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Flat knot 6.952

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,1,1,2,3,3,-1,-1,0,0,0,0,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.952']
Arrow polynomial of the knot is: -2K1K2 - 2K1K3 + K1 - 2K22 + K2 + K3 + 2K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160']
Outer characteristic polynomial of the knot is: t^7+39t^5+74t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.952']
2-strand cable arrow polynomial of the knot is: -64K14 + 32K1*3K3K4 + 128K1*2K2K32 - 32K1*2K2K4 + 584K1*2K2 - 3216K12K3*2 - 80K1*2K4*2 - 48K1*2K6*2 - 2776K1*2 - 352K1K22K3 - 576K1K2K3K4 + 4784K1K2K3 - 64K1K32K5 - 64K1K3K4K6 + 3848K1K3K4 + 232K1K4K5 + 104K1K5K6 + 72K1K6K7 - 288K22K3*2 - 8K2*2K4*2 + 584K2*2K4 - 8K22K6*2 - 2110K2*2 - 32K2K3K4K5 + 496K2K3K5 - 32K2K4*2K6 + 104K2K4K6 + 8K2K5K7 + 16K2K6K8 - 16K3*4 - 32K3*2K4*2 + 112K3*2K6 - 2628K32 + 64K3K4K7 - 8K44 + 16K4*2K8 - 1322K42 - 168K5*2 - 130K6*2 - 28K7*2 - 4K8**2 + 2900
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.952']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4650', 'vk6.4933', 'vk6.6092', 'vk6.6585', 'vk6.8107', 'vk6.8503', 'vk6.9491', 'vk6.9854', 'vk6.20627', 'vk6.22056', 'vk6.28109', 'vk6.29552', 'vk6.39529', 'vk6.41754', 'vk6.46136', 'vk6.47780', 'vk6.48690', 'vk6.48893', 'vk6.49442', 'vk6.49663', 'vk6.50700', 'vk6.50897', 'vk6.51185', 'vk6.51390', 'vk6.57511', 'vk6.58701', 'vk6.62203', 'vk6.63151', 'vk6.67021', 'vk6.67896', 'vk6.69646', 'vk6.70329']
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: [-3,0,0,0,1,2,1,1,2,3,3,-1,-1,0,0,0,0,1,0,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.952']

Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160']

Outer characteristic polynomial of the knot is: t^7+39t^5+74t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.952']

2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K3*K4 + 128*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 584*K1**2*K2 - 3216*K1**2*K3**2 - 80*K1**2*K4**2 - 48*K1**2*K6**2 - 2776*K1**2 - 352*K1*K2**2*K3 - 576*K1*K2*K3*K4 + 4784*K1*K2*K3 - 64*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 3848*K1*K3*K4 + 232*K1*K4*K5 + 104*K1*K5*K6 + 72*K1*K6*K7 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 584*K2**2*K4 - 8*K2**2*K6**2 - 2110*K2**2 - 32*K2*K3*K4*K5 + 496*K2*K3*K5 - 32*K2*K4**2*K6 + 104*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 112*K3**2*K6 - 2628*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1322*K4**2 - 168*K5**2 - 130*K6**2 - 28*K7**2 - 4*K8**2 + 2900

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.952']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4650', 'vk6.4933', 'vk6.6092', 'vk6.6585', 'vk6.8107', 'vk6.8503', 'vk6.9491', 'vk6.9854', 'vk6.20627', 'vk6.22056', 'vk6.28109', 'vk6.29552', 'vk6.39529', 'vk6.41754', 'vk6.46136', 'vk6.47780', 'vk6.48690', 'vk6.48893', 'vk6.49442', 'vk6.49663', 'vk6.50700', 'vk6.50897', 'vk6.51185', 'vk6.51390', 'vk6.57511', 'vk6.58701', 'vk6.62203', 'vk6.63151', 'vk6.67021', 'vk6.67896', 'vk6.69646', 'vk6.70329']

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	O1O2O3O4U5U1O6O5U6U3U2U4
R3 orbit	{'O1O2O3O4U5U1O6O5U6U3U2U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U3U2U5O6O5U4U6
Gauss code of K*	O1O2O3O4U5U3U2U4O6O5U1U6
Gauss code of -K*	O1O2O3O4U5U4O6O5U1U3U2U6
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 0 3 0 -1],[ 2 0 1 0 2 2 -1],[ 0 -1 0 0 2 1 -1],[ 0 0 0 0 1 1 -1],[-3 -2 -2 -1 0 -2 -1],[ 0 -2 -1 -1 2 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix	[[ 0 3 0 0 0 -1 -2],[-3 0 -1 -2 -2 -1 -2],[ 0 1 0 1 0 -1 0],[ 0 2 -1 0 -1 -1 -2],[ 0 2 0 1 0 -1 -1],[ 1 1 1 1 1 0 1],[ 2 2 0 2 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,0,0,0,1,2,1,2,2,1,2,-1,0,1,0,1,1,2,1,1,-1]
Phi over symmetry	[-3,0,0,0,1,2,1,1,2,3,3,-1,-1,0,0,0,0,1,0,2,2]
Phi of -K	[-2,-1,0,0,0,3,2,0,1,2,3,0,0,0,3,1,1,1,0,1,2]
Phi of K*	[-3,0,0,0,1,2,1,1,2,3,3,-1,-1,0,0,0,0,1,0,2,2]
Phi of -K*	[-2,-1,0,0,0,3,-1,0,1,2,2,1,1,1,1,0,1,1,1,2,2]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	4z^2+21z+27
Enhanced Jones-Krushkal polynomial	4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial	t^6+25t^4+17t^2+1
Outer characteristic polynomial	t^7+39t^5+74t^3+9t
Flat arrow polynomial	-2K1K2 - 2K1K3 + K1 - 2K22 + K2 + K3 + 2K4 + 2
2-strand cable arrow polynomial	-64K14 + 32K1*3K3K4 + 128K1*2K2K32 - 32K1*2K2K4 + 584K1*2K2 - 3216K12K3*2 - 80K1*2K4*2 - 48K1*2K6*2 - 2776K1*2 - 352K1K22K3 - 576K1K2K3K4 + 4784K1K2K3 - 64K1K32K5 - 64K1K3K4K6 + 3848K1K3K4 + 232K1K4K5 + 104K1K5K6 + 72K1K6K7 - 288K22K3*2 - 8K2*2K4*2 + 584K2*2K4 - 8K22K6*2 - 2110K2*2 - 32K2K3K4K5 + 496K2K3K5 - 32K2K4*2K6 + 104K2K4K6 + 8K2K5K7 + 16K2K6K8 - 16K3*4 - 32K3*2K4*2 + 112K3*2K6 - 2628K32 + 64K3K4K7 - 8K44 + 16K4*2K8 - 1322K42 - 168K5*2 - 130K6*2 - 28K7*2 - 4K8**2 + 2900
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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

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