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Flat knot 6.1927

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,2,2,3,1,1,2,3,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1927']
Arrow polynomial of the knot is: 4K12K2 - 8K12 - 4K1K2 - 2K1K3 + 2K1 - 2K22 + 3K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1213', '6.1927']
Outer characteristic polynomial of the knot is: t^7+35t^5+72t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1927']
2-strand cable arrow polynomial of the knot is: -64K14 + 160K1*3K2K3 - 448K1*3K3 + 64K12K2*3 - 64K1*2K2*2K3*2 - 2336K1*2K2*2 + 160K1*2K2K32 - 416K1*2K2K4 + 5488K1*2K2 - 576K12K3*2 - 160K1*2K3K5 - 16K1*2K4*2 - 32K1*2K5*2 - 6228K1*2 + 576K1K23K3 + 192K1K2*2K3K4 - 1824K1K22K3 + 32K1K2*2K4K5 - 192K1K22K5 + 32K1K2K33 - 320K1K2K3K4 + 7024K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 - 64K1K2K5K6 + 2016K1K3K4 + 528K1K4K5 + 136K1K5K6 + 40K1K6K7 - 32K24K4*2 + 128K2*4K4 - 1008K24 + 32K2*3K4K6 - 32K2*3K6 - 912K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 368K2*2K4*2 - 32K2*2K4K8 + 2248K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 4924K2*2 - 192K2K32K4 + 960K2K3K5 + 408K2K4K6 + 104K2K5K7 + 8K2K6K8 + 24K32K6 - 2964K32 + 24K3K4K7 - 8K44 + 8K4*2K8 - 1362K42 - 480K5*2 - 132K6*2 - 48K7*2 - 2K8**2 + 5042
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1927']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71477', 'vk6.71498', 'vk6.71538', 'vk6.71555', 'vk6.72015', 'vk6.72028', 'vk6.72068', 'vk6.72078', 'vk6.72521', 'vk6.72527', 'vk6.72640', 'vk6.72659', 'vk6.72916', 'vk6.72956', 'vk6.73116', 'vk6.73137', 'vk6.73648', 'vk6.73684', 'vk6.73696', 'vk6.77098', 'vk6.77124', 'vk6.77152', 'vk6.77177', 'vk6.77447', 'vk6.77474', 'vk6.77941', 'vk6.77961', 'vk6.78582', 'vk6.81427', 'vk6.86904', 'vk6.87249', 'vk6.89355']
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,-2,0,1,1,2,0,0,2,2,3,1,1,2,3,1,0,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1927']

Arrow polynomial of the knot is: 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1213', '6.1927']

Outer characteristic polynomial of the knot is: t^7+35t^5+72t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1927']

2-strand cable arrow polynomial of the knot is: -64*K1**4 + 160*K1**3*K2*K3 - 448*K1**3*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 2336*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 5488*K1**2*K2 - 576*K1**2*K3**2 - 160*K1**2*K3*K5 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 6228*K1**2 + 576*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 7024*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 2016*K1*K3*K4 + 528*K1*K4*K5 + 136*K1*K5*K6 + 40*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1008*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 912*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 368*K2**2*K4**2 - 32*K2**2*K4*K8 + 2248*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4924*K2**2 - 192*K2*K3**2*K4 + 960*K2*K3*K5 + 408*K2*K4*K6 + 104*K2*K5*K7 + 8*K2*K6*K8 + 24*K3**2*K6 - 2964*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1362*K4**2 - 480*K5**2 - 132*K6**2 - 48*K7**2 - 2*K8**2 + 5042

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1927']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71477', 'vk6.71498', 'vk6.71538', 'vk6.71555', 'vk6.72015', 'vk6.72028', 'vk6.72068', 'vk6.72078', 'vk6.72521', 'vk6.72527', 'vk6.72640', 'vk6.72659', 'vk6.72916', 'vk6.72956', 'vk6.73116', 'vk6.73137', 'vk6.73648', 'vk6.73684', 'vk6.73696', 'vk6.77098', 'vk6.77124', 'vk6.77152', 'vk6.77177', 'vk6.77447', 'vk6.77474', 'vk6.77941', 'vk6.77961', 'vk6.78582', 'vk6.81427', 'vk6.86904', 'vk6.87249', 'vk6.89355']

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	O1O2O3U1U4U2O5O4O6U5U3U6
R3 orbit	{'O1O2O3U1U4U2O5O4O6U5U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U1U5O4O6O5U2U6U3
Gauss code of K*	O1O2O3U4U2U5O4O6O5U1U3U6
Gauss code of -K*	O1O2O3U1U4U3O5O4O6U2U5U6
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 1 1 0 -2 2],[ 2 0 1 2 1 0 1],[-1 -1 0 0 -1 -1 1],[-1 -2 0 0 0 -1 2],[ 0 -1 1 0 0 -2 1],[ 2 0 1 1 2 0 1],[-2 -1 -1 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 -1 -2 -1 -1 -1],[-1 1 0 0 -1 -1 -1],[-1 2 0 0 0 -1 -2],[ 0 1 1 0 0 -2 -1],[ 2 1 1 1 2 0 0],[ 2 1 1 2 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,1,2,1,1,1,0,1,1,1,0,1,2,2,1,0]
Phi over symmetry	[-2,-2,0,1,1,2,0,0,2,2,3,1,1,2,3,1,0,1,0,-1,0]
Phi of -K	[-2,-2,0,1,1,2,0,0,2,2,3,1,1,2,3,1,0,1,0,-1,0]
Phi of K*	[-2,-1,-1,0,2,2,-1,0,1,3,3,0,1,1,2,0,2,2,1,0,0]
Phi of -K*	[-2,-2,0,1,1,2,0,1,1,2,1,2,1,1,1,1,0,1,0,1,2]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	5z^2+24z+29
Enhanced Jones-Krushkal polynomial	5w^3z^2+24w^2z+29w
Inner characteristic polynomial	t^6+21t^4+37t^2+1
Outer characteristic polynomial	t^7+35t^5+72t^3+6t
Flat arrow polynomial	4K12K2 - 8K12 - 4K1K2 - 2K1K3 + 2K1 - 2K22 + 3K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial	-64K14 + 160K1*3K2K3 - 448K1*3K3 + 64K12K2*3 - 64K1*2K2*2K3*2 - 2336K1*2K2*2 + 160K1*2K2K32 - 416K1*2K2K4 + 5488K1*2K2 - 576K12K3*2 - 160K1*2K3K5 - 16K1*2K4*2 - 32K1*2K5*2 - 6228K1*2 + 576K1K23K3 + 192K1K2*2K3K4 - 1824K1K22K3 + 32K1K2*2K4K5 - 192K1K22K5 + 32K1K2K33 - 320K1K2K3K4 + 7024K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 - 64K1K2K5K6 + 2016K1K3K4 + 528K1K4K5 + 136K1K5K6 + 40K1K6K7 - 32K24K4*2 + 128K2*4K4 - 1008K24 + 32K2*3K4K6 - 32K2*3K6 - 912K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 368K2*2K4*2 - 32K2*2K4K8 + 2248K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 4924K2*2 - 192K2K32K4 + 960K2K3K5 + 408K2K4K6 + 104K2K5K7 + 8K2K6K8 + 24K32K6 - 2964K32 + 24K3K4K7 - 8K44 + 8K4*2K8 - 1362K42 - 480K5*2 - 132K6*2 - 48K7*2 - 2K8**2 + 5042
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}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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