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Flat knot 6.961

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,1,3,-1,0,1,1,1,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.961']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 12K12 - 6K1K2 - 4K1K3 - 3K1 + 6*K2 + K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.961', '6.1154']
Outer characteristic polynomial of the knot is: t^7+36t^5+77t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.961']
2-strand cable arrow polynomial of the knot is: -448K16 - 384K1*4K2*2 + 1408K1*4K2 - 4096K14 + 1056K1*3K2K3 + 32K1*3K3K4 - 1088K1*3K3 - 256K12K2*4 + 800K1*2K2*3 - 128K1*2K2*2K3*2 + 96K1*2K2*2K4 - 7600K12K2*2 + 192K1*2K2K32 - 1152K1*2K2K4 + 11816K1*2K2 - 1376K12K3*2 - 160K1*2K3K5 - 112K1*2K4*2 - 7164K1*2 + 1440K1K23K3 + 320K1K2*2K3K4 - 1568K1K22K3 + 32K1K2*2K4K5 - 704K1K22K5 + 128K1K2K33 - 512K1K2K3K4 - 224K1K2K3K6 + 10904K1K2K3 - 32K1K2K4K5 - 32K1K2K5K6 + 2320K1K3K4 + 344K1K4K5 + 40K1K5K6 + 8K1K6K7 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1824K24 + 128K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 1600K22K3*2 - 32K2*2K3K7 - 264K2*2K4*2 + 2400K2*2K4 - 80K22K5*2 - 48K2*2K6*2 - 5778K2*2 - 64K2K32K4 + 1296K2K3K5 + 256K2K4K6 + 40K2K5K7 + 16K2K6K8 - 32K34 + 40K3*2K6 - 3288K32 + 8K3K4K7 - 1032K42 - 244K5*2 - 54K6*2 - 8K7*2 - 2K8**2 + 6296
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.961']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4216', 'vk6.4296', 'vk6.5476', 'vk6.5587', 'vk6.7577', 'vk6.7668', 'vk6.9078', 'vk6.9158', 'vk6.11173', 'vk6.12261', 'vk6.12368', 'vk6.19374', 'vk6.19669', 'vk6.19786', 'vk6.26154', 'vk6.26225', 'vk6.26572', 'vk6.26668', 'vk6.30763', 'vk6.31968', 'vk6.38154', 'vk6.38201', 'vk6.44811', 'vk6.44946', 'vk6.48538', 'vk6.49234', 'vk6.49345', 'vk6.50319', 'vk6.52747', 'vk6.63579', 'vk6.66310', 'vk6.66341']
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: [-3,0,0,1,1,1,1,2,1,1,3,-1,0,1,1,1,1,1,1,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.961']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 6*K2 + K3 + K4 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.961', '6.1154']

Outer characteristic polynomial of the knot is: t^7+36t^5+77t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.961']

2-strand cable arrow polynomial of the knot is: -448*K1**6 - 384*K1**4*K2**2 + 1408*K1**4*K2 - 4096*K1**4 + 1056*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1088*K1**3*K3 - 256*K1**2*K2**4 + 800*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 96*K1**2*K2**2*K4 - 7600*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 1152*K1**2*K2*K4 + 11816*K1**2*K2 - 1376*K1**2*K3**2 - 160*K1**2*K3*K5 - 112*K1**2*K4**2 - 7164*K1**2 + 1440*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 + 128*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 10904*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 2320*K1*K3*K4 + 344*K1*K4*K5 + 40*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1824*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1600*K2**2*K3**2 - 32*K2**2*K3*K7 - 264*K2**2*K4**2 + 2400*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5778*K2**2 - 64*K2*K3**2*K4 + 1296*K2*K3*K5 + 256*K2*K4*K6 + 40*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 40*K3**2*K6 - 3288*K3**2 + 8*K3*K4*K7 - 1032*K4**2 - 244*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 6296

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.961']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4216', 'vk6.4296', 'vk6.5476', 'vk6.5587', 'vk6.7577', 'vk6.7668', 'vk6.9078', 'vk6.9158', 'vk6.11173', 'vk6.12261', 'vk6.12368', 'vk6.19374', 'vk6.19669', 'vk6.19786', 'vk6.26154', 'vk6.26225', 'vk6.26572', 'vk6.26668', 'vk6.30763', 'vk6.31968', 'vk6.38154', 'vk6.38201', 'vk6.44811', 'vk6.44946', 'vk6.48538', 'vk6.49234', 'vk6.49345', 'vk6.50319', 'vk6.52747', 'vk6.63579', 'vk6.66310', 'vk6.66341']

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	O1O2O3O4U5U2O6O5U6U3U1U4
R3 orbit	{'O1O2O3O4U5U2O6O5U6U3U1U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U4U2U5O6O5U3U6
Gauss code of K*	O1O2O3O4U3U5U2U4O6O5U1U6
Gauss code of -K*	O1O2O3O4U5U4O6O5U1U3U6U2
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 -1 0 3 0 -1],[ 1 0 0 1 3 1 -1],[ 1 0 0 0 1 1 -1],[ 0 -1 0 0 1 1 -1],[-3 -3 -1 -1 0 -2 -1],[ 0 -1 -1 -1 2 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix	[[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -2 -1 -1 -3],[ 0 1 0 1 0 -1 -1],[ 0 2 -1 0 -1 -1 -1],[ 1 1 0 1 0 -1 0],[ 1 1 1 1 1 0 1],[ 1 3 1 1 0 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,0,0,1,1,1,1,2,1,1,3,-1,0,1,1,1,1,1,1,0,-1]
Phi over symmetry	[-3,0,0,1,1,1,1,2,1,1,3,-1,0,1,1,1,1,1,1,0,-1]
Phi of -K	[-1,-1,-1,0,0,3,-1,-1,0,0,3,0,0,0,1,0,1,3,1,1,2]
Phi of K*	[-3,0,0,1,1,1,1,2,1,3,3,-1,0,0,0,0,0,1,-1,0,1]
Phi of -K*	[-1,-1,-1,0,0,3,-1,0,0,1,1,1,1,1,1,1,1,3,1,1,2]
Symmetry type of based matrix	c
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	3z^2+24z+37
Enhanced Jones-Krushkal polynomial	3w^3z^2+24w^2z+37w
Inner characteristic polynomial	t^6+24t^4+37t^2+4
Outer characteristic polynomial	t^7+36t^5+77t^3+11t
Flat arrow polynomial	8K13 + 4K1*2K2 - 12K12 - 6K1K2 - 4K1K3 - 3K1 + 6*K2 + K3 + K4 + 6
2-strand cable arrow polynomial	-448K16 - 384K1*4K2*2 + 1408K1*4K2 - 4096K14 + 1056K1*3K2K3 + 32K1*3K3K4 - 1088K1*3K3 - 256K12K2*4 + 800K1*2K2*3 - 128K1*2K2*2K3*2 + 96K1*2K2*2K4 - 7600K12K2*2 + 192K1*2K2K32 - 1152K1*2K2K4 + 11816K1*2K2 - 1376K12K3*2 - 160K1*2K3K5 - 112K1*2K4*2 - 7164K1*2 + 1440K1K23K3 + 320K1K2*2K3K4 - 1568K1K22K3 + 32K1K2*2K4K5 - 704K1K22K5 + 128K1K2K33 - 512K1K2K3K4 - 224K1K2K3K6 + 10904K1K2K3 - 32K1K2K4K5 - 32K1K2K5K6 + 2320K1K3K4 + 344K1K4K5 + 40K1K5K6 + 8K1K6K7 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1824K24 + 128K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 1600K22K3*2 - 32K2*2K3K7 - 264K2*2K4*2 + 2400K2*2K4 - 80K22K5*2 - 48K2*2K6*2 - 5778K2*2 - 64K2K32K4 + 1296K2K3K5 + 256K2K4K6 + 40K2K5K7 + 16K2K6K8 - 32K34 + 40K3*2K6 - 3288K32 + 8K3K4K7 - 1032K42 - 244K5*2 - 54K6*2 - 8K7*2 - 2K8**2 + 6296
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}, {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}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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