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Flat knot 6.466

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,3,1,0,0,1,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.466']
Arrow polynomial of the knot is: -6K1K2 - 2K1K3 + 3K1 - 2K2*2 + K2 + 3K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.466', '6.871', '6.1186']
Outer characteristic polynomial of the knot is: t^7+52t^5+40t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.466']
2-strand cable arrow polynomial of the knot is: -2928K14 + 352K1*3K2K3 + 160K1*3K3K4 - 224K1*3K3 + 96K12K2*2K4 - 1856K12K2*2 - 1088K1*2K2K4 + 5552K1*2K2 - 1280K12K3*2 - 608K1*2K3K5 - 528K1*2K4*2 - 32K1*2K5*2 - 4392K1*2 - 576K1K22K3 - 384K1K2*2K5 - 256K1K2K3K4 - 96K1K2K3K6 + 5248K1K2K3 + 4112K1K3K4 + 1416K1K4K5 + 136K1K5K6 + 8K1K6K7 - 32K2*4 - 32K2*3K6 - 64K22K3*2 - 56K2*2K4*2 + 1096K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3634K2*2 + 792K2K3K5 + 152K2K4K6 + 24K2K5K7 + 16K2K6K8 - 16K3*4 - 64K3*2K4*2 + 96K3*2K6 - 2912K32 + 64K3K4K7 - 8K44 + 16K4*2K8 - 1966K42 - 708K5*2 - 142K6*2 - 28K7*2 - 12K8**2 + 4608
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.466']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4871', 'vk6.5216', 'vk6.6457', 'vk6.6878', 'vk6.8422', 'vk6.8843', 'vk6.9770', 'vk6.10063', 'vk6.11671', 'vk6.12022', 'vk6.13017', 'vk6.20488', 'vk6.20777', 'vk6.21843', 'vk6.27888', 'vk6.29396', 'vk6.29742', 'vk6.32660', 'vk6.33001', 'vk6.39329', 'vk6.39809', 'vk6.46369', 'vk6.47599', 'vk6.47946', 'vk6.48837', 'vk6.49108', 'vk6.51360', 'vk6.51573', 'vk6.53278', 'vk6.57357', 'vk6.64339', 'vk6.66914']
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,-1,0,1,1,2,0,2,1,2,3,1,0,0,1,1,1,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.466']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.466', '6.871', '6.1186']

Outer characteristic polynomial of the knot is: t^7+52t^5+40t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.466']

2-strand cable arrow polynomial of the knot is: -2928*K1**4 + 352*K1**3*K2*K3 + 160*K1**3*K3*K4 - 224*K1**3*K3 + 96*K1**2*K2**2*K4 - 1856*K1**2*K2**2 - 1088*K1**2*K2*K4 + 5552*K1**2*K2 - 1280*K1**2*K3**2 - 608*K1**2*K3*K5 - 528*K1**2*K4**2 - 32*K1**2*K5**2 - 4392*K1**2 - 576*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5248*K1*K2*K3 + 4112*K1*K3*K4 + 1416*K1*K4*K5 + 136*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4 - 32*K2**3*K6 - 64*K2**2*K3**2 - 56*K2**2*K4**2 + 1096*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3634*K2**2 + 792*K2*K3*K5 + 152*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 16*K3**4 - 64*K3**2*K4**2 + 96*K3**2*K6 - 2912*K3**2 + 64*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1966*K4**2 - 708*K5**2 - 142*K6**2 - 28*K7**2 - 12*K8**2 + 4608

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.466']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4871', 'vk6.5216', 'vk6.6457', 'vk6.6878', 'vk6.8422', 'vk6.8843', 'vk6.9770', 'vk6.10063', 'vk6.11671', 'vk6.12022', 'vk6.13017', 'vk6.20488', 'vk6.20777', 'vk6.21843', 'vk6.27888', 'vk6.29396', 'vk6.29742', 'vk6.32660', 'vk6.33001', 'vk6.39329', 'vk6.39809', 'vk6.46369', 'vk6.47599', 'vk6.47946', 'vk6.48837', 'vk6.49108', 'vk6.51360', 'vk6.51573', 'vk6.53278', 'vk6.57357', 'vk6.64339', 'vk6.66914']

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	O1O2O3O4O5U6U4O6U1U5U3U2
R3 orbit	{'O1O2O3O4O5U6U4O6U1U5U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U3U1U5O6U2U6
Gauss code of K*	O1O2O3O4U1U4U3U5U2O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U3U6U2U1U4
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 1 1 0 2 -1],[ 3 0 3 2 1 2 2],[-1 -3 0 0 0 1 -2],[-1 -2 0 0 0 1 -2],[ 0 -1 0 0 0 0 0],[-2 -2 -1 -1 0 0 -2],[ 1 -2 2 2 0 2 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 0 -2 -2],[-1 1 0 0 0 -2 -2],[-1 1 0 0 0 -2 -3],[ 0 0 0 0 0 0 -1],[ 1 2 2 2 0 0 -2],[ 3 2 2 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,1,1,0,2,2,0,0,2,2,0,2,3,0,1,2]
Phi over symmetry	[-3,-1,0,1,1,2,0,2,1,2,3,1,0,0,1,1,1,2,0,0,0]
Phi of -K	[-3,-1,0,1,1,2,0,2,1,2,3,1,0,0,1,1,1,2,0,0,0]
Phi of K*	[-2,-1,-1,0,1,3,0,0,2,1,3,0,1,0,1,1,0,2,1,2,0]
Phi of -K*	[-3,-1,0,1,1,2,2,1,2,3,2,0,2,2,2,0,0,0,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+36t^4+23t^2
Outer characteristic polynomial	t^7+52t^5+40t^3+3t
Flat arrow polynomial	-6K1K2 - 2K1K3 + 3K1 - 2K2*2 + K2 + 3K3 + 2*K4 + 2
2-strand cable arrow polynomial	-2928K14 + 352K1*3K2K3 + 160K1*3K3K4 - 224K1*3K3 + 96K12K2*2K4 - 1856K12K2*2 - 1088K1*2K2K4 + 5552K1*2K2 - 1280K12K3*2 - 608K1*2K3K5 - 528K1*2K4*2 - 32K1*2K5*2 - 4392K1*2 - 576K1K22K3 - 384K1K2*2K5 - 256K1K2K3K4 - 96K1K2K3K6 + 5248K1K2K3 + 4112K1K3K4 + 1416K1K4K5 + 136K1K5K6 + 8K1K6K7 - 32K2*4 - 32K2*3K6 - 64K22K3*2 - 56K2*2K4*2 + 1096K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3634K2*2 + 792K2K3K5 + 152K2K4K6 + 24K2K5K7 + 16K2K6K8 - 16K3*4 - 64K3*2K4*2 + 96K3*2K6 - 2912K32 + 64K3K4K7 - 8K44 + 16K4*2K8 - 1966K42 - 708K5*2 - 142K6*2 - 28K7*2 - 12K8**2 + 4608
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice	False

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