figure 3 of https://arxiv.org/abs/1804.09169 or the paper they cite:
          Swenton, Frank J., On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10, No. 8, 1133-1141 (2001). ZBL1001.57044.). I wonder what additional moves you might need to go between any two ribbon disks.



          Anyway, there is a nice way to calculate a group that is the homomorphic image of the fundamental group of the knot complement from such a band presentation, as you are probably aware:



          Johnson, Dennis, Homomorphs of knot groups, Proc. Am. Math. Soc. 78, 135-138 (1980). ZBL0435.57003.



          Each unknot is assigned a generator, and each band sum introduces a single additional relation depending on which unknots the band passes through. This implies that $I(K)geq operatorname{rank}(pi_1(S^3-K))$, where the rank is the minimal number of generators over all presentations of the finitely generated group $pi_1(S^3-K)$.



          The following paper implies that there are knot groups of arbitrary rank:



          Weidmann, Richard, On the rank of amalgamated products and product knot groups, Math. Ann. 312, No. 4, 761-771 (1998). ZBL0926.20019.



          In particular, the $n$-fold connect sum of any nontrivial ribbon knot $K$ has $I(mathop{#}_nK)geq operatorname{rank}(pi_1(S^3-mathop{#}_nK))geq n+1$.



          (If there were a way to modify Johnson's construction to generate prime knots, then one could get an infinite family of prime ribbon knots with arbitrary rank.)






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
            $endgroup$
            – user101010
            Feb 5 at 0:29










          • $begingroup$
            @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
            $endgroup$
            – Kyle Miller
            Feb 7 at 0:05



















          0












          $begingroup$

          Since the Johnson paper only gives a quotient of the knot group, it is not clear why $I(#_n K)geq n$, and in particular, it is not clear if $I$ can be arbitrarily large. Am I missing something obvious?






          share|cite|improve this answer









          $endgroup$














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            2 Answers
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            2 Answers
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            active

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            2












            $begingroup$

            A handle decomposition of a ribbon disk gives a band presentation of the knot. If I understand how this goes correctly, the band presentation is an unlink along with attached bands between them such that




            1. performing embedded arc surgery along these bands (which carry framing information for the surgery) gives the knot, and


            2. if the unknots are thought of as vertices and the bands as edges, the graph forms a tree (since loops in the graph would correspond to a maxima, which don't exist).



            The number of components in the unlink is the number of minima, and so your $I(K)$ is the minimal number of such components over all band presentations of the ribbon knot $K$.



            For example, here is a band presentation of a square knot:



            square knot band decomposition



            By the way, there appear to be some moves that can go between any two band presentations for the same ribbon disk of a ribbon knot:



            band presentation moves



            (See at least Figure 3 of https://arxiv.org/abs/1804.09169 or the paper they cite:
            Swenton, Frank J., On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10, No. 8, 1133-1141 (2001). ZBL1001.57044.). I wonder what additional moves you might need to go between any two ribbon disks.



            Anyway, there is a nice way to calculate a group that is the homomorphic image of the fundamental group of the knot complement from such a band presentation, as you are probably aware:



            Johnson, Dennis, Homomorphs of knot groups, Proc. Am. Math. Soc. 78, 135-138 (1980). ZBL0435.57003.



            Each unknot is assigned a generator, and each band sum introduces a single additional relation depending on which unknots the band passes through. This implies that $I(K)geq operatorname{rank}(pi_1(S^3-K))$, where the rank is the minimal number of generators over all presentations of the finitely generated group $pi_1(S^3-K)$.



            The following paper implies that there are knot groups of arbitrary rank:



            Weidmann, Richard, On the rank of amalgamated products and product knot groups, Math. Ann. 312, No. 4, 761-771 (1998). ZBL0926.20019.



            In particular, the $n$-fold connect sum of any nontrivial ribbon knot $K$ has $I(mathop{#}_nK)geq operatorname{rank}(pi_1(S^3-mathop{#}_nK))geq n+1$.



            (If there were a way to modify Johnson's construction to generate prime knots, then one could get an infinite family of prime ribbon knots with arbitrary rank.)






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
              $endgroup$
              – user101010
              Feb 5 at 0:29










            • $begingroup$
              @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
              $endgroup$
              – Kyle Miller
              Feb 7 at 0:05
















            2












            $begingroup$

            A handle decomposition of a ribbon disk gives a band presentation of the knot. If I understand how this goes correctly, the band presentation is an unlink along with attached bands between them such that




            1. performing embedded arc surgery along these bands (which carry framing information for the surgery) gives the knot, and


            2. if the unknots are thought of as vertices and the bands as edges, the graph forms a tree (since loops in the graph would correspond to a maxima, which don't exist).



            The number of components in the unlink is the number of minima, and so your $I(K)$ is the minimal number of such components over all band presentations of the ribbon knot $K$.



            For example, here is a band presentation of a square knot:



            square knot band decomposition



            By the way, there appear to be some moves that can go between any two band presentations for the same ribbon disk of a ribbon knot:



            band presentation moves



            (See at least Figure 3 of https://arxiv.org/abs/1804.09169 or the paper they cite:
            Swenton, Frank J., On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10, No. 8, 1133-1141 (2001). ZBL1001.57044.). I wonder what additional moves you might need to go between any two ribbon disks.



            Anyway, there is a nice way to calculate a group that is the homomorphic image of the fundamental group of the knot complement from such a band presentation, as you are probably aware:



            Johnson, Dennis, Homomorphs of knot groups, Proc. Am. Math. Soc. 78, 135-138 (1980). ZBL0435.57003.



            Each unknot is assigned a generator, and each band sum introduces a single additional relation depending on which unknots the band passes through. This implies that $I(K)geq operatorname{rank}(pi_1(S^3-K))$, where the rank is the minimal number of generators over all presentations of the finitely generated group $pi_1(S^3-K)$.



            The following paper implies that there are knot groups of arbitrary rank:



            Weidmann, Richard, On the rank of amalgamated products and product knot groups, Math. Ann. 312, No. 4, 761-771 (1998). ZBL0926.20019.



            In particular, the $n$-fold connect sum of any nontrivial ribbon knot $K$ has $I(mathop{#}_nK)geq operatorname{rank}(pi_1(S^3-mathop{#}_nK))geq n+1$.



            (If there were a way to modify Johnson's construction to generate prime knots, then one could get an infinite family of prime ribbon knots with arbitrary rank.)






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
              $endgroup$
              – user101010
              Feb 5 at 0:29










            • $begingroup$
              @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
              $endgroup$
              – Kyle Miller
              Feb 7 at 0:05














            2












            2








            2





            $begingroup$

            A handle decomposition of a ribbon disk gives a band presentation of the knot. If I understand how this goes correctly, the band presentation is an unlink along with attached bands between them such that




            1. performing embedded arc surgery along these bands (which carry framing information for the surgery) gives the knot, and


            2. if the unknots are thought of as vertices and the bands as edges, the graph forms a tree (since loops in the graph would correspond to a maxima, which don't exist).



            The number of components in the unlink is the number of minima, and so your $I(K)$ is the minimal number of such components over all band presentations of the ribbon knot $K$.



            For example, here is a band presentation of a square knot:



            square knot band decomposition



            By the way, there appear to be some moves that can go between any two band presentations for the same ribbon disk of a ribbon knot:



            band presentation moves



            (See at least Figure 3 of https://arxiv.org/abs/1804.09169 or the paper they cite:
            Swenton, Frank J., On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10, No. 8, 1133-1141 (2001). ZBL1001.57044.). I wonder what additional moves you might need to go between any two ribbon disks.



            Anyway, there is a nice way to calculate a group that is the homomorphic image of the fundamental group of the knot complement from such a band presentation, as you are probably aware:



            Johnson, Dennis, Homomorphs of knot groups, Proc. Am. Math. Soc. 78, 135-138 (1980). ZBL0435.57003.



            Each unknot is assigned a generator, and each band sum introduces a single additional relation depending on which unknots the band passes through. This implies that $I(K)geq operatorname{rank}(pi_1(S^3-K))$, where the rank is the minimal number of generators over all presentations of the finitely generated group $pi_1(S^3-K)$.



            The following paper implies that there are knot groups of arbitrary rank:



            Weidmann, Richard, On the rank of amalgamated products and product knot groups, Math. Ann. 312, No. 4, 761-771 (1998). ZBL0926.20019.



            In particular, the $n$-fold connect sum of any nontrivial ribbon knot $K$ has $I(mathop{#}_nK)geq operatorname{rank}(pi_1(S^3-mathop{#}_nK))geq n+1$.



            (If there were a way to modify Johnson's construction to generate prime knots, then one could get an infinite family of prime ribbon knots with arbitrary rank.)






            share|cite|improve this answer











            $endgroup$



            A handle decomposition of a ribbon disk gives a band presentation of the knot. If I understand how this goes correctly, the band presentation is an unlink along with attached bands between them such that




            1. performing embedded arc surgery along these bands (which carry framing information for the surgery) gives the knot, and


            2. if the unknots are thought of as vertices and the bands as edges, the graph forms a tree (since loops in the graph would correspond to a maxima, which don't exist).



            The number of components in the unlink is the number of minima, and so your $I(K)$ is the minimal number of such components over all band presentations of the ribbon knot $K$.



            For example, here is a band presentation of a square knot:



            square knot band decomposition



            By the way, there appear to be some moves that can go between any two band presentations for the same ribbon disk of a ribbon knot:



            band presentation moves



            (See at least Figure 3 of https://arxiv.org/abs/1804.09169 or the paper they cite:
            Swenton, Frank J., On a calculus for 2-knots and surfaces in 4-space, J. Knot Theory Ramifications 10, No. 8, 1133-1141 (2001). ZBL1001.57044.). I wonder what additional moves you might need to go between any two ribbon disks.



            Anyway, there is a nice way to calculate a group that is the homomorphic image of the fundamental group of the knot complement from such a band presentation, as you are probably aware:



            Johnson, Dennis, Homomorphs of knot groups, Proc. Am. Math. Soc. 78, 135-138 (1980). ZBL0435.57003.



            Each unknot is assigned a generator, and each band sum introduces a single additional relation depending on which unknots the band passes through. This implies that $I(K)geq operatorname{rank}(pi_1(S^3-K))$, where the rank is the minimal number of generators over all presentations of the finitely generated group $pi_1(S^3-K)$.



            The following paper implies that there are knot groups of arbitrary rank:



            Weidmann, Richard, On the rank of amalgamated products and product knot groups, Math. Ann. 312, No. 4, 761-771 (1998). ZBL0926.20019.



            In particular, the $n$-fold connect sum of any nontrivial ribbon knot $K$ has $I(mathop{#}_nK)geq operatorname{rank}(pi_1(S^3-mathop{#}_nK))geq n+1$.



            (If there were a way to modify Johnson's construction to generate prime knots, then one could get an infinite family of prime ribbon knots with arbitrary rank.)







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Feb 5 at 18:47

























            answered Feb 4 at 1:48









            Kyle MillerKyle Miller

            10.1k930




            10.1k930








            • 1




              $begingroup$
              Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
              $endgroup$
              – user101010
              Feb 5 at 0:29










            • $begingroup$
              @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
              $endgroup$
              – Kyle Miller
              Feb 7 at 0:05














            • 1




              $begingroup$
              Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
              $endgroup$
              – user101010
              Feb 5 at 0:29










            • $begingroup$
              @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
              $endgroup$
              – Kyle Miller
              Feb 7 at 0:05








            1




            1




            $begingroup$
            Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
            $endgroup$
            – user101010
            Feb 5 at 0:29




            $begingroup$
            Thanks Kyle, this is great! I had forgotten about the Johnson paper and I did not know about that result of Weidmann - that is exactly the sort of fact that I was looking for!!
            $endgroup$
            – user101010
            Feb 5 at 0:29












            $begingroup$
            @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
            $endgroup$
            – Kyle Miller
            Feb 7 at 0:05




            $begingroup$
            @user101010 I've realized I misunderstood the kind of group presentation you get from Johnson's construction and that you only get particular quotients of the fundamental group; the ranks of these groups are lower bounds for the rank of the fundamental group itself. I think $I(mathop{#}_n K)geq n+1$ still applies, though.
            $endgroup$
            – Kyle Miller
            Feb 7 at 0:05











            0












            $begingroup$

            Since the Johnson paper only gives a quotient of the knot group, it is not clear why $I(#_n K)geq n$, and in particular, it is not clear if $I$ can be arbitrarily large. Am I missing something obvious?






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              Since the Johnson paper only gives a quotient of the knot group, it is not clear why $I(#_n K)geq n$, and in particular, it is not clear if $I$ can be arbitrarily large. Am I missing something obvious?






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                Since the Johnson paper only gives a quotient of the knot group, it is not clear why $I(#_n K)geq n$, and in particular, it is not clear if $I$ can be arbitrarily large. Am I missing something obvious?






                share|cite|improve this answer









                $endgroup$



                Since the Johnson paper only gives a quotient of the knot group, it is not clear why $I(#_n K)geq n$, and in particular, it is not clear if $I$ can be arbitrarily large. Am I missing something obvious?







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Mar 22 at 21:16









                Sucharit SarkarSucharit Sarkar

                101




                101






























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