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
Part 1: Knots from a combinatorial point of view J. Purcell Diagrammatically maximal and geometrically maximal knots
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
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
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
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
Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots
Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots
Reduced diagrams and Tait graphs K reduced ⇔ Γ K has no loops, no bridges. J. Purcell Diagrammatically maximal and geometrically maximal knots
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
Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots
Example: Twist knot J. Purcell Diagrammatically maximal and geometrically maximal knots
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
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
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
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
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
Part 1.5: Geometric Interlude J. Purcell Diagrammatically maximal and geometrically maximal knots
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
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
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
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
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
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
Part 2: Sequences of knots. J. Purcell Diagrammatically maximal and geometrically maximal knots
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