Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 6.471

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,0,3,2,4,4,1,1,0,1,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.471']
Arrow polynomial of the knot is: 12K13 + 4K1*2K2 - 12K12 - 8K1K2 - 2K1K3 - 5K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.471']
Outer characteristic polynomial of the knot is: t^7+73t^5+161t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.471']
2-strand cable arrow polynomial of the knot is: -448K14K2*2 + 896K1*4K2 - 2800K14 + 480K1*3K2K3 - 256K1*3K3 - 1536K12K2*4 + 4096K1*2K2*3 + 128K1*2K2*2K4 - 11456K12K2*2 - 800K1*2K2K4 + 10760K1*2K2 - 240K12K3*2 - 5728K1*2 - 512K1K24K3 + 3136K1K2*3K3 + 384K1K2*2K3K4 - 2560K1K22K3 + 128K1K2*2K4K5 - 544K1K22K5 - 352K1K2K3K4 - 32K1K2K3K6 + 9552K1K2K3 - 32K1K2K4K5 + 1000K1K3K4 + 176K1K4K5 + 24K1K5K6 - 352K26 - 192K2*4K3*2 - 32K2*4K4*2 + 736K2*4K4 - 4016K24 + 160K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 1680K22K3*2 - 544K2*2K4*2 + 3256K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 3370K2*2 + 888K2K3K5 + 136K2K4K6 + 8K2K5K7 + 16K32K6 - 2348K32 - 826K4*2 - 204K5*2 - 22K6**2 + 5072
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.471']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4679', 'vk6.4976', 'vk6.6153', 'vk6.6632', 'vk6.8156', 'vk6.8570', 'vk6.9544', 'vk6.9889', 'vk6.20690', 'vk6.22130', 'vk6.28211', 'vk6.29636', 'vk6.39671', 'vk6.41912', 'vk6.46255', 'vk6.47862', 'vk6.48719', 'vk6.48936', 'vk6.49503', 'vk6.49710', 'vk6.50745', 'vk6.50956', 'vk6.51230', 'vk6.51421', 'vk6.57625', 'vk6.58783', 'vk6.62305', 'vk6.63238', 'vk6.67099', 'vk6.67963', 'vk6.69699', 'vk6.70382']
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: [-4,0,0,1,1,2,0,3,2,4,4,1,1,0,1,0,1,1,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.471']

Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 5*K2 + K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.471']

Outer characteristic polynomial of the knot is: t^7+73t^5+161t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.471']

2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 896*K1**4*K2 - 2800*K1**4 + 480*K1**3*K2*K3 - 256*K1**3*K3 - 1536*K1**2*K2**4 + 4096*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11456*K1**2*K2**2 - 800*K1**2*K2*K4 + 10760*K1**2*K2 - 240*K1**2*K3**2 - 5728*K1**2 - 512*K1*K2**4*K3 + 3136*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 2560*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 544*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9552*K1*K2*K3 - 32*K1*K2*K4*K5 + 1000*K1*K3*K4 + 176*K1*K4*K5 + 24*K1*K5*K6 - 352*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 736*K2**4*K4 - 4016*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1680*K2**2*K3**2 - 544*K2**2*K4**2 + 3256*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 3370*K2**2 + 888*K2*K3*K5 + 136*K2*K4*K6 + 8*K2*K5*K7 + 16*K3**2*K6 - 2348*K3**2 - 826*K4**2 - 204*K5**2 - 22*K6**2 + 5072

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.471']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4679', 'vk6.4976', 'vk6.6153', 'vk6.6632', 'vk6.8156', 'vk6.8570', 'vk6.9544', 'vk6.9889', 'vk6.20690', 'vk6.22130', 'vk6.28211', 'vk6.29636', 'vk6.39671', 'vk6.41912', 'vk6.46255', 'vk6.47862', 'vk6.48719', 'vk6.48936', 'vk6.49503', 'vk6.49710', 'vk6.50745', 'vk6.50956', 'vk6.51230', 'vk6.51421', 'vk6.57625', 'vk6.58783', 'vk6.62305', 'vk6.63238', 'vk6.67099', 'vk6.67963', 'vk6.69699', 'vk6.70382']

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	O1O2O3O4O5U6U4O6U3U1U2U5
R3 orbit	{'O1O2O3O4O5U6U4O6U3U1U2U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U4U5U3O6U2U6
Gauss code of K*	O1O2O3O4U2U3U1U5U4O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U1U6U4U2U3
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 -2 0 -1 0 4 -1],[ 2 0 1 0 1 4 1],[ 0 -1 0 0 1 3 -1],[ 1 0 0 0 1 2 0],[ 0 -1 -1 -1 0 0 0],[-4 -4 -3 -2 0 0 -4],[ 1 -1 1 0 0 4 0]]
Primitive based matrix	[[ 0 4 0 0 -1 -1 -2],[-4 0 0 -3 -2 -4 -4],[ 0 0 0 -1 -1 0 -1],[ 0 3 1 0 0 -1 -1],[ 1 2 1 0 0 0 0],[ 1 4 0 1 0 0 -1],[ 2 4 1 1 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,0,0,1,1,2,0,3,2,4,4,1,1,0,1,0,1,1,0,0,1]
Phi over symmetry	[-4,0,0,1,1,2,0,3,2,4,4,1,1,0,1,0,1,1,0,0,1]
Phi of -K	[-2,-1,-1,0,0,4,0,1,1,1,2,0,0,1,1,1,0,3,-1,1,4]
Phi of K*	[-4,0,0,1,1,2,1,4,1,3,2,1,0,1,1,1,0,1,0,0,1]
Phi of -K*	[-2,-1,-1,0,0,4,0,1,1,1,4,0,0,1,2,1,0,4,1,3,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^2+2t
Normalized Jones-Krushkal polynomial	4z^2+23z+31
Enhanced Jones-Krushkal polynomial	4w^3z^2-2w^3z+25w^2z+31w
Inner characteristic polynomial	t^6+51t^4+82t^2+1
Outer characteristic polynomial	t^7+73t^5+161t^3+10t
Flat arrow polynomial	12K13 + 4K1*2K2 - 12K12 - 8K1K2 - 2K1K3 - 5K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial	-448K14K2*2 + 896K1*4K2 - 2800K14 + 480K1*3K2K3 - 256K1*3K3 - 1536K12K2*4 + 4096K1*2K2*3 + 128K1*2K2*2K4 - 11456K12K2*2 - 800K1*2K2K4 + 10760K1*2K2 - 240K12K3*2 - 5728K1*2 - 512K1K24K3 + 3136K1K2*3K3 + 384K1K2*2K3K4 - 2560K1K22K3 + 128K1K2*2K4K5 - 544K1K22K5 - 352K1K2K3K4 - 32K1K2K3K6 + 9552K1K2K3 - 32K1K2K4K5 + 1000K1K3K4 + 176K1K4K5 + 24K1K5K6 - 352K26 - 192K2*4K3*2 - 32K2*4K4*2 + 736K2*4K4 - 4016K24 + 160K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 1680K22K3*2 - 544K2*2K4*2 + 3256K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 3370K2*2 + 888K2K3K5 + 136K2K4K6 + 8K2K5K7 + 16K32K6 - 2348K32 - 826K4*2 - 204K5*2 - 22K6**2 + 5072
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}], [{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}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

Contact