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Flat knot 6.158

Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,1,1,2,4,3,1,2,3,3,0,-1,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.158']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K1K3 - 3K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.143', '6.158', '6.264', '6.282', '6.501']
Outer characteristic polynomial of the knot is: t^7+92t^5+67t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.158']
2-strand cable arrow polynomial of the knot is: 512K14K2*3 - 1152K1*4K2*2 + 1280K1*4K2 - 1664K14 - 384K1*3K2*2K3 + 768K13K2K3 + 32K1*3K3K4 - 480K1*3K3 + 640K12K2*5 - 2240K1*2K2*4 + 256K1*2K2*3K3*2 + 4512K1*2K2*3 - 384K1*2K2*2K3*2 + 64K1*2K2*2K4 - 7632K12K2*2 + 192K1*2K2K32 + 32K1*2K2K3K5 - 832K12K2K4 + 6360K1*2K2 - 352K12K3*2 - 32K1*2K4*2 - 3980K1*2 - 1024K1K24K3 - 128K1K2*3K3K4 + 2752K1K23K3 + 416K1K2*2K3K4 - 1056K1K22K3 - 128K1K2*2K5 - 128K1K2K3K4 + 5120K1K2K3 + 752K1K3K4 + 32K1K4K5 - 576K2*6 - 320K2*4K3*2 - 32K2*4K4*2 + 672K2*4K4 - 2224K24 + 224K2*3K3K5 + 32K2*3K4K6 - 976K2*2K3*2 - 216K2*2K4*2 + 1024K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 1330K2*2 - 32K2K32K4 + 272K2K3K5 + 48K2K4K6 + 8K2K5K7 - 1204K3*2 - 378K4*2 - 48K5*2 - 6K6**2 + 2992
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.158']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16958', 'vk6.16967', 'vk6.17199', 'vk6.17209', 'vk6.20847', 'vk6.20852', 'vk6.22248', 'vk6.22255', 'vk6.23356', 'vk6.23650', 'vk6.23665', 'vk6.28308', 'vk6.35405', 'vk6.35824', 'vk6.35848', 'vk6.39918', 'vk6.39933', 'vk6.42014', 'vk6.43155', 'vk6.43165', 'vk6.46468', 'vk6.46479', 'vk6.55118', 'vk6.55121', 'vk6.55379', 'vk6.57658', 'vk6.57663', 'vk6.58842', 'vk6.59824', 'vk6.59830', 'vk6.68396', 'vk6.69714']
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: [-4,-2,1,1,1,3,1,1,2,4,3,1,2,3,3,0,-1,0,0,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.158']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.143', '6.158', '6.264', '6.282', '6.501']

Outer characteristic polynomial of the knot is: t^7+92t^5+67t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.158']

2-strand cable arrow polynomial of the knot is: 512*K1**4*K2**3 - 1152*K1**4*K2**2 + 1280*K1**4*K2 - 1664*K1**4 - 384*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 32*K1**3*K3*K4 - 480*K1**3*K3 + 640*K1**2*K2**5 - 2240*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 4512*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7632*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 832*K1**2*K2*K4 + 6360*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 3980*K1**2 - 1024*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 2752*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5120*K1*K2*K3 + 752*K1*K3*K4 + 32*K1*K4*K5 - 576*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 672*K2**4*K4 - 2224*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 976*K2**2*K3**2 - 216*K2**2*K4**2 + 1024*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1330*K2**2 - 32*K2*K3**2*K4 + 272*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 1204*K3**2 - 378*K4**2 - 48*K5**2 - 6*K6**2 + 2992

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.158']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16958', 'vk6.16967', 'vk6.17199', 'vk6.17209', 'vk6.20847', 'vk6.20852', 'vk6.22248', 'vk6.22255', 'vk6.23356', 'vk6.23650', 'vk6.23665', 'vk6.28308', 'vk6.35405', 'vk6.35824', 'vk6.35848', 'vk6.39918', 'vk6.39933', 'vk6.42014', 'vk6.43155', 'vk6.43165', 'vk6.46468', 'vk6.46479', 'vk6.55118', 'vk6.55121', 'vk6.55379', 'vk6.57658', 'vk6.57663', 'vk6.58842', 'vk6.59824', 'vk6.59830', 'vk6.68396', 'vk6.69714']

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	O1O2O3O4O5U1O6U3U6U4U5U2
R3 orbit	{'O1O2O3O4O5U1O6U3U6U4U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U1U2U6U3O6U5
Gauss code of K*	O1O2O3O4O5U6U5U1U3U4O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U2U3U5U1U6
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 -4 1 -2 1 3 1],[ 4 0 4 1 2 3 1],[-1 -4 0 -3 0 2 1],[ 2 -1 3 0 2 3 1],[-1 -2 0 -2 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 3 1 1 1 -2 -4],[-3 0 0 -1 -2 -3 -3],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 0 -2 -2],[-1 2 1 0 0 -3 -4],[ 2 3 1 2 3 0 -1],[ 4 3 1 2 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,-1,2,4,0,1,2,3,3,0,1,1,1,0,2,2,3,4,1]
Phi over symmetry	[-4,-2,1,1,1,3,1,1,2,4,3,1,2,3,3,0,-1,0,0,1,2]
Phi of -K	[-4,-2,1,1,1,3,1,1,3,4,4,0,1,2,2,0,-1,0,0,1,2]
Phi of K*	[-3,-1,-1,-1,2,4,0,1,2,2,4,0,1,0,1,0,1,3,2,4,1]
Phi of -K*	[-4,-2,1,1,1,3,1,1,2,4,3,1,2,3,3,0,-1,0,0,1,2]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3+t^2-3t
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+60t^4+20t^2
Outer characteristic polynomial	t^7+92t^5+67t^3+7t
Flat arrow polynomial	8K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K1K3 - 3K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial	512K14K2*3 - 1152K1*4K2*2 + 1280K1*4K2 - 1664K14 - 384K1*3K2*2K3 + 768K13K2K3 + 32K1*3K3K4 - 480K1*3K3 + 640K12K2*5 - 2240K1*2K2*4 + 256K1*2K2*3K3*2 + 4512K1*2K2*3 - 384K1*2K2*2K3*2 + 64K1*2K2*2K4 - 7632K12K2*2 + 192K1*2K2K32 + 32K1*2K2K3K5 - 832K12K2K4 + 6360K1*2K2 - 352K12K3*2 - 32K1*2K4*2 - 3980K1*2 - 1024K1K24K3 - 128K1K2*3K3K4 + 2752K1K23K3 + 416K1K2*2K3K4 - 1056K1K22K3 - 128K1K2*2K5 - 128K1K2K3K4 + 5120K1K2K3 + 752K1K3K4 + 32K1K4K5 - 576K2*6 - 320K2*4K3*2 - 32K2*4K4*2 + 672K2*4K4 - 2224K24 + 224K2*3K3K5 + 32K2*3K4K6 - 976K2*2K3*2 - 216K2*2K4*2 + 1024K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 1330K2*2 - 32K2K32K4 + 272K2K3K5 + 48K2K4K6 + 8K2K5K7 - 1204K3*2 - 378K4*2 - 48K5*2 - 6K6**2 + 2992
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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