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Flat knot 6.286

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,0,2,4,4,0,1,2,2,0,1,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.286']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 2K1K3 - 2K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']
Outer characteristic polynomial of the knot is: t^7+84t^5+76t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.286']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 224K1*4K2 - 1824K14 + 128K1*3K2*3K3 + 224K13K2K3 - 320K1*3K3 - 1152K12K2*4 - 640K1*2K2*3K4 + 2208K12K2*3 - 128K1*2K2*2K3*2 - 5168K1*2K2*2 - 576K1*2K2K4 + 5352K1*2K2 - 384K12K3*2 - 32K1*2K4*2 - 2960K1*2 + 2080K1K23K3 + 640K1K2*2K3K4 - 800K1K22K3 - 32K1K2*2K5 + 32K1K2K33 - 160K1K2K3K4 - 64K1K2K3K6 + 4568K1K2K3 + 936K1K3K4 + 152K1K4K5 - 32K26 - 192K2*4K3*2 - 288K2*4K4*2 + 832K2*4K4 - 2152K24 + 128K2*3K3K5 + 160K2*3K4K6 - 32K2*3K6 - 896K22K3*2 - 616K2*2K4*2 + 1264K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1364K2*2 + 272K2K3K5 + 112K2K4K6 - 1312K3*2 - 584K4*2 - 96K5*2 - 4K6**2 + 2710
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.286']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17084', 'vk6.17327', 'vk6.20252', 'vk6.21557', 'vk6.23466', 'vk6.23806', 'vk6.27481', 'vk6.29076', 'vk6.35603', 'vk6.36055', 'vk6.38896', 'vk6.41096', 'vk6.42976', 'vk6.43290', 'vk6.45653', 'vk6.47386', 'vk6.55217', 'vk6.55472', 'vk6.57082', 'vk6.58236', 'vk6.59615', 'vk6.59961', 'vk6.61629', 'vk6.62810', 'vk6.65020', 'vk6.65226', 'vk6.66715', 'vk6.67572', 'vk6.68290', 'vk6.68442', 'vk6.69362', 'vk6.70104']
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: [-4,-1,0,1,1,3,1,0,2,4,4,0,1,2,2,0,1,0,0,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.286']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']

Outer characteristic polynomial of the knot is: t^7+84t^5+76t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.286']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 1824*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 - 320*K1**3*K3 - 1152*K1**2*K2**4 - 640*K1**2*K2**3*K4 + 2208*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5168*K1**2*K2**2 - 576*K1**2*K2*K4 + 5352*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K4**2 - 2960*K1**2 + 2080*K1*K2**3*K3 + 640*K1*K2**2*K3*K4 - 800*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4568*K1*K2*K3 + 936*K1*K3*K4 + 152*K1*K4*K5 - 32*K2**6 - 192*K2**4*K3**2 - 288*K2**4*K4**2 + 832*K2**4*K4 - 2152*K2**4 + 128*K2**3*K3*K5 + 160*K2**3*K4*K6 - 32*K2**3*K6 - 896*K2**2*K3**2 - 616*K2**2*K4**2 + 1264*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1364*K2**2 + 272*K2*K3*K5 + 112*K2*K4*K6 - 1312*K3**2 - 584*K4**2 - 96*K5**2 - 4*K6**2 + 2710

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.286']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17084', 'vk6.17327', 'vk6.20252', 'vk6.21557', 'vk6.23466', 'vk6.23806', 'vk6.27481', 'vk6.29076', 'vk6.35603', 'vk6.36055', 'vk6.38896', 'vk6.41096', 'vk6.42976', 'vk6.43290', 'vk6.45653', 'vk6.47386', 'vk6.55217', 'vk6.55472', 'vk6.57082', 'vk6.58236', 'vk6.59615', 'vk6.59961', 'vk6.61629', 'vk6.62810', 'vk6.65020', 'vk6.65226', 'vk6.66715', 'vk6.67572', 'vk6.68290', 'vk6.68442', 'vk6.69362', 'vk6.70104']

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	O1O2O3O4O5U1U5O6U3U4U2U6
R3 orbit	{'O1O2O3O4O5U1U5O6U3U4U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U4U2U3O6U1U5
Gauss code of K*	O1O2O3O4U5U3U1U2U6O5O6U4
Gauss code of -K*	O1O2O3O4U1O5O6U5U3U4U2U6
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 -4 0 -1 1 1 3],[ 4 0 4 2 3 1 3],[ 0 -4 0 -1 1 0 3],[ 1 -2 1 0 1 0 2],[-1 -3 -1 -1 0 0 1],[-1 -1 0 0 0 0 0],[-3 -3 -3 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 0 -1 -3 -2 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -1 -3],[ 0 3 0 1 0 -1 -4],[ 1 2 0 1 1 0 -2],[ 4 3 1 3 4 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,0,1,3,2,3,0,0,0,1,1,1,3,1,4,2]
Phi over symmetry	[-4,-1,0,1,1,3,1,0,2,4,4,0,1,2,2,0,1,0,0,1,2]
Phi of -K	[-4,-1,0,1,1,3,1,0,2,4,4,0,1,2,2,0,1,0,0,1,2]
Phi of K*	[-3,-1,-1,0,1,4,1,2,0,2,4,0,0,1,2,1,2,4,0,0,1]
Phi of -K*	[-4,-1,0,1,1,3,2,4,1,3,3,1,0,1,2,0,1,3,0,0,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	-2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial	t^6+56t^4+24t^2
Outer characteristic polynomial	t^7+84t^5+76t^3+7t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 2K1K3 - 2K1 + 2*K2 + 3
2-strand cable arrow polynomial	-192K14K2*2 + 224K1*4K2 - 1824K14 + 128K1*3K2*3K3 + 224K13K2K3 - 320K1*3K3 - 1152K12K2*4 - 640K1*2K2*3K4 + 2208K12K2*3 - 128K1*2K2*2K3*2 - 5168K1*2K2*2 - 576K1*2K2K4 + 5352K1*2K2 - 384K12K3*2 - 32K1*2K4*2 - 2960K1*2 + 2080K1K23K3 + 640K1K2*2K3K4 - 800K1K22K3 - 32K1K2*2K5 + 32K1K2K33 - 160K1K2K3K4 - 64K1K2K3K6 + 4568K1K2K3 + 936K1K3K4 + 152K1K4K5 - 32K26 - 192K2*4K3*2 - 288K2*4K4*2 + 832K2*4K4 - 2152K24 + 128K2*3K3K5 + 160K2*3K4K6 - 32K2*3K6 - 896K22K3*2 - 616K2*2K4*2 + 1264K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1364K2*2 + 272K2K3K5 + 112K2K4K6 - 1312K3*2 - 584K4*2 - 96K5*2 - 4K6**2 + 2710
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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