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Flat knot 6.416

Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,0,1,1,2,3,2,1,3,4,1,1,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.416']
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+96t^5+139t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.416']
2-strand cable arrow polynomial of the knot is: -144K14 + 384K1*3K2K3 - 352K1*3K3 - 256K12K2*2K3*2 - 896K1*2K2*2 + 256K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 2024K1*2K2 - 432K12K3*2 - 1972K1*2 + 288K1K23K3 + 224K1K2*2K3K4 - 576K1K22K3 - 32K1K2*2K5 - 128K1K2K3K4 + 2296K1K2K3 + 512K1K3K4 - 72K24 - 400K2*2K3*2 - 72K2*2K4*2 + 280K2*2K4 - 1318K22 + 144K2K3K5 + 8K2K4K6 - 776K3*2 - 170K4*2 - 4K5*2 - 2K6**2 + 1344
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.416']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16921', 'vk6.17163', 'vk6.19963', 'vk6.20229', 'vk6.21128', 'vk6.21524', 'vk6.23309', 'vk6.26824', 'vk6.26966', 'vk6.27437', 'vk6.28606', 'vk6.29047', 'vk6.35343', 'vk6.38258', 'vk6.38376', 'vk6.38857', 'vk6.40378', 'vk6.41046', 'vk6.42832', 'vk6.45123', 'vk6.45250', 'vk6.45610', 'vk6.46983', 'vk6.47367', 'vk6.55081', 'vk6.56689', 'vk6.56765', 'vk6.57768', 'vk6.58188', 'vk6.59474', 'vk6.61086', 'vk6.61241', 'vk6.62347', 'vk6.62760', 'vk6.64921', 'vk6.66471', 'vk6.67526', 'vk6.68220', 'vk6.69123', 'vk6.69831']
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,-3,0,1,2,3,0,1,1,2,3,2,1,3,4,1,1,2,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.416']

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+96t^5+139t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.416']

2-strand cable arrow polynomial of the knot is: -144*K1**4 + 384*K1**3*K2*K3 - 352*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 896*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 2024*K1**2*K2 - 432*K1**2*K3**2 - 1972*K1**2 + 288*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2296*K1*K2*K3 + 512*K1*K3*K4 - 72*K2**4 - 400*K2**2*K3**2 - 72*K2**2*K4**2 + 280*K2**2*K4 - 1318*K2**2 + 144*K2*K3*K5 + 8*K2*K4*K6 - 776*K3**2 - 170*K4**2 - 4*K5**2 - 2*K6**2 + 1344

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.416']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16921', 'vk6.17163', 'vk6.19963', 'vk6.20229', 'vk6.21128', 'vk6.21524', 'vk6.23309', 'vk6.26824', 'vk6.26966', 'vk6.27437', 'vk6.28606', 'vk6.29047', 'vk6.35343', 'vk6.38258', 'vk6.38376', 'vk6.38857', 'vk6.40378', 'vk6.41046', 'vk6.42832', 'vk6.45123', 'vk6.45250', 'vk6.45610', 'vk6.46983', 'vk6.47367', 'vk6.55081', 'vk6.56689', 'vk6.56765', 'vk6.57768', 'vk6.58188', 'vk6.59474', 'vk6.61086', 'vk6.61241', 'vk6.62347', 'vk6.62760', 'vk6.64921', 'vk6.66471', 'vk6.67526', 'vk6.68220', 'vk6.69123', 'vk6.69831']

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	O1O2O3O4O5U6U1O6U2U3U5U4
R3 orbit	{'O1O2O3O4O5U6U1O6U2U3U5U4', 'O1O2O3O4O5U2U6U1O6U3U5U4'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U2U1U3U4O6U5U6
Gauss code of K*	O1O2O3O4U5U1U2U4U3O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U2U1U3U4U6
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 -3 -2 0 3 3 -1],[ 3 0 0 1 3 2 3],[ 2 0 0 1 3 2 2],[ 0 -1 -1 0 2 1 0],[-3 -3 -3 -2 0 0 -3],[-3 -2 -2 -1 0 0 -3],[ 1 -3 -2 0 3 3 0]]
Primitive based matrix	[[ 0 3 3 0 -1 -2 -3],[-3 0 0 -1 -3 -2 -2],[-3 0 0 -2 -3 -3 -3],[ 0 1 2 0 0 -1 -1],[ 1 3 3 0 0 -2 -3],[ 2 2 3 1 2 0 0],[ 3 2 3 1 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,0,1,2,3,0,1,3,2,2,2,3,3,3,0,1,1,2,3,0]
Phi over symmetry	[-3,-3,0,1,2,3,0,1,1,2,3,2,1,3,4,1,1,2,-1,-1,1]
Phi of -K	[-3,-2,-1,0,3,3,1,-1,2,3,4,-1,1,2,3,1,1,1,1,2,0]
Phi of K*	[-3,-3,0,1,2,3,0,1,1,2,3,2,1,3,4,1,1,2,-1,-1,1]
Phi of -K*	[-3,-2,-1,0,3,3,0,3,1,2,3,2,1,2,3,0,3,3,1,2,0]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2+17w^2z+19w
Inner characteristic polynomial	t^6+64t^4+54t^2+1
Outer characteristic polynomial	t^7+96t^5+139t^3+4t
Flat arrow polynomial	-2K12 - 2K1*K2 + K1 + K2 + K3 + 2
2-strand cable arrow polynomial	-144K14 + 384K1*3K2K3 - 352K1*3K3 - 256K12K2*2K3*2 - 896K1*2K2*2 + 256K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 2024K1*2K2 - 432K12K3*2 - 1972K1*2 + 288K1K23K3 + 224K1K2*2K3K4 - 576K1K22K3 - 32K1K2*2K5 - 128K1K2K3K4 + 2296K1K2K3 + 512K1K3K4 - 72K24 - 400K2*2K3*2 - 72K2*2K4*2 + 280K2*2K4 - 1318K22 + 144K2K3K5 + 8K2K4K6 - 776K3*2 - 170K4*2 - 4K5*2 - 2K6**2 + 1344
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}], [{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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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