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Flat knot 6.1366

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1366']
Arrow polynomial of the knot is: 8K13 - 4K1*2 - 10K1K2 - K1 + 2K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.770', '6.1299', '6.1366']
Outer characteristic polynomial of the knot is: t^7+43t^5+79t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1366']
2-strand cable arrow polynomial of the knot is: 4992K14K2 - 8640K14 + 2176K1*3K2K3 - 2688K1*3K3 - 256K12K2*4 + 1792K1*2K2*3 + 512K1*2K2*2K4 - 11616K12K2*2 - 2176K1*2K2K4 + 13728K1*2K2 - 1472K12K3*2 - 128K1*2K4*2 - 4360K1*2 + 896K1K23K3 - 2816K1K2*2K3 - 768K1K2*2K5 - 704K1K2K3K4 + 11408K1K2K3 + 2448K1K3K4 + 464K1K4K5 - 64K2*6 + 256K2*4K4 - 1744K24 - 128K2*3K6 - 928K22K3*2 - 296K2*2K4*2 + 2776K2*2K4 - 5122K22 + 1072K2K3K5 + 264K2K4K6 - 2640K3*2 - 1056K4*2 - 296K5*2 - 54K6**2 + 5166
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1366']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13814', 'vk6.13824', 'vk6.13833', 'vk6.13846', 'vk6.13853', 'vk6.13858', 'vk6.14885', 'vk6.14896', 'vk6.14905', 'vk6.14908', 'vk6.14916', 'vk6.14927', 'vk6.14932', 'vk6.14934', 'vk6.34238', 'vk6.34242', 'vk6.53835', 'vk6.53838', 'vk6.54377', 'vk6.54381']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,0,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1366']

Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 10*K1*K2 - K1 + 2*K2 + 3*K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.770', '6.1299', '6.1366']

Outer characteristic polynomial of the knot is: t^7+43t^5+79t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1366']

2-strand cable arrow polynomial of the knot is: 4992*K1**4*K2 - 8640*K1**4 + 2176*K1**3*K2*K3 - 2688*K1**3*K3 - 256*K1**2*K2**4 + 1792*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 11616*K1**2*K2**2 - 2176*K1**2*K2*K4 + 13728*K1**2*K2 - 1472*K1**2*K3**2 - 128*K1**2*K4**2 - 4360*K1**2 + 896*K1*K2**3*K3 - 2816*K1*K2**2*K3 - 768*K1*K2**2*K5 - 704*K1*K2*K3*K4 + 11408*K1*K2*K3 + 2448*K1*K3*K4 + 464*K1*K4*K5 - 64*K2**6 + 256*K2**4*K4 - 1744*K2**4 - 128*K2**3*K6 - 928*K2**2*K3**2 - 296*K2**2*K4**2 + 2776*K2**2*K4 - 5122*K2**2 + 1072*K2*K3*K5 + 264*K2*K4*K6 - 2640*K3**2 - 1056*K4**2 - 296*K5**2 - 54*K6**2 + 5166

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1366']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13814', 'vk6.13824', 'vk6.13833', 'vk6.13846', 'vk6.13853', 'vk6.13858', 'vk6.14885', 'vk6.14896', 'vk6.14905', 'vk6.14908', 'vk6.14916', 'vk6.14927', 'vk6.14932', 'vk6.14934', 'vk6.34238', 'vk6.34242', 'vk6.53835', 'vk6.53838', 'vk6.54377', 'vk6.54381']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3U2O4O5U4U6U3O6U1U5
R3 orbit	{'O1O2O3U2O4O5U4U6U3O6U1U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U3O5U1U5U6O4O6U2
Gauss code of K*	Same
Gauss code of -K*	O1O2O3U4U3O5U1U5U6O4O6U2
Diagrammatic symmetry type	+
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 -1 1 -1 3 -1],[ 1 0 -1 2 0 3 0],[ 1 1 0 1 0 1 0],[-1 -2 -1 0 0 1 -1],[ 1 0 0 0 0 1 1],[-3 -3 -1 -1 -1 0 -3],[ 1 0 0 1 -1 3 0]]
Primitive based matrix	[[ 0 3 1 -1 -1 -1 -1],[-3 0 -1 -1 -1 -3 -3],[-1 1 0 0 -1 -1 -2],[ 1 1 0 0 0 1 0],[ 1 1 1 0 0 0 1],[ 1 3 1 -1 0 0 0],[ 1 3 2 0 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,0,0,-1,0]
Phi over symmetry	[-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,0,0,-1,0]
Phi of -K	[-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,0,1,-1,2,3,1,1,1]
Phi of K*	[-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,0,0,-1,0]
Phi of -K*	[-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,0,1,-1,2,3,1,1,1]
Symmetry type of based matrix	+
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+29t^4+49t^2+1
Outer characteristic polynomial	t^7+43t^5+79t^3+5t
Flat arrow polynomial	8K13 - 4K1*2 - 10K1K2 - K1 + 2K2 + 3*K3 + 3
2-strand cable arrow polynomial	4992K14K2 - 8640K14 + 2176K1*3K2K3 - 2688K1*3K3 - 256K12K2*4 + 1792K1*2K2*3 + 512K1*2K2*2K4 - 11616K12K2*2 - 2176K1*2K2K4 + 13728K1*2K2 - 1472K12K3*2 - 128K1*2K4*2 - 4360K1*2 + 896K1K23K3 - 2816K1K2*2K3 - 768K1K2*2K5 - 704K1K2K3K4 + 11408K1K2K3 + 2448K1K3K4 + 464K1K4K5 - 64K2*6 + 256K2*4K4 - 1744K24 - 128K2*3K6 - 928K22K3*2 - 296K2*2K4*2 + 2776K2*2K4 - 5122K22 + 1072K2K3K5 + 264K2K4K6 - 2640K3*2 - 1056K4*2 - 296K5*2 - 54K6**2 + 5166
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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