L10n25{0} - HFL and FL

details/L10n25_0_b328b96e.json

F_L expansions

Newt(Δ_L) vertex Exposing functional Leading x-power Leading q-power Inversion datum Inversion cycle F_L

Newt(Δ_L) vertex	Exposing functional	Leading x-power	Leading q-power	Inversion datum	Inversion cycle	F_L
`(1/2, -5/2)`	`(1, 1)`	`x1^(-1/2)x2^(5/2)`	`q^(-23/2)`	`{"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, 1, -1, 1, 1, 1, 1]}`		(q^(19/2))x1^(-7/2)x2^(17/2) + (q^(25/2) + q^(23/2) + q^(21/2) + 2q^(19/2))x1^(-7/2)x2^(19/2) + (-q^(11/2))x1^(-5/2)x2^(13/2) + (-q^(15/2) - q^(13/2) - 2q^(11/2))x1^(-5/2)x2^(15/2) + (-q^(21/2) - 3q^(15/2) - 3q^(13/2) - 4q^(11/2) - q^(9/2))x1^(-5/2)x2^(17/2) + (-q^(25/2) - 3q^(23/2) - 2q^(21/2) - q^(17/2) - 6q^(15/2) - 7q^(13/2) - 9q^(11/2) - 4q^(9/2) - 2q^(7/2))x1^(-5/2)x2^(19/2) + (q^(5/2))x1^(-3/2)x2^(9/2) + (q^(7/2) + 2q^(5/2))x1^(-3/2)x2^(11/2) + (q^(11/2) + 2q^(7/2) + 4q^(5/2) + q^(3/2))x1^(-3/2)x2^(13/2) + (2q^(13/2) + 3q^(11/2) + 3q^(7/2) + 7q^(5/2) + 4q^(3/2) + 2q^(1/2))x1^(-3/2)x2^(15/2) + (q^(17/2) + 2q^(15/2) + 5q^(13/2) + 7q^(11/2) + 2q^(9/2) + 4q^(7/2) + 10q^(5/2) + 9q^(3/2) + 8q^(1/2) + 4q^(-1/2) + q^(-3/2))x1^(-3/2)x2^(17/2) + (q^(21/2) + 2q^(19/2) + 5q^(17/2) + 6q^(15/2) + 11q^(13/2) + 15q^(11/2) + 9q^(9/2) + 9q^(7/2) + 13q^(5/2) + 14q^(3/2) + 17q^(1/2) + 15q^(-1/2) + 10q^(-3/2) + 4q^(-5/2) + 2q^(-7/2))x1^(-3/2)x2^(19/2) + (-q^(1/2))x1^(-1/2)x2^(5/2) + (-2q^(1/2))x1^(-1/2)x2^(7/2) + (-q^(3/2) - 3q^(1/2) - q^(-1/2))x1^(-1/2)x2^(9/2) + (-q^(5/2) - 2q^(3/2) - 4q^(1/2) - 2q^(-1/2) - 2q^(-3/2))x1^(-1/2)x2^(11/2) + (-q^(7/2) - 3q^(5/2) - 4q^(3/2) - 6q^(1/2) - 3q^(-1/2) - 4q^(-3/2) - 3q^(-5/2) - q^(-7/2))x1^(-1/2)x2^(13/2) + (-2q^(9/2) - 3q^(7/2) - 6q^(5/2) - 7q^(3/2) - 10q^(1/2) - 6q^(-1/2) - 6q^(-3/2) - 6q^(-5/2) - 6q^(-7/2) - 2q^(-9/2) - 2q^(-11/2))x1^(-1/2)x2^(15/2) + (-q^(13/2) - 2q^(11/2) - 5q^(9/2) - 8q^(7/2) - 11q^(5/2) - 13q^(3/2) - 16q^(1/2) - 13q^(-1/2) - 12q^(-3/2) - 10q^(-5/2) - 11q^(-7/2) - 9q^(-9/2) - 7q^(-11/2) - 4q^(-13/2) - 3q^(-15/2) - q^(-17/2))x1^(-1/2)x2^(17/2) + (-q^(17/2) - q^(15/2) - 5q^(13/2) - 7q^(11/2) - 12q^(9/2) - 16q^(7/2) - 21q^(5/2) - 23q^(3/2) - 28q^(1/2) - 24q^(-1/2) - 25q^(-3/2) - 22q^(-5/2) - 20q^(-7/2) - 18q^(-9/2) - 18q^(-11/2) - 12q^(-13/2) - 12q^(-15/2) - 8q^(-17/2) - 6q^(-19/2) - 2q^(-21/2) - 2q^(-23/2))x1^(-1/2)x2^(19/2)
`(-1/2, 5/2)`		`x1^(1/2)x2^(-5/2)`	`q^(-23/2)`			(q^(19/2))x1^(7/2)x2^(-17/2) + (q^(25/2) + q^(23/2) + q^(21/2) + 2q^(19/2))x1^(7/2)x2^(-19/2) + (-q^(11/2))x1^(5/2)x2^(-13/2) + (-q^(15/2) - q^(13/2) - 2q^(11/2))x1^(5/2)x2^(-15/2) + (-q^(21/2) - 3q^(15/2) - 3q^(13/2) - 4q^(11/2) - q^(9/2))x1^(5/2)x2^(-17/2) + (-q^(25/2) - 3q^(23/2) - 2q^(21/2) - q^(17/2) - 6q^(15/2) - 7q^(13/2) - 9q^(11/2) - 4q^(9/2) - 2q^(7/2))x1^(5/2)x2^(-19/2) + (q^(5/2))x1^(3/2)x2^(-9/2) + (q^(7/2) + 2q^(5/2))x1^(3/2)x2^(-11/2) + (q^(11/2) + 2q^(7/2) + 4q^(5/2) + q^(3/2))x1^(3/2)x2^(-13/2) + (2q^(13/2) + 3q^(11/2) + 3q^(7/2) + 7q^(5/2) + 4q^(3/2) + 2q^(1/2))x1^(3/2)x2^(-15/2) + (q^(17/2) + 2q^(15/2) + 5q^(13/2) + 7q^(11/2) + 2q^(9/2) + 4q^(7/2) + 10q^(5/2) + 9q^(3/2) + 8q^(1/2) + 4q^(-1/2) + q^(-3/2))x1^(3/2)x2^(-17/2) + (q^(21/2) + 2q^(19/2) + 5q^(17/2) + 6q^(15/2) + 11q^(13/2) + 15q^(11/2) + 9q^(9/2) + 9q^(7/2) + 13q^(5/2) + 14q^(3/2) + 17q^(1/2) + 15q^(-1/2) + 10q^(-3/2) + 4q^(-5/2) + 2q^(-7/2))x1^(3/2)x2^(-19/2) + (-q^(1/2))x1^(1/2)x2^(-5/2) + (-2q^(1/2))x1^(1/2)x2^(-7/2) + (-q^(3/2) - 3q^(1/2) - q^(-1/2))x1^(1/2)x2^(-9/2) + (-q^(5/2) - 2q^(3/2) - 4q^(1/2) - 2q^(-1/2) - 2q^(-3/2))x1^(1/2)x2^(-11/2) + (-q^(7/2) - 3q^(5/2) - 4q^(3/2) - 6q^(1/2) - 3q^(-1/2) - 4q^(-3/2) - 3q^(-5/2) - q^(-7/2))x1^(1/2)x2^(-13/2) + (-2q^(9/2) - 3q^(7/2) - 6q^(5/2) - 7q^(3/2) - 10q^(1/2) - 6q^(-1/2) - 6q^(-3/2) - 6q^(-5/2) - 6q^(-7/2) - 2q^(-9/2) - 2q^(-11/2))x1^(1/2)x2^(-15/2) + (-q^(13/2) - 2q^(11/2) - 5q^(9/2) - 8q^(7/2) - 11q^(5/2) - 13q^(3/2) - 16q^(1/2) - 13q^(-1/2) - 12q^(-3/2) - 10q^(-5/2) - 11q^(-7/2) - 9q^(-9/2) - 7q^(-11/2) - 4q^(-13/2) - 3q^(-15/2) - q^(-17/2))x1^(1/2)x2^(-17/2) + (-q^(17/2) - q^(15/2) - 5q^(13/2) - 7q^(11/2) - 12q^(9/2) - 16q^(7/2) - 21q^(5/2) - 23q^(3/2) - 28q^(1/2) - 24q^(-1/2) - 25q^(-3/2) - 22q^(-5/2) - 20q^(-7/2) - 18q^(-9/2) - 18q^(-11/2) - 12q^(-13/2) - 12q^(-15/2) - 8q^(-17/2) - 6q^(-19/2) - 2q^(-21/2) - 2q^(-23/2))x1^(1/2)x2^(-19/2)

(1/2, -5/2)

(1, 1)

x1^(-1/2)x2^(5/2)

q^(-23/2)

{"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, 1, -1, 1, 1, 1, 1]}

(q^(19/2))*x1^(-7/2)*x2^(17/2) + (q^(25/2) + q^(23/2) + q^(21/2) + 2*q^(19/2))*x1^(-7/2)*x2^(19/2) + (-q^(11/2))*x1^(-5/2)*x2^(13/2) + (-q^(15/2) - q^(13/2) - 2*q^(11/2))*x1^(-5/2)*x2^(15/2) + (-q^(21/2) - 3*q^(15/2) - 3*q^(13/2) - 4*q^(11/2) - q^(9/2))*x1^(-5/2)*x2^(17/2) + (-q^(25/2) - 3*q^(23/2) - 2*q^(21/2) - q^(17/2) - 6*q^(15/2) - 7*q^(13/2) - 9*q^(11/2) - 4*q^(9/2) - 2*q^(7/2))*x1^(-5/2)*x2^(19/2) + (q^(5/2))*x1^(-3/2)*x2^(9/2) + (q^(7/2) + 2*q^(5/2))*x1^(-3/2)*x2^(11/2) + (q^(11/2) + 2*q^(7/2) + 4*q^(5/2) + q^(3/2))*x1^(-3/2)*x2^(13/2) + (2*q^(13/2) + 3*q^(11/2) + 3*q^(7/2) + 7*q^(5/2) + 4*q^(3/2) + 2*q^(1/2))*x1^(-3/2)*x2^(15/2) + (q^(17/2) + 2*q^(15/2) + 5*q^(13/2) + 7*q^(11/2) + 2*q^(9/2) + 4*q^(7/2) + 10*q^(5/2) + 9*q^(3/2) + 8*q^(1/2) + 4*q^(-1/2) + q^(-3/2))*x1^(-3/2)*x2^(17/2) + (q^(21/2) + 2*q^(19/2) + 5*q^(17/2) + 6*q^(15/2) + 11*q^(13/2) + 15*q^(11/2) + 9*q^(9/2) + 9*q^(7/2) + 13*q^(5/2) + 14*q^(3/2) + 17*q^(1/2) + 15*q^(-1/2) + 10*q^(-3/2) + 4*q^(-5/2) + 2*q^(-7/2))*x1^(-3/2)*x2^(19/2) + (-q^(1/2))*x1^(-1/2)*x2^(5/2) + (-2*q^(1/2))*x1^(-1/2)*x2^(7/2) + (-q^(3/2) - 3*q^(1/2) - q^(-1/2))*x1^(-1/2)*x2^(9/2) + (-q^(5/2) - 2*q^(3/2) - 4*q^(1/2) - 2*q^(-1/2) - 2*q^(-3/2))*x1^(-1/2)*x2^(11/2) + (-q^(7/2) - 3*q^(5/2) - 4*q^(3/2) - 6*q^(1/2) - 3*q^(-1/2) - 4*q^(-3/2) - 3*q^(-5/2) - q^(-7/2))*x1^(-1/2)*x2^(13/2) + (-2*q^(9/2) - 3*q^(7/2) - 6*q^(5/2) - 7*q^(3/2) - 10*q^(1/2) - 6*q^(-1/2) - 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 2*q^(-9/2) - 2*q^(-11/2))*x1^(-1/2)*x2^(15/2) + (-q^(13/2) - 2*q^(11/2) - 5*q^(9/2) - 8*q^(7/2) - 11*q^(5/2) - 13*q^(3/2) - 16*q^(1/2) - 13*q^(-1/2) - 12*q^(-3/2) - 10*q^(-5/2) - 11*q^(-7/2) - 9*q^(-9/2) - 7*q^(-11/2) - 4*q^(-13/2) - 3*q^(-15/2) - q^(-17/2))*x1^(-1/2)*x2^(17/2) + (-q^(17/2) - q^(15/2) - 5*q^(13/2) - 7*q^(11/2) - 12*q^(9/2) - 16*q^(7/2) - 21*q^(5/2) - 23*q^(3/2) - 28*q^(1/2) - 24*q^(-1/2) - 25*q^(-3/2) - 22*q^(-5/2) - 20*q^(-7/2) - 18*q^(-9/2) - 18*q^(-11/2) - 12*q^(-13/2) - 12*q^(-15/2) - 8*q^(-17/2) - 6*q^(-19/2) - 2*q^(-21/2) - 2*q^(-23/2))*x1^(-1/2)*x2^(19/2)

(-1/2, 5/2)

x1^(1/2)x2^(-5/2)

q^(-23/2)

(q^(19/2))*x1^(7/2)*x2^(-17/2) + (q^(25/2) + q^(23/2) + q^(21/2) + 2*q^(19/2))*x1^(7/2)*x2^(-19/2) + (-q^(11/2))*x1^(5/2)*x2^(-13/2) + (-q^(15/2) - q^(13/2) - 2*q^(11/2))*x1^(5/2)*x2^(-15/2) + (-q^(21/2) - 3*q^(15/2) - 3*q^(13/2) - 4*q^(11/2) - q^(9/2))*x1^(5/2)*x2^(-17/2) + (-q^(25/2) - 3*q^(23/2) - 2*q^(21/2) - q^(17/2) - 6*q^(15/2) - 7*q^(13/2) - 9*q^(11/2) - 4*q^(9/2) - 2*q^(7/2))*x1^(5/2)*x2^(-19/2) + (q^(5/2))*x1^(3/2)*x2^(-9/2) + (q^(7/2) + 2*q^(5/2))*x1^(3/2)*x2^(-11/2) + (q^(11/2) + 2*q^(7/2) + 4*q^(5/2) + q^(3/2))*x1^(3/2)*x2^(-13/2) + (2*q^(13/2) + 3*q^(11/2) + 3*q^(7/2) + 7*q^(5/2) + 4*q^(3/2) + 2*q^(1/2))*x1^(3/2)*x2^(-15/2) + (q^(17/2) + 2*q^(15/2) + 5*q^(13/2) + 7*q^(11/2) + 2*q^(9/2) + 4*q^(7/2) + 10*q^(5/2) + 9*q^(3/2) + 8*q^(1/2) + 4*q^(-1/2) + q^(-3/2))*x1^(3/2)*x2^(-17/2) + (q^(21/2) + 2*q^(19/2) + 5*q^(17/2) + 6*q^(15/2) + 11*q^(13/2) + 15*q^(11/2) + 9*q^(9/2) + 9*q^(7/2) + 13*q^(5/2) + 14*q^(3/2) + 17*q^(1/2) + 15*q^(-1/2) + 10*q^(-3/2) + 4*q^(-5/2) + 2*q^(-7/2))*x1^(3/2)*x2^(-19/2) + (-q^(1/2))*x1^(1/2)*x2^(-5/2) + (-2*q^(1/2))*x1^(1/2)*x2^(-7/2) + (-q^(3/2) - 3*q^(1/2) - q^(-1/2))*x1^(1/2)*x2^(-9/2) + (-q^(5/2) - 2*q^(3/2) - 4*q^(1/2) - 2*q^(-1/2) - 2*q^(-3/2))*x1^(1/2)*x2^(-11/2) + (-q^(7/2) - 3*q^(5/2) - 4*q^(3/2) - 6*q^(1/2) - 3*q^(-1/2) - 4*q^(-3/2) - 3*q^(-5/2) - q^(-7/2))*x1^(1/2)*x2^(-13/2) + (-2*q^(9/2) - 3*q^(7/2) - 6*q^(5/2) - 7*q^(3/2) - 10*q^(1/2) - 6*q^(-1/2) - 6*q^(-3/2) - 6*q^(-5/2) - 6*q^(-7/2) - 2*q^(-9/2) - 2*q^(-11/2))*x1^(1/2)*x2^(-15/2) + (-q^(13/2) - 2*q^(11/2) - 5*q^(9/2) - 8*q^(7/2) - 11*q^(5/2) - 13*q^(3/2) - 16*q^(1/2) - 13*q^(-1/2) - 12*q^(-3/2) - 10*q^(-5/2) - 11*q^(-7/2) - 9*q^(-9/2) - 7*q^(-11/2) - 4*q^(-13/2) - 3*q^(-15/2) - q^(-17/2))*x1^(1/2)*x2^(-17/2) + (-q^(17/2) - q^(15/2) - 5*q^(13/2) - 7*q^(11/2) - 12*q^(9/2) - 16*q^(7/2) - 21*q^(5/2) - 23*q^(3/2) - 28*q^(1/2) - 24*q^(-1/2) - 25*q^(-3/2) - 22*q^(-5/2) - 20*q^(-7/2) - 18*q^(-9/2) - 18*q^(-11/2) - 12*q^(-13/2) - 12*q^(-15/2) - 8*q^(-17/2) - 6*q^(-19/2) - 2*q^(-21/2) - 2*q^(-23/2))*x1^(1/2)*x2^(-19/2)

Braid

Braid word	Components	Crossings	Strands
`[-2, -3, 4, -3, 2, -3, -4, -1, -1, 2, -3, 2, 3, 3, 1]`	2	15	5

Multivariable Alexander polynomial

Δ_L
`A1^(1/2)A2^(1/2) - A1^(1/2)A2^(-1/2) - 2A1^(1/2)A2^(-3/2) + A1^(1/2)A2^(-5/2) + A1^(-1/2)A2^(5/2) - 2A1^(-1/2)A2^(3/2) - A1^(-1/2)A2^(1/2) + A1^(-1/2)A2^(-1/2)`

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

Dual Thurston polytope

Vertex	Dual to fibered face
`(-1, -1)`	yes
`(1, -5)`	yes
`(1, 1)`	yes
`(-1, 5)`	yes

HFL polytope

Poincaré polynomial
`m^(-4)A2^(-3) + 3m^(-3)A2^(-2) + m^(-3)A1A2^(-3) + m^(-2)A1^(-1)A2^(-1) + 2m^(-2)A2^(-1) + 3m^(-2)A1A2^(-2) + 2m^(-1)A1^(-1) + 2m^(-1)A1^(-1)A2 + 2m^(-1)A2^(-1) + m^(-1) + 2m^(-1)A1A2^(-1) + A1^(-1)A2 + 3A1^(-1)A2^2 + 5 + 2A2 + A1A2^(-1) + mA1^(-1)A2^3 + 2mA2 + 3mA2^2 + 2mA1 + m^2A2^3 + m^2A1A2`

Vertex	Maslov	Rank
`(-1, -1)`	-2	1
`(0, -3)`	-4	1
`(1, -3)`	-3	1
`(1, 1)`	2	1
`(0, 3)`	2	1
`(-1, 3)`	1	1

Record

FL/HFL Links

`orderOption(int)`	`513`
`namestring`	`L10n25{0}`
`statusstring`	`success`
`slugstring`	`L10n25_0_b328b96e`
`result_filestring`	`details/L10n25_0_b328b96e.json`
`complexity_scoreOption(int)`	`200`
`strandsOption(int)`	`null`
`crossingsOption(int)`	`10`
`component_countOption(int)`	`2`
`girthOption(int)`	`6`
`total_rankOption(int)`	`42`
`wall_secondsOption(float)`	`0.003216`
`backend_secondsOption(float)`	`0.0`
`errorstring`
`dimensionOption(int)`	`2`
`support_sizeOption(int)`	`17`
`vertex_countOption(int)`	`6`
`fibered_vertex_countOption(int)`	`4`
`polytope_statusstring`	`computed`
`alexander_support_sizeOption(int)`	`8`
`alexander_vertex_countOption(int)`	`4`
`alexander_relationstring`	`not_computed`
`fk_countint`	`1`
`detailsRowRef(FLHFLLinks)`	{ "dataset": "FLHFLLinks", "key": "L10n25_0_b328b96e" }
`detail_assetstring`	`/api/v1/datasets/FLHFLLinks/assets/L10n25_0_b328b96e`