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Diagrammatically maximal and geometrically maximal knots Jessica Purcell Monash University, School of Mathematical Sciences Joint work with Abhijit Champanerkar, Ilya Kofman J. Purcell Diagrammatically maximal and geometrically maximal knots


  1. Diagrammatically maximal and geometrically maximal knots Jessica Purcell Monash University, School of Mathematical Sciences Joint work with Abhijit Champanerkar, Ilya Kofman J. Purcell Diagrammatically maximal and geometrically maximal knots

  2. Part 1: Knots from a combinatorial point of view J. Purcell Diagrammatically maximal and geometrically maximal knots

  3. Knot diagram Knot diagram: 4–valent planar graph with over–under crossing info at each vertex. Alternating knot: Crossings alternate between over and under. Diagram graph: Graph obtained by dropping crossing decoration on K , denoted G ( K ). J. Purcell Diagrammatically maximal and geometrically maximal knots

  4. Diagram graph of alternating knot Given any 4–valent graph G , ∃ ! alternating knot K with G ( K ) = G (up to reflection). J. Purcell Diagrammatically maximal and geometrically maximal knots

  5. Reduced alternating diagram Throughout, all diagrams are connected (i.e. G ( K ) connected) Diagram is reduced if it has no nugatory crosings : Undo nugatory crossings J. Purcell Diagrammatically maximal and geometrically maximal knots

  6. Tait graph Associated to any diagram K is a Tait graph Γ K : Checkerboard color complementary regions of K . Assign a vertex to every shaded region, edge to every crossing, ± sign to every edge: ����������� ������ ����������� ����������� ������ ����������� ����������� ����������� ����������� ����������� ����� ����� ����������� ����������� ����������� ����������� ������ ������ ����������� ����������� ����������� ����������� ����� ����� ����������� ����������� ������ ����������� ������ ����������� ����������� ����������� ����������� ����������� ����� ����� ����������� ������ ����������� ����������� ������ ����������� ����������� ����������� ����������� ����������� ����� ����� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ����� ����������� ����������� ����������� ����������� ����� ������ ����� ����� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ����� ����� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ����� ����� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ ����� ����� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ������ ������ Note: e (Γ K ) = c ( K ) crossing number of diagram of K = v ( G ( K )). Signs agree on all edges of Γ K ⇔ K is alternating. J. Purcell Diagrammatically maximal and geometrically maximal knots

  7. Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots

  8. Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots

  9. Reduced diagrams and Tait graphs K reduced ⇔ Γ K has no loops, no bridges. J. Purcell Diagrammatically maximal and geometrically maximal knots

  10. Determinant of a knot Let τ ( K ) = number spanning trees of Tait graph Γ K . Definition If K is alternating, det( K ) = τ ( K ) . More generally, let s n ( K ) be number of spanning trees of Γ K with n positive edges. � � � � � ( − 1) n s n ( K ) det( K ) = � � � � � n � (Not usual definition of determinant, but equivalent) J. Purcell Diagrammatically maximal and geometrically maximal knots

  11. Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots

  12. Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots

  13. Determinant under crossing change Proposition Let K be a reduced alternating link diagram, K ′ obtained by changing any proper subset of crossings of K. Then det( K ′ ) < det( K ) . J. Purcell Diagrammatically maximal and geometrically maximal knots

  14. Determinant under crossing change Proposition Let K be a reduced alternating link diagram, K ′ obtained by changing any proper subset of crossings of K. Then det( K ′ ) < det( K ) . Proof. If only one crossing is switched, let e be corresponding edge. Note e is only negative edge in Γ K ′ . K has no nugatory crossings ⇒ e is not a bridge or loop ⇒ ∃ spanning trees T 1 , T 2 such that e ∈ T 1 and e / ∈ T 2 . Add 2 to det( K ) for T 1 , T 2 . Add 1 subtract 1 to det( K ′ ). Similarly if more than one crossing is switched. J. Purcell Diagrammatically maximal and geometrically maximal knots

  15. How big can det( K ) be? det( K ) can be arbitrarily large. However, note it grows by crossing number. J. Purcell Diagrammatically maximal and geometrically maximal knots

  16. Determinant density conjecture Conjecture If K is any knot or link, 2 π log det( K ) ≤ v oct . c ( K ) Here v oct is the volume of a regular hyperbolic ideal octahedron. J. Purcell Diagrammatically maximal and geometrically maximal knots

  17. Equivalent to Conjecture of Kenyon Conjecture (Kenyon, 1996) If G is any finite planar graph, log τ ( G ) ≤ 2 C /π, e ( G ) where C ≈ 0 . 916 is Catalan’s constant. Equivalence: 4 C = v oct Any finite planar graph G can be realized as the Tait graph Γ K of an anternating link K e (Γ K ) = c ( K ) J. Purcell Diagrammatically maximal and geometrically maximal knots

  18. Part 1.5: Geometric Interlude J. Purcell Diagrammatically maximal and geometrically maximal knots

  19. Some geometry of knots Can build the complement of K out of octahedra: S 3 − K = � octahedra . crossings J. Purcell Diagrammatically maximal and geometrically maximal knots

  20. Some geometry of knots v oct = vol of regular hyperbolic ideal octahedron = max vol of any hyperbolic octahedron vol ( K ) = ⇒ < v oct . c ( K ) J. Purcell Diagrammatically maximal and geometrically maximal knots

  21. Relations between vol ( K ) and det( K ) Known Conjectured vol ( K ) 2 π log det( K ) c ( K ) ≤ v oct ≤ v oct c ( K ) det( K ′ ) < det( K ) vol ( K ′ ) < vol ( K ) 2nd line: K ′ obtained from alternating K by changing crossings. Conjectures verified experimentally for 10.7 million knots. J. Purcell Diagrammatically maximal and geometrically maximal knots

  22. Relations between vol ( K ) and det( K ) Known Conjectured vol ( K ) 2 π log det( K ) c ( K ) ≤ v oct ≤ v oct c ( K ) det( K ′ ) < det( K ) vol ( K ′ ) < vol ( K ) 2nd line: K ′ obtained from alternating K by changing crossings. Conjectures verified experimentally for 10.7 million knots. Conjecture For any alternating hyperbolic knot, vol ( K ) < 2 π log det( K ) J. Purcell Diagrammatically maximal and geometrically maximal knots

  23. How sharp is the upper bound? Is v oct the best possible? I.e. does ∃ sequence of knots K n with 2 π log det( K n ) lim → v oct ? c ( K n ) n →∞ J. Purcell Diagrammatically maximal and geometrically maximal knots

  24. How sharp is the upper bound? Is v oct the best possible? I.e. does ∃ sequence of knots K n with 2 π log det( K n ) lim → v oct ? c ( K n ) n →∞ Answer: Yes. J. Purcell Diagrammatically maximal and geometrically maximal knots

  25. Part 2: Sequences of knots. J. Purcell Diagrammatically maximal and geometrically maximal knots

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