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Flat knot 6.987

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,0,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.987']
Arrow polynomial of the knot is: -4K12 - 6K1K2 + 3K1 + 2K2 + 3K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']
Outer characteristic polynomial of the knot is: t^7+32t^5+49t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.987']
2-strand cable arrow polynomial of the knot is: 96K14K2 - 2448K14 + 352K1*3K2K3 + 64K1*3K3K4 - 1568K1*3K3 + 128K12K2*2K4 - 1472K12K2*2 - 1088K1*2K2K4 + 5928K1*2K2 - 672K12K3*2 - 208K1*2K4*2 - 3904K1*2 - 128K1K22K3 - 192K1K2*2K5 + 32K1K2K33 - 192K1K2K3K4 + 5056K1K2K3 - 32K1K3*2K5 + 1696K1K3K4 + 320K1K4K5 + 8K1K5K6 - 80K2*4 - 64K2*2K3*2 - 24K2*2K4*2 + 920K2*2K4 - 3286K22 + 352K2K3K5 + 40K2K4K6 - 16K3*4 + 16K3*2K6 - 1868K32 - 888K4*2 - 196K5*2 - 18K6**2 + 3334
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.987']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4344', 'vk6.4375', 'vk6.5666', 'vk6.5697', 'vk6.7735', 'vk6.7766', 'vk6.9217', 'vk6.9248', 'vk6.10502', 'vk6.10548', 'vk6.10645', 'vk6.10721', 'vk6.10752', 'vk6.10834', 'vk6.14621', 'vk6.15309', 'vk6.15434', 'vk6.16244', 'vk6.17984', 'vk6.24428', 'vk6.30181', 'vk6.30227', 'vk6.30324', 'vk6.30453', 'vk6.33947', 'vk6.34348', 'vk6.34402', 'vk6.43855', 'vk6.50438', 'vk6.50468', 'vk6.54207', 'vk6.63437']
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: [-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,0,1,0,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.987']

Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']

Outer characteristic polynomial of the knot is: t^7+32t^5+49t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.987']

2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 2448*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 + 128*K1**2*K2**2*K4 - 1472*K1**2*K2**2 - 1088*K1**2*K2*K4 + 5928*K1**2*K2 - 672*K1**2*K3**2 - 208*K1**2*K4**2 - 3904*K1**2 - 128*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 5056*K1*K2*K3 - 32*K1*K3**2*K5 + 1696*K1*K3*K4 + 320*K1*K4*K5 + 8*K1*K5*K6 - 80*K2**4 - 64*K2**2*K3**2 - 24*K2**2*K4**2 + 920*K2**2*K4 - 3286*K2**2 + 352*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 1868*K3**2 - 888*K4**2 - 196*K5**2 - 18*K6**2 + 3334

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.987']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4344', 'vk6.4375', 'vk6.5666', 'vk6.5697', 'vk6.7735', 'vk6.7766', 'vk6.9217', 'vk6.9248', 'vk6.10502', 'vk6.10548', 'vk6.10645', 'vk6.10721', 'vk6.10752', 'vk6.10834', 'vk6.14621', 'vk6.15309', 'vk6.15434', 'vk6.16244', 'vk6.17984', 'vk6.24428', 'vk6.30181', 'vk6.30227', 'vk6.30324', 'vk6.30453', 'vk6.33947', 'vk6.34348', 'vk6.34402', 'vk6.43855', 'vk6.50438', 'vk6.50468', 'vk6.54207', 'vk6.63437']

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	O1O2O3O4U1U4O5U3O6U5U6U2
R3 orbit	{'O1O2O3O4U1U4O5U3O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U5U6O5U2O6U1U4
Gauss code of K*	O1O2O3U4U3U5U6O4O6U1O5U2
Gauss code of -K*	O1O2O3U2O4U3O5O6U5U4U1U6
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 0 1 0 1],[ 3 0 3 2 1 1 0],[-1 -3 0 -1 0 0 1],[ 0 -2 1 0 0 1 1],[-1 -1 0 0 0 0 0],[ 0 -1 0 -1 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 -1 -3],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 0 -1],[ 0 0 1 0 0 -1 -1],[ 0 1 1 0 1 0 -2],[ 3 3 0 1 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,0,3,-1,0,0,1,3,0,1,1,0,0,0,1,1,1,2]
Phi over symmetry	[-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,0,1,0,1,0,-1,0]
Phi of -K	[-3,0,0,1,1,1,1,2,1,3,4,-1,0,1,0,1,1,0,0,-1,0]
Phi of K*	[-1,-1,-1,0,0,3,-1,0,0,0,4,0,0,1,1,1,1,3,1,1,2]
Phi of -K*	[-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,0,1,0,1,0,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	4z^2+21z+27
Enhanced Jones-Krushkal polynomial	4w^3z^2+21w^2z+27w
Inner characteristic polynomial	t^6+20t^4+15t^2
Outer characteristic polynomial	t^7+32t^5+49t^3+4t
Flat arrow polynomial	-4K12 - 6K1K2 + 3K1 + 2K2 + 3K3 + 3
2-strand cable arrow polynomial	96K14K2 - 2448K14 + 352K1*3K2K3 + 64K1*3K3K4 - 1568K1*3K3 + 128K12K2*2K4 - 1472K12K2*2 - 1088K1*2K2K4 + 5928K1*2K2 - 672K12K3*2 - 208K1*2K4*2 - 3904K1*2 - 128K1K22K3 - 192K1K2*2K5 + 32K1K2K33 - 192K1K2K3K4 + 5056K1K2K3 - 32K1K3*2K5 + 1696K1K3K4 + 320K1K4K5 + 8K1K5K6 - 80K2*4 - 64K2*2K3*2 - 24K2*2K4*2 + 920K2*2K4 - 3286K22 + 352K2K3K5 + 40K2K4K6 - 16K3*4 + 16K3*2K6 - 1868K32 - 888K4*2 - 196K5*2 - 18K6**2 + 3334
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice	False

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