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Flat knot 6.912

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,1,1,2,1,1,0,1,0,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.912']
Arrow polynomial of the knot is: -8K12 - 8K1K2 + 4K1 + 4K2 + 4K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.899', '6.912', '6.1806']
Outer characteristic polynomial of the knot is: t^7+31t^5+33t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.912']
2-strand cable arrow polynomial of the knot is: -512K16 - 192K1*4K2*2 + 1248K1*4K2 - 5040K14 + 160K1*3K2K3 + 32K1*3K3K4 - 288K1*3K3 - 3344K12K2*2 - 512K1*2K2K4 + 8432K1*2K2 - 2192K12K3*2 - 160K1*2K3K5 - 832K1*2K4*2 - 32K1*2K5*2 - 5116K1*2 - 288K1K22K3 - 32K1K2*2K5 - 224K1K2K3K4 + 6808K1K2K3 - 64K1K2K4K5 + 4280K1K3K4 + 1184K1K4K5 + 64K1K5K6 - 256K24 - 288K2*2K3*2 - 96K2*2K4*2 + 1064K2*2K4 - 4736K22 + 632K2K3K5 + 128K2K4K6 - 3196K3*2 - 1904K4*2 - 528K5*2 - 48K6**2 + 5846
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.912']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3652', 'vk6.3747', 'vk6.3942', 'vk6.4037', 'vk6.4488', 'vk6.4583', 'vk6.5874', 'vk6.6001', 'vk6.7135', 'vk6.7314', 'vk6.7405', 'vk6.7927', 'vk6.8046', 'vk6.9361', 'vk6.17929', 'vk6.18024', 'vk6.18748', 'vk6.24468', 'vk6.24871', 'vk6.25334', 'vk6.37495', 'vk6.43895', 'vk6.44226', 'vk6.44531', 'vk6.48276', 'vk6.48339', 'vk6.50057', 'vk6.50171', 'vk6.50588', 'vk6.50651', 'vk6.55872', 'vk6.60710']
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: [-2,-1,0,0,1,2,0,1,1,1,2,1,1,0,1,0,1,1,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.912']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.899', '6.912', '6.1806']

Outer characteristic polynomial of the knot is: t^7+31t^5+33t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.912']

2-strand cable arrow polynomial of the knot is: -512*K1**6 - 192*K1**4*K2**2 + 1248*K1**4*K2 - 5040*K1**4 + 160*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 3344*K1**2*K2**2 - 512*K1**2*K2*K4 + 8432*K1**2*K2 - 2192*K1**2*K3**2 - 160*K1**2*K3*K5 - 832*K1**2*K4**2 - 32*K1**2*K5**2 - 5116*K1**2 - 288*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6808*K1*K2*K3 - 64*K1*K2*K4*K5 + 4280*K1*K3*K4 + 1184*K1*K4*K5 + 64*K1*K5*K6 - 256*K2**4 - 288*K2**2*K3**2 - 96*K2**2*K4**2 + 1064*K2**2*K4 - 4736*K2**2 + 632*K2*K3*K5 + 128*K2*K4*K6 - 3196*K3**2 - 1904*K4**2 - 528*K5**2 - 48*K6**2 + 5846

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.912']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3652', 'vk6.3747', 'vk6.3942', 'vk6.4037', 'vk6.4488', 'vk6.4583', 'vk6.5874', 'vk6.6001', 'vk6.7135', 'vk6.7314', 'vk6.7405', 'vk6.7927', 'vk6.8046', 'vk6.9361', 'vk6.17929', 'vk6.18024', 'vk6.18748', 'vk6.24468', 'vk6.24871', 'vk6.25334', 'vk6.37495', 'vk6.43895', 'vk6.44226', 'vk6.44531', 'vk6.48276', 'vk6.48339', 'vk6.50057', 'vk6.50171', 'vk6.50588', 'vk6.50651', 'vk6.55872', 'vk6.60710']

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	O1O2O3O4U2U5O6O5U4U6U1U3
R3 orbit	{'O1O2O3O4U2U5O6O5U4U6U1U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U4U5U1O6O5U6U3
Gauss code of K*	O1O2O3O4U3U5U4U1O5O6U2U6
Gauss code of -K*	O1O2O3O4U5U3O5O6U4U1U6U2
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 -1 -2 2 0 1 0],[ 1 0 -1 2 0 2 0],[ 2 1 0 2 1 2 1],[-2 -2 -2 0 -1 -1 0],[ 0 0 -1 1 0 0 0],[-1 -2 -2 1 0 0 0],[ 0 0 -1 0 0 0 0]]
Primitive based matrix	[[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -2 -2],[-1 1 0 0 0 -2 -2],[ 0 0 0 0 0 0 -1],[ 0 1 0 0 0 0 -1],[ 1 2 2 0 0 0 -1],[ 2 2 2 1 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,0,1,2,1,0,1,2,2,0,0,2,2,0,0,1,0,1,1]
Phi over symmetry	[-2,-1,0,0,1,2,0,1,1,1,2,1,1,0,1,0,1,1,1,2,0]
Phi of -K	[-2,-1,0,0,1,2,0,1,1,1,2,1,1,0,1,0,1,1,1,2,0]
Phi of K*	[-2,-1,0,0,1,2,0,1,2,1,2,1,1,0,1,0,1,1,1,1,0]
Phi of -K*	[-2,-1,0,0,1,2,1,1,1,2,2,0,0,2,2,0,0,0,0,1,1]
Symmetry type of based matrix	c
u-polynomial	0
Normalized Jones-Krushkal polynomial	21z+43
Enhanced Jones-Krushkal polynomial	21w^2z+43w
Inner characteristic polynomial	t^6+21t^4+21t^2+4
Outer characteristic polynomial	t^7+31t^5+33t^3+7t
Flat arrow polynomial	-8K12 - 8K1K2 + 4K1 + 4K2 + 4K3 + 5
2-strand cable arrow polynomial	-512K16 - 192K1*4K2*2 + 1248K1*4K2 - 5040K14 + 160K1*3K2K3 + 32K1*3K3K4 - 288K1*3K3 - 3344K12K2*2 - 512K1*2K2K4 + 8432K1*2K2 - 2192K12K3*2 - 160K1*2K3K5 - 832K1*2K4*2 - 32K1*2K5*2 - 5116K1*2 - 288K1K22K3 - 32K1K2*2K5 - 224K1K2K3K4 + 6808K1K2K3 - 64K1K2K4K5 + 4280K1K3K4 + 1184K1K4K5 + 64K1K5K6 - 256K24 - 288K2*2K3*2 - 96K2*2K4*2 + 1064K2*2K4 - 4736K22 + 632K2K3K5 + 128K2K4K6 - 3196K3*2 - 1904K4*2 - 528K5*2 - 48K6**2 + 5846
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice	False

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