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Flat knot 6.349

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,1,3,3,1,0,1,2,0,0,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.349']
Arrow polynomial of the knot is: -2K12 - 2K1*K2 + K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314']
Outer characteristic polynomial of the knot is: t^7+81t^5+85t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.349']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 224K1*4K2 - 272K14 + 384K1*3K2K3 - 224K1*3K3 - 512K12K2*4 - 384K1*2K2*3K4 + 1792K12K2*3 - 5216K1*2K2*2 - 864K1*2K2K4 + 4528K1*2K2 - 144K12K3*2 - 3204K1*2 + 1184K1K23K3 + 608K1K2*2K3K4 - 1312K1K22K3 - 128K1K2K3K4 - 32K1K2K3K6 + 4840K1K2K3 + 728K1K3K4 + 8K1K4K5 - 128K2*6 + 416K2*4K4 - 2216K24 - 688K2*2K3*2 - 488K2*2K4*2 + 2040K2*2K4 - 1574K22 + 136K2K3K5 + 72K2K4K6 - 1192K3*2 - 562K4*2 - 4K5*2 - 2K6**2 + 2440
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.349']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11425', 'vk6.11720', 'vk6.12735', 'vk6.13080', 'vk6.20612', 'vk6.22029', 'vk6.28085', 'vk6.29533', 'vk6.31170', 'vk6.31511', 'vk6.32332', 'vk6.32754', 'vk6.39492', 'vk6.41702', 'vk6.46088', 'vk6.47744', 'vk6.52181', 'vk6.52431', 'vk6.53005', 'vk6.53323', 'vk6.57484', 'vk6.58651', 'vk6.62162', 'vk6.63117', 'vk6.63750', 'vk6.63858', 'vk6.64172', 'vk6.64362', 'vk6.67008', 'vk6.67876', 'vk6.69629', 'vk6.70316']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,1,3,3,1,0,1,2,0,0,1,-1,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.349']

Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314']

Outer characteristic polynomial of the knot is: t^7+81t^5+85t^3+12t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.349']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 224*K1**4*K2 - 272*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 - 512*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 1792*K1**2*K2**3 - 5216*K1**2*K2**2 - 864*K1**2*K2*K4 + 4528*K1**2*K2 - 144*K1**2*K3**2 - 3204*K1**2 + 1184*K1*K2**3*K3 + 608*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 128*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4840*K1*K2*K3 + 728*K1*K3*K4 + 8*K1*K4*K5 - 128*K2**6 + 416*K2**4*K4 - 2216*K2**4 - 688*K2**2*K3**2 - 488*K2**2*K4**2 + 2040*K2**2*K4 - 1574*K2**2 + 136*K2*K3*K5 + 72*K2*K4*K6 - 1192*K3**2 - 562*K4**2 - 4*K5**2 - 2*K6**2 + 2440

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.349']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11425', 'vk6.11720', 'vk6.12735', 'vk6.13080', 'vk6.20612', 'vk6.22029', 'vk6.28085', 'vk6.29533', 'vk6.31170', 'vk6.31511', 'vk6.32332', 'vk6.32754', 'vk6.39492', 'vk6.41702', 'vk6.46088', 'vk6.47744', 'vk6.52181', 'vk6.52431', 'vk6.53005', 'vk6.53323', 'vk6.57484', 'vk6.58651', 'vk6.62162', 'vk6.63117', 'vk6.63750', 'vk6.63858', 'vk6.64172', 'vk6.64362', 'vk6.67008', 'vk6.67876', 'vk6.69629', 'vk6.70316']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is c.

The reverse -K is

The mirror image K* is

The reversed mirror image -K* is

The fillings (up to the first 10) associated to the algebraic genus:

Or click here to check the fillings

invariant	value
Gauss code	O1O2O3O4O5U3U2O6U4U5U1U6
R3 orbit	{'O1O2O3O4O5U3U2O6U4U5U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U5U1U2O6U4U3
Gauss code of K*	O1O2O3O4U3U5U6U1U2O6O5U4
Gauss code of -K*	O1O2O3O4U1O5O6U3U4U6U5U2
Diagrammatic symmetry type	c
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 -2 -2 0 2 3],[ 1 0 -2 -2 1 3 3],[ 2 2 0 0 2 3 2],[ 2 2 0 0 1 2 2],[ 0 -1 -2 -1 0 1 2],[-2 -3 -3 -2 -1 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix	[[ 0 3 2 0 -1 -2 -2],[-3 0 -1 -2 -3 -2 -2],[-2 1 0 -1 -3 -2 -3],[ 0 2 1 0 -1 -1 -2],[ 1 3 3 1 0 -2 -2],[ 2 2 2 1 2 0 0],[ 2 2 3 2 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,0,1,2,2,1,2,3,2,2,1,3,2,3,1,1,2,2,2,0]
Phi over symmetry	[-3,-2,0,1,2,2,0,1,1,3,3,1,0,1,2,0,0,1,-1,-1,0]
Phi of -K	[-2,-2,-1,0,2,3,0,-1,0,1,3,-1,1,2,3,0,0,1,1,1,0]
Phi of K*	[-3,-2,0,1,2,2,0,1,1,3,3,1,0,1,2,0,0,1,-1,-1,0]
Phi of -K*	[-2,-2,-1,0,2,3,0,2,1,2,2,2,2,3,2,1,3,3,1,2,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	5z^2+18z+17
Enhanced Jones-Krushkal polynomial	-4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w
Inner characteristic polynomial	t^6+59t^4+42t^2+1
Outer characteristic polynomial	t^7+81t^5+85t^3+12t
Flat arrow polynomial	-2K12 - 2K1*K2 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial	-256K14K2*2 + 224K1*4K2 - 272K14 + 384K1*3K2K3 - 224K1*3K3 - 512K12K2*4 - 384K1*2K2*3K4 + 1792K12K2*3 - 5216K1*2K2*2 - 864K1*2K2K4 + 4528K1*2K2 - 144K12K3*2 - 3204K1*2 + 1184K1K23K3 + 608K1K2*2K3K4 - 1312K1K22K3 - 128K1K2K3K4 - 32K1K2K3K6 + 4840K1K2K3 + 728K1K3K4 + 8K1K4K5 - 128K2*6 + 416K2*4K4 - 2216K24 - 688K2*2K3*2 - 488K2*2K4*2 + 2040K2*2K4 - 1574K22 + 136K2K3K5 + 72K2K4K6 - 1192K3*2 - 562K4*2 - 4K5*2 - 2K6**2 + 2440
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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