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Flat knot 6.1003

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,1,1,2,4,0,1,0,1,0,1,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1003']
Arrow polynomial of the knot is: 12K13 - 8K1*2 - 6K1K2 - 6K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.360', '6.988', '6.1003']
Outer characteristic polynomial of the knot is: t^7+47t^5+57t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1003']
2-strand cable arrow polynomial of the knot is: 256K14K2*3 - 1280K1*4K2*2 + 1952K1*4K2 - 2816K14 + 128K1*3K2*3K3 - 128K13K2*2K3 + 960K13K2K3 + 32K1*3K3K4 - 320K1*3K3 - 832K12K2*4 + 2752K1*2K2*3 - 192K1*2K2*2K3*2 + 128K1*2K2*2K4 - 11088K12K2*2 + 64K1*2K2K32 - 768K1*2K2K4 + 11072K1*2K2 - 352K12K3*2 - 32K1*2K4*2 - 6224K1*2 + 1952K1K23K3 + 96K1K2*2K3K4 - 1760K1K22K3 - 384K1K2*2K5 - 64K1K2K3K4 + 8728K1K2K3 + 600K1K3K4 + 8K1K4K5 - 96K2*6 + 160K2*4K4 - 2544K24 - 912K2*2K3*2 - 72K2*2K4*2 + 1712K2*2K4 - 3624K22 + 296K2K3K5 - 1848K32 - 288K4*2 - 24K5**2 + 4750
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1003']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4744', 'vk6.5071', 'vk6.6282', 'vk6.6721', 'vk6.8247', 'vk6.8696', 'vk6.9637', 'vk6.9952', 'vk6.20404', 'vk6.21759', 'vk6.27748', 'vk6.29280', 'vk6.39182', 'vk6.41420', 'vk6.45920', 'vk6.47551', 'vk6.48784', 'vk6.48995', 'vk6.49600', 'vk6.49803', 'vk6.50804', 'vk6.51019', 'vk6.51287', 'vk6.51482', 'vk6.57265', 'vk6.58488', 'vk6.61915', 'vk6.63018', 'vk6.66882', 'vk6.67762', 'vk6.69516', 'vk6.70228']
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,-1,0,0,2,2,-1,1,1,2,4,0,1,0,1,0,1,1,0,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1003']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.360', '6.988', '6.1003']

Outer characteristic polynomial of the knot is: t^7+47t^5+57t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1003']

2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1280*K1**4*K2**2 + 1952*K1**4*K2 - 2816*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 960*K1**3*K2*K3 + 32*K1**3*K3*K4 - 320*K1**3*K3 - 832*K1**2*K2**4 + 2752*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 11088*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 11072*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 6224*K1**2 + 1952*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 384*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 8728*K1*K2*K3 + 600*K1*K3*K4 + 8*K1*K4*K5 - 96*K2**6 + 160*K2**4*K4 - 2544*K2**4 - 912*K2**2*K3**2 - 72*K2**2*K4**2 + 1712*K2**2*K4 - 3624*K2**2 + 296*K2*K3*K5 - 1848*K3**2 - 288*K4**2 - 24*K5**2 + 4750

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1003']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4744', 'vk6.5071', 'vk6.6282', 'vk6.6721', 'vk6.8247', 'vk6.8696', 'vk6.9637', 'vk6.9952', 'vk6.20404', 'vk6.21759', 'vk6.27748', 'vk6.29280', 'vk6.39182', 'vk6.41420', 'vk6.45920', 'vk6.47551', 'vk6.48784', 'vk6.48995', 'vk6.49600', 'vk6.49803', 'vk6.50804', 'vk6.51019', 'vk6.51287', 'vk6.51482', 'vk6.57265', 'vk6.58488', 'vk6.61915', 'vk6.63018', 'vk6.66882', 'vk6.67762', 'vk6.69516', 'vk6.70228']

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	O1O2O3O4U2U3O5U1O6U5U6U4
R3 orbit	{'O1O2O3O4U2U3O5U1O6U5U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U5U6O5U4O6U2U3
Gauss code of K*	O1O2O3U4U5U6U3O5O6U1O4U2
Gauss code of -K*	O1O2O3U2O4U3O5O6U1U5U6U4
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 -2 0 3 0 1],[ 2 0 -1 1 4 1 1],[ 2 1 0 1 2 0 0],[ 0 -1 -1 0 1 0 0],[-3 -4 -2 -1 0 -1 1],[ 0 -1 0 0 1 0 1],[-1 -1 0 0 -1 -1 0]]
Primitive based matrix	[[ 0 3 1 0 0 -2 -2],[-3 0 1 -1 -1 -2 -4],[-1 -1 0 0 -1 0 -1],[ 0 1 0 0 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 2 2 0 1 0 0 1],[ 2 4 1 1 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,0,2,2,-1,1,1,2,4,0,1,0,1,0,1,1,0,1,-1]
Phi over symmetry	[-3,-1,0,0,2,2,-1,1,1,2,4,0,1,0,1,0,1,1,0,1,-1]
Phi of -K	[-2,-2,0,0,1,3,-1,1,2,3,3,1,1,2,1,0,1,2,0,2,3]
Phi of K*	[-3,-1,0,0,2,2,3,2,2,1,3,0,1,2,3,0,1,2,1,1,-1]
Phi of -K*	[-2,-2,0,0,1,3,-1,1,1,1,4,0,1,0,2,0,1,1,0,1,-1]
Symmetry type of based matrix	c
u-polynomial	-t^3+2t^2-t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+29t^4+31t^2+1
Outer characteristic polynomial	t^7+47t^5+57t^3+4t
Flat arrow polynomial	12K13 - 8K1*2 - 6K1K2 - 6K1 + 4*K2 + 5
2-strand cable arrow polynomial	256K14K2*3 - 1280K1*4K2*2 + 1952K1*4K2 - 2816K14 + 128K1*3K2*3K3 - 128K13K2*2K3 + 960K13K2K3 + 32K1*3K3K4 - 320K1*3K3 - 832K12K2*4 + 2752K1*2K2*3 - 192K1*2K2*2K3*2 + 128K1*2K2*2K4 - 11088K12K2*2 + 64K1*2K2K32 - 768K1*2K2K4 + 11072K1*2K2 - 352K12K3*2 - 32K1*2K4*2 - 6224K1*2 + 1952K1K23K3 + 96K1K2*2K3K4 - 1760K1K22K3 - 384K1K2*2K5 - 64K1K2K3K4 + 8728K1K2K3 + 600K1K3K4 + 8K1K4K5 - 96K2*6 + 160K2*4K4 - 2544K24 - 912K2*2K3*2 - 72K2*2K4*2 + 1712K2*2K4 - 3624K22 + 296K2K3K5 - 1848K32 - 288K4*2 - 24K5**2 + 4750
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice	False

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