details/L10n4_0_002b1a55.json

FL expansions

Newt(ΔL) vertex Exposing functional Leading x-power Leading q-power Inversion datum Inversion cycle FL
(-1/2, -5/2) (1, 1) x1^(1/2)x2^(5/2) q^(-1/2) 
{"0": [-1, 1, 1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1], "1": [1, -1, -1, -1, -1, -1]}
 
(-q^(-1/2))*x1^(1/2)*x2^(5/2) + (-2*q^(-1/2))*x1^(1/2)*x2^(7/2) + (q^(1/2) - 3*q^(-1/2))*x1^(1/2)*x2^(9/2) + (-2*q^(-5/2))*x1^(3/2)*x2^(9/2) + (2*q^(1/2) - 4*q^(-1/2))*x1^(1/2)*x2^(11/2) + (-4*q^(-5/2) - 2*q^(-7/2))*x1^(3/2)*x2^(11/2) + (-2*q^(5/2) + 3*q^(1/2) - 5*q^(-1/2))*x1^(1/2)*x2^(13/2) + (2*q^(-3/2) - 6*q^(-5/2) - 4*q^(-7/2) - 2*q^(-9/2))*x1^(3/2)*x2^(13/2) + (-q^(-11/2) - 3*q^(-13/2))*x1^(5/2)*x2^(13/2) + (-2*q^(7/2) - 4*q^(5/2) + 4*q^(1/2) - 6*q^(-1/2))*x1^(1/2)*x2^(15/2) + (-q^(1/2) + 4*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 4*q^(-9/2) - 2*q^(-11/2))*x1^(3/2)*x2^(15/2) + (-2*q^(-11/2) - 8*q^(-13/2) - 4*q^(-15/2) - 2*q^(-17/2))*x1^(5/2)*x2^(15/2) + (2*q^(11/2) - 2*q^(9/2) - 4*q^(7/2) - 6*q^(5/2) + 5*q^(1/2) - 7*q^(-1/2))*x1^(1/2)*x2^(17/2) + (-2*q^(3/2) - 6*q^(1/2) - 2*q^(-1/2) + 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 6*q^(-9/2) - 4*q^(-11/2) - 2*q^(-13/2))*x1^(3/2)*x2^(17/2) + (q^(-9/2) - 13*q^(-13/2) - 11*q^(-15/2) - 9*q^(-17/2) - 2*q^(-19/2) - 2*q^(-21/2))*x1^(5/2)*x2^(17/2) + (-2*q^(-21/2) - 2*q^(-23/2) - 4*q^(-25/2))*x1^(7/2)*x2^(17/2) + (2*q^(13/2) + 2*q^(11/2) - 4*q^(9/2) - 6*q^(7/2) - 8*q^(5/2) + 6*q^(1/2) - 8*q^(-1/2))*x1^(1/2)*x2^(19/2) + (q^(7/2) - 2*q^(5/2) - 8*q^(3/2) - 15*q^(1/2) - 8*q^(-1/2) + 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 6*q^(-9/2) - 6*q^(-11/2) - 4*q^(-13/2) - 2*q^(-15/2))*x1^(3/2)*x2^(19/2) + (-2*q^(-5/2) - 2*q^(-7/2) + 2*q^(-9/2) + 4*q^(-11/2) - 14*q^(-13/2) - 16*q^(-15/2) - 20*q^(-17/2) - 10*q^(-19/2) - 6*q^(-21/2) - 2*q^(-23/2) - 2*q^(-25/2))*x1^(5/2)*x2^(19/2) + (-4*q^(-21/2) - 8*q^(-23/2) - 14*q^(-25/2) - 8*q^(-27/2) - 4*q^(-29/2) - 2*q^(-31/2))*x1^(7/2)*x2^(19/2)
(-1/2, -5/2) (1, 1) x1^(1/2)x2^(5/2) q^(-1/2) 
{"0": [1, 1, 1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1], "1": [1, 1, -1, -1, -1, -1]}
 
(-q^(-1/2))*x1^(1/2)*x2^(5/2) + (-2*q^(-1/2))*x1^(1/2)*x2^(7/2) + (q^(1/2) - 3*q^(-1/2))*x1^(1/2)*x2^(9/2) + (-2*q^(-5/2))*x1^(3/2)*x2^(9/2) + (2*q^(1/2) - 4*q^(-1/2))*x1^(1/2)*x2^(11/2) + (-4*q^(-5/2) - 2*q^(-7/2))*x1^(3/2)*x2^(11/2) + (-2*q^(5/2) + 3*q^(1/2) - 5*q^(-1/2))*x1^(1/2)*x2^(13/2) + (2*q^(-3/2) - 6*q^(-5/2) - 4*q^(-7/2) - 2*q^(-9/2))*x1^(3/2)*x2^(13/2) + (-q^(-11/2) - 3*q^(-13/2))*x1^(5/2)*x2^(13/2) + (-2*q^(7/2) - 4*q^(5/2) + 4*q^(1/2) - 6*q^(-1/2))*x1^(1/2)*x2^(15/2) + (-q^(1/2) + 4*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 4*q^(-9/2) - 2*q^(-11/2))*x1^(3/2)*x2^(15/2) + (-2*q^(-11/2) - 8*q^(-13/2) - 4*q^(-15/2) - 2*q^(-17/2))*x1^(5/2)*x2^(15/2) + (2*q^(11/2) - 2*q^(9/2) - 4*q^(7/2) - 6*q^(5/2) + 5*q^(1/2) - 7*q^(-1/2))*x1^(1/2)*x2^(17/2) + (-2*q^(3/2) - 6*q^(1/2) - 2*q^(-1/2) + 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 6*q^(-9/2) - 4*q^(-11/2) - 2*q^(-13/2))*x1^(3/2)*x2^(17/2) + (q^(-9/2) - 13*q^(-13/2) - 11*q^(-15/2) - 9*q^(-17/2) - 2*q^(-19/2) - 2*q^(-21/2))*x1^(5/2)*x2^(17/2) + (-2*q^(-21/2) - 2*q^(-23/2) - 4*q^(-25/2))*x1^(7/2)*x2^(17/2) + (2*q^(13/2) + 2*q^(11/2) - 4*q^(9/2) - 6*q^(7/2) - 8*q^(5/2) + 6*q^(1/2) - 8*q^(-1/2))*x1^(1/2)*x2^(19/2) + (q^(7/2) - 2*q^(5/2) - 8*q^(3/2) - 15*q^(1/2) - 8*q^(-1/2) + 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 6*q^(-9/2) - 6*q^(-11/2) - 4*q^(-13/2) - 2*q^(-15/2))*x1^(3/2)*x2^(19/2) + (-2*q^(-5/2) - 2*q^(-7/2) + 2*q^(-9/2) + 4*q^(-11/2) - 14*q^(-13/2) - 16*q^(-15/2) - 20*q^(-17/2) - 10*q^(-19/2) - 6*q^(-21/2) - 2*q^(-23/2) - 2*q^(-25/2))*x1^(5/2)*x2^(19/2) + (-4*q^(-21/2) - 8*q^(-23/2) - 14*q^(-25/2) - 8*q^(-27/2) - 4*q^(-29/2) - 2*q^(-31/2))*x1^(7/2)*x2^(19/2)
(1/2, 5/2) x1^(-1/2)x2^(-5/2) q^(-1/2) 
(-q^(-1/2))*x1^(-1/2)*x2^(-5/2) + (-2*q^(-1/2))*x1^(-1/2)*x2^(-7/2) + (q^(1/2) - 3*q^(-1/2))*x1^(-1/2)*x2^(-9/2) + (-2*q^(-5/2))*x1^(-3/2)*x2^(-9/2) + (2*q^(1/2) - 4*q^(-1/2))*x1^(-1/2)*x2^(-11/2) + (-4*q^(-5/2) - 2*q^(-7/2))*x1^(-3/2)*x2^(-11/2) + (-2*q^(5/2) + 3*q^(1/2) - 5*q^(-1/2))*x1^(-1/2)*x2^(-13/2) + (2*q^(-3/2) - 6*q^(-5/2) - 4*q^(-7/2) - 2*q^(-9/2))*x1^(-3/2)*x2^(-13/2) + (-q^(-11/2) - 3*q^(-13/2))*x1^(-5/2)*x2^(-13/2) + (-2*q^(7/2) - 4*q^(5/2) + 4*q^(1/2) - 6*q^(-1/2))*x1^(-1/2)*x2^(-15/2) + (-q^(1/2) + 4*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 4*q^(-9/2) - 2*q^(-11/2))*x1^(-3/2)*x2^(-15/2) + (-2*q^(-11/2) - 8*q^(-13/2) - 4*q^(-15/2) - 2*q^(-17/2))*x1^(-5/2)*x2^(-15/2) + (2*q^(11/2) - 2*q^(9/2) - 4*q^(7/2) - 6*q^(5/2) + 5*q^(1/2) - 7*q^(-1/2))*x1^(-1/2)*x2^(-17/2) + (-2*q^(3/2) - 6*q^(1/2) - 2*q^(-1/2) + 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 6*q^(-9/2) - 4*q^(-11/2) - 2*q^(-13/2))*x1^(-3/2)*x2^(-17/2) + (q^(-9/2) - 13*q^(-13/2) - 11*q^(-15/2) - 9*q^(-17/2) - 2*q^(-19/2) - 2*q^(-21/2))*x1^(-5/2)*x2^(-17/2) + (-2*q^(-21/2) - 2*q^(-23/2) - 4*q^(-25/2))*x1^(-7/2)*x2^(-17/2) + (2*q^(13/2) + 2*q^(11/2) - 4*q^(9/2) - 6*q^(7/2) - 8*q^(5/2) + 6*q^(1/2) - 8*q^(-1/2))*x1^(-1/2)*x2^(-19/2) + (q^(7/2) - 2*q^(5/2) - 8*q^(3/2) - 15*q^(1/2) - 8*q^(-1/2) + 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 6*q^(-9/2) - 6*q^(-11/2) - 4*q^(-13/2) - 2*q^(-15/2))*x1^(-3/2)*x2^(-19/2) + (-2*q^(-5/2) - 2*q^(-7/2) + 2*q^(-9/2) + 4*q^(-11/2) - 14*q^(-13/2) - 16*q^(-15/2) - 20*q^(-17/2) - 10*q^(-19/2) - 6*q^(-21/2) - 2*q^(-23/2) - 2*q^(-25/2))*x1^(-5/2)*x2^(-19/2) + (-4*q^(-21/2) - 8*q^(-23/2) - 14*q^(-25/2) - 8*q^(-27/2) - 4*q^(-29/2) - 2*q^(-31/2))*x1^(-7/2)*x2^(-19/2)

Braid

Braid word Components Crossings Strands
[1, 2, 2, 2, -3, 2, -1, -2, -3, -2, -3, -2] 2 12 4

Multivariable Alexander polynomial

ΔL
A1^(1/2)*A2^(5/2) - 2*A1^(1/2)*A2^(3/2) + 2*A1^(1/2)*A2^(1/2) - 2*A1^(1/2)*A2^(-1/2) - 2*A1^(-1/2)*A2^(1/2) + 2*A1^(-1/2)*A2^(-1/2) - 2*A1^(-1/2)*A2^(-3/2) + A1^(-1/2)*A2^(-5/2)

Warning: the A1, A2, … basis used for the HFL and Alexander data might not match the x1, x2, … basis used for FL.

Newton Polygon of Alexander-0.500.5-202A1A2

Dual Thurston polytope

VertexDual to fibered face
(-1, -5) yes
(1, -1) no
(1, 5) yes
(-1, 1) no
Dual Thurston polygon-101-505h1h2

HFL polytope

Poincaré polynomial
m^(-4)*A1^(-1)*A2^(-3) + 3*m^(-3)*A1^(-1)*A2^(-2) + m^(-3)*A2^(-3) + 4*m^(-2)*A1^(-1)*A2^(-1) + 3*m^(-2)*A2^(-2) + 4*m^(-1)*A1^(-1) + 6*m^(-1)*A2^(-1) + 2*A1^(-1)*A2 + 8 + 2*A1*A2^(-1) + 6*m*A2 + 4*m*A1 + 3*m^2*A2^2 + 4*m^2*A1*A2 + m^3*A2^3 + 3*m^3*A1*A2^2 + m^4*A1*A2^3
VertexMaslovRank
(-1, -3)-41
(0, -3)-31
(1, -1)02
(1, 3)41
(0, 3)31
(-1, 1)02
HFL polytope-10105A1A2

Record

FL/HFL Links

orderOption(int) 567
namestring L10n4{0}
statusstring success
slugstring L10n4_0_002b1a55
result_filestring details/L10n4_0_002b1a55.json
complexity_scoreOption(int) 200
strandsOption(int) null
crossingsOption(int) 10
component_countOption(int) 2
girthOption(int) 6
total_rankOption(int) 56
wall_secondsOption(float) 0.002901
backend_secondsOption(float) 0.0
errorstring
dimensionOption(int) 2
support_sizeOption(int) 17
vertex_countOption(int) 6
fibered_vertex_countOption(int) 2
polytope_statusstring computed
alexander_support_sizeOption(int) 8
alexander_vertex_countOption(int) 4
alexander_relationstring not_computed
fk_countint 2
detailsRowRef(FLHFLLinks) 
{
  "dataset": "FLHFLLinks",
  "key": "L10n4_0_002b1a55"
}
detail_assetstring /api/v1/datasets/FLHFLLinks/assets/L10n4_0_002b1a55