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Flat knot 6.1770

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,2,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['4.6', '6.1770', '6.1951', '7.43053', '7.44462']
Arrow polynomial of the knot is: 8K13 - 10K1*2 - 8K1K2 - 2K1 + 5K2 + 2K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']
Outer characteristic polynomial of the knot is: t^5+13t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1770', '6.1951', '7.38625', '7.43063']
2-strand cable arrow polynomial of the knot is: -128K14K2*2 + 1440K1*4K2 - 4416K14 + 832K1*3K2K3 - 1984K1*3K3 + 736K12K2*3 + 256K1*2K2*2K4 - 7408K12K2*2 + 64K1*2K2K32 - 896K1*2K2K4 + 13456K1*2K2 - 1248K12K3*2 - 64K1*2K3K5 - 8612K1*2 + 480K1K23K3 - 1504K1K2*2K3 - 416K1K2*2K5 - 480K1K2K3K4 + 11448K1K2K3 + 1760K1K3K4 + 160K1K4K5 - 64K2*6 + 192K2*4K4 - 1256K24 - 64K2*3K6 - 544K22K3*2 - 128K2*2K4*2 + 2024K2*2K4 - 6884K22 + 728K2K3K5 + 80K2K4K6 - 3492K3*2 - 818K4*2 - 168K5*2 - 12K6**2 + 6928
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1770']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17093', 'vk6.17336', 'vk6.20581', 'vk6.21989', 'vk6.23478', 'vk6.23817', 'vk6.28046', 'vk6.29504', 'vk6.35631', 'vk6.36074', 'vk6.39460', 'vk6.41659', 'vk6.42997', 'vk6.43309', 'vk6.46048', 'vk6.47714', 'vk6.55244', 'vk6.55496', 'vk6.57466', 'vk6.58629', 'vk6.59644', 'vk6.59992', 'vk6.62141', 'vk6.63102', 'vk6.65048', 'vk6.65245', 'vk6.66997', 'vk6.67861', 'vk6.68311', 'vk6.68461', 'vk6.69615', 'vk6.70307']
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: [-2,0,1,1,1,1,2,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['4.6', '6.1770', '6.1951', '7.43053', '7.44462']

Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794']

Outer characteristic polynomial of the knot is: t^5+13t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1770', '6.1951', '7.38625', '7.43063']

2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1440*K1**4*K2 - 4416*K1**4 + 832*K1**3*K2*K3 - 1984*K1**3*K3 + 736*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7408*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 13456*K1**2*K2 - 1248*K1**2*K3**2 - 64*K1**2*K3*K5 - 8612*K1**2 + 480*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 416*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 11448*K1*K2*K3 + 1760*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1256*K2**4 - 64*K2**3*K6 - 544*K2**2*K3**2 - 128*K2**2*K4**2 + 2024*K2**2*K4 - 6884*K2**2 + 728*K2*K3*K5 + 80*K2*K4*K6 - 3492*K3**2 - 818*K4**2 - 168*K5**2 - 12*K6**2 + 6928

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1770']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17093', 'vk6.17336', 'vk6.20581', 'vk6.21989', 'vk6.23478', 'vk6.23817', 'vk6.28046', 'vk6.29504', 'vk6.35631', 'vk6.36074', 'vk6.39460', 'vk6.41659', 'vk6.42997', 'vk6.43309', 'vk6.46048', 'vk6.47714', 'vk6.55244', 'vk6.55496', 'vk6.57466', 'vk6.58629', 'vk6.59644', 'vk6.59992', 'vk6.62141', 'vk6.63102', 'vk6.65048', 'vk6.65245', 'vk6.66997', 'vk6.67861', 'vk6.68311', 'vk6.68461', 'vk6.69615', 'vk6.70307']

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	O1O2O3U4U2O5O4U1U3O6U5U6
R3 orbit	{'O1O2O3U4U2O5O4U1U3O6U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O4U1U3O6O5U2U6
Gauss code of K*	O1O2U3O4O3U1U5U2O6O5U4U6
Gauss code of -K*	O1O2U1O3O4U5U2O6O5U3U6U4
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 0 1],[ 2 0 1 2 1 1 1],[ 0 -1 0 0 0 0 0],[-1 -2 0 0 -1 0 1],[ 0 -1 0 1 0 0 0],[ 0 -1 0 0 0 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix	[[ 0 1 1 0 -2],[-1 0 1 0 -2],[-1 -1 0 0 -1],[ 0 0 0 0 -1],[ 2 2 1 1 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-1,-1,0,2,-1,0,2,0,1,1]
Phi over symmetry	[-2,0,1,1,1,1,2,0,0,-1]
Phi of -K	[-2,0,1,1,1,1,2,1,1,-1]
Phi of K*	[-1,-1,0,2,-1,1,2,1,1,1]
Phi of -K*	[-2,0,1,1,1,1,2,0,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	2z^2+23z+39
Enhanced Jones-Krushkal polynomial	2w^3z^2+23w^2z+39w
Inner characteristic polynomial	t^4+7t^2+1
Outer characteristic polynomial	t^5+13t^3+4t
Flat arrow polynomial	8K13 - 10K1*2 - 8K1K2 - 2K1 + 5K2 + 2K3 + 6
2-strand cable arrow polynomial	-128K14K2*2 + 1440K1*4K2 - 4416K14 + 832K1*3K2K3 - 1984K1*3K3 + 736K12K2*3 + 256K1*2K2*2K4 - 7408K12K2*2 + 64K1*2K2K32 - 896K1*2K2K4 + 13456K1*2K2 - 1248K12K3*2 - 64K1*2K3K5 - 8612K1*2 + 480K1K23K3 - 1504K1K2*2K3 - 416K1K2*2K5 - 480K1K2K3K4 + 11448K1K2K3 + 1760K1K3K4 + 160K1K4K5 - 64K2*6 + 192K2*4K4 - 1256K24 - 64K2*3K6 - 544K22K3*2 - 128K2*2K4*2 + 2024K2*2K4 - 6884K22 + 728K2K3K5 + 80K2K4K6 - 3492K3*2 - 818K4*2 - 168K5*2 - 12K6**2 + 6928
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice	False

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