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Flat knot 6.873

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,1,2,2,0,1,1,2,1,0,0,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.873']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 8K1*2K2 - 8K12 - 2K1K2 - 2K1K3 - 2K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.873']
Outer characteristic polynomial of the knot is: t^7+62t^5+50t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.873']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 608K1*4K2 - 2592K14 + 128K1*3K2*3K3 + 224K13K2K3 - 256K1*3K3 + 256K12K2*5 - 1664K1*2K2*4 + 3744K1*2K2*3 - 128K1*2K2*2K3*2 - 9040K1*2K2*2 - 448K1*2K2K4 + 8224K1*2K2 - 192K12K3*2 - 32K1*2K4*2 - 3908K1*2 + 128K1K25K3 - 896K1K2*4K3 - 128K1K2*4K5 + 2400K1K2*3K3 + 256K1K2*2K3K4 - 1120K1K22K3 - 64K1K2*2K5 + 32K1K2K33 - 160K1K2K3K4 + 6360K1K2K3 + 592K1K3K4 + 96K1K4K5 - 128K28 + 256K2*6K4 - 928K26 - 128K2*5K6 - 64K24K3*2 - 64K2*4K4*2 + 1408K2*4K4 - 3712K24 + 96K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 976K22K3*2 - 424K2*2K4*2 + 2048K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1404K2*2 - 32K2K32K4 + 264K2K3K5 + 80K2K4K6 - 1440K32 - 510K4*2 - 60K5*2 - 4K6**2 + 3516
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.873']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17087', 'vk6.17328', 'vk6.20247', 'vk6.21554', 'vk6.23473', 'vk6.23810', 'vk6.27474', 'vk6.29071', 'vk6.35614', 'vk6.36060', 'vk6.38889', 'vk6.41091', 'vk6.42987', 'vk6.43297', 'vk6.45650', 'vk6.47385', 'vk6.55232', 'vk6.55482', 'vk6.57074', 'vk6.58227', 'vk6.59632', 'vk6.59977', 'vk6.61610', 'vk6.62786', 'vk6.65031', 'vk6.65231', 'vk6.66702', 'vk6.67561', 'vk6.68301', 'vk6.68449', 'vk6.69356', 'vk6.70100']
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,-1,0,1,1,2,1,0,1,2,2,0,1,1,2,1,0,0,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.873']

Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 5*K2 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.873']

Outer characteristic polynomial of the knot is: t^7+62t^5+50t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.873']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 608*K1**4*K2 - 2592*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 256*K1**3*K3 + 256*K1**2*K2**5 - 1664*K1**2*K2**4 + 3744*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 9040*K1**2*K2**2 - 448*K1**2*K2*K4 + 8224*K1**2*K2 - 192*K1**2*K3**2 - 32*K1**2*K4**2 - 3908*K1**2 + 128*K1*K2**5*K3 - 896*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2400*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 6360*K1*K2*K3 + 592*K1*K3*K4 + 96*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 928*K2**6 - 128*K2**5*K6 - 64*K2**4*K3**2 - 64*K2**4*K4**2 + 1408*K2**4*K4 - 3712*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 976*K2**2*K3**2 - 424*K2**2*K4**2 + 2048*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1404*K2**2 - 32*K2*K3**2*K4 + 264*K2*K3*K5 + 80*K2*K4*K6 - 1440*K3**2 - 510*K4**2 - 60*K5**2 - 4*K6**2 + 3516

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.873']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17087', 'vk6.17328', 'vk6.20247', 'vk6.21554', 'vk6.23473', 'vk6.23810', 'vk6.27474', 'vk6.29071', 'vk6.35614', 'vk6.36060', 'vk6.38889', 'vk6.41091', 'vk6.42987', 'vk6.43297', 'vk6.45650', 'vk6.47385', 'vk6.55232', 'vk6.55482', 'vk6.57074', 'vk6.58227', 'vk6.59632', 'vk6.59977', 'vk6.61610', 'vk6.62786', 'vk6.65031', 'vk6.65231', 'vk6.66702', 'vk6.67561', 'vk6.68301', 'vk6.68449', 'vk6.69356', 'vk6.70100']

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	O1O2O3O4U1U5O6O5U3U4U2U6
R3 orbit	{'O1O2O3O4U1U5O6O5U3U4U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U3U1U2O6O5U6U4
Gauss code of K*	O1O2O3O4U5U3U1U2O5O6U4U6
Gauss code of -K*	O1O2O3O4U5U1O5O6U3U4U2U6
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 0 -1 1 1 2],[ 3 0 3 1 2 3 3],[ 0 -3 0 -1 1 0 2],[ 1 -1 1 0 1 1 1],[-1 -2 -1 -1 0 -1 0],[-1 -3 0 -1 1 0 2],[-2 -3 -2 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -2 -1 -3],[-1 0 0 -1 -1 -1 -2],[-1 2 1 0 0 -1 -3],[ 0 2 1 0 0 -1 -3],[ 1 1 1 1 1 0 -1],[ 3 3 2 3 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,0,2,2,1,3,1,1,1,2,0,1,3,1,3,1]
Phi over symmetry	[-3,-1,0,1,1,2,1,0,1,2,2,0,1,1,2,1,0,0,-1,-1,1]
Phi of -K	[-3,-1,0,1,1,2,1,0,1,2,2,0,1,1,2,1,0,0,-1,-1,1]
Phi of K*	[-2,-1,-1,0,1,3,-1,1,0,2,2,1,1,1,1,0,1,2,0,0,1]
Phi of -K*	[-3,-1,0,1,1,2,1,3,2,3,3,1,1,1,1,1,0,2,-1,0,2]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	-2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial	t^6+46t^4+13t^2
Outer characteristic polynomial	t^7+62t^5+50t^3+7t
Flat arrow polynomial	-8K14 + 4K1*3 + 8K1*2K2 - 8K12 - 2K1K2 - 2K1K3 - 2K1 + 5*K2 + 6
2-strand cable arrow polynomial	-192K14K2*2 + 608K1*4K2 - 2592K14 + 128K1*3K2*3K3 + 224K13K2K3 - 256K1*3K3 + 256K12K2*5 - 1664K1*2K2*4 + 3744K1*2K2*3 - 128K1*2K2*2K3*2 - 9040K1*2K2*2 - 448K1*2K2K4 + 8224K1*2K2 - 192K12K3*2 - 32K1*2K4*2 - 3908K1*2 + 128K1K25K3 - 896K1K2*4K3 - 128K1K2*4K5 + 2400K1K2*3K3 + 256K1K2*2K3K4 - 1120K1K22K3 - 64K1K2*2K5 + 32K1K2K33 - 160K1K2K3K4 + 6360K1K2K3 + 592K1K3K4 + 96K1K4K5 - 128K28 + 256K2*6K4 - 928K26 - 128K2*5K6 - 64K24K3*2 - 64K2*4K4*2 + 1408K2*4K4 - 3712K24 + 96K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 976K22K3*2 - 424K2*2K4*2 + 2048K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1404K2*2 - 32K2K32K4 + 264K2K3K5 + 80K2K4K6 - 1440K32 - 510K4*2 - 60K5*2 - 4K6**2 + 3516
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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {4}, {2, 3}, {1}]]
If K is slice	False

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