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Flat knot 6.1239

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,1,1,2,1,1,1,2,-1,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1239']
Arrow polynomial of the knot is: -8K12 - 4K1K2 + 2K1 + 4K2 + 2K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']
Outer characteristic polynomial of the knot is: t^7+42t^5+33t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1239']
2-strand cable arrow polynomial of the knot is: -320K16 - 192K1*4K2*2 + 544K1*4K2 - 2096K14 + 128K1*3K2K3 - 2272K1*2K2*2 + 3112K1*2K2 - 816K12K3*2 - 112K1*2K4*2 - 604K1*2 + 2624K1K2K3 + 784K1K3K4 + 88K1K4K5 - 640K24 - 416K2*2K3*2 - 48K2*2K4*2 + 536K2*2K4 - 940K22 + 296K2K3K5 + 32K2K4K6 - 688K3*2 - 280K4*2 - 60K5*2 - 4K6**2 + 1342
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1239']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13888', 'vk6.13983', 'vk6.14129', 'vk6.14352', 'vk6.14963', 'vk6.15084', 'vk6.15585', 'vk6.16055', 'vk6.16304', 'vk6.16327', 'vk6.17418', 'vk6.22619', 'vk6.22650', 'vk6.23930', 'vk6.33707', 'vk6.33782', 'vk6.34136', 'vk6.34265', 'vk6.34595', 'vk6.36203', 'vk6.36230', 'vk6.42294', 'vk6.53862', 'vk6.53903', 'vk6.54090', 'vk6.54407', 'vk6.54585', 'vk6.55564', 'vk6.59032', 'vk6.59061', 'vk6.60056', 'vk6.64544']
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,-2,1,1,1,1,-1,1,1,1,2,1,1,1,2,-1,-1,-1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1239']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']

Outer characteristic polynomial of the knot is: t^7+42t^5+33t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1239']

2-strand cable arrow polynomial of the knot is: -320*K1**6 - 192*K1**4*K2**2 + 544*K1**4*K2 - 2096*K1**4 + 128*K1**3*K2*K3 - 2272*K1**2*K2**2 + 3112*K1**2*K2 - 816*K1**2*K3**2 - 112*K1**2*K4**2 - 604*K1**2 + 2624*K1*K2*K3 + 784*K1*K3*K4 + 88*K1*K4*K5 - 640*K2**4 - 416*K2**2*K3**2 - 48*K2**2*K4**2 + 536*K2**2*K4 - 940*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 688*K3**2 - 280*K4**2 - 60*K5**2 - 4*K6**2 + 1342

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1239']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13888', 'vk6.13983', 'vk6.14129', 'vk6.14352', 'vk6.14963', 'vk6.15084', 'vk6.15585', 'vk6.16055', 'vk6.16304', 'vk6.16327', 'vk6.17418', 'vk6.22619', 'vk6.22650', 'vk6.23930', 'vk6.33707', 'vk6.33782', 'vk6.34136', 'vk6.34265', 'vk6.34595', 'vk6.36203', 'vk6.36230', 'vk6.42294', 'vk6.53862', 'vk6.53903', 'vk6.54090', 'vk6.54407', 'vk6.54585', 'vk6.55564', 'vk6.59032', 'vk6.59061', 'vk6.60056', 'vk6.64544']

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	O1O2O3O4U3U5U1O5O6U2U4U6
R3 orbit	{'O1O2O3O4U3U5U1O5O6U2U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U1U3O5O6U4U6U2
Gauss code of K*	O1O2O3U2U4O5O6O4U3U5U1U6
Gauss code of -K*	O1O2O3U1U4O5O4O6U2U6U3U5
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 -1 -1 2 -1 2],[ 1 0 0 0 2 1 2],[ 1 0 0 0 2 1 2],[ 1 0 0 0 1 1 1],[-2 -2 -2 -1 0 -2 1],[ 1 -1 -1 -1 2 0 2],[-2 -2 -2 -1 -1 -2 0]]
Primitive based matrix	[[ 0 2 2 -1 -1 -1 -1],[-2 0 1 -1 -2 -2 -2],[-2 -1 0 -1 -2 -2 -2],[ 1 1 1 0 1 0 0],[ 1 2 2 -1 0 -1 -1],[ 1 2 2 0 1 0 0],[ 1 2 2 0 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,1,1,1,1,-1,1,2,2,2,1,2,2,2,-1,0,0,1,1,0]
Phi over symmetry	[-2,-2,1,1,1,1,-1,1,1,1,2,1,1,1,2,-1,-1,-1,0,0,0]
Phi of -K	[-1,-1,-1,-1,2,2,-1,0,0,1,1,1,1,1,1,0,1,1,2,2,-1]
Phi of K*	[-2,-2,1,1,1,1,-1,1,1,1,2,1,1,1,2,-1,-1,-1,0,0,0]
Phi of -K*	[-1,-1,-1,-1,2,2,-1,-1,-1,2,2,0,0,1,1,0,2,2,2,2,-1]
Symmetry type of based matrix	c
u-polynomial	-2t^2+4t
Normalized Jones-Krushkal polynomial	12z+25
Enhanced Jones-Krushkal polynomial	12w^2z+25w
Inner characteristic polynomial	t^6+30t^4+7t^2
Outer characteristic polynomial	t^7+42t^5+33t^3
Flat arrow polynomial	-8K12 - 4K1K2 + 2K1 + 4K2 + 2K3 + 5
2-strand cable arrow polynomial	-320K16 - 192K1*4K2*2 + 544K1*4K2 - 2096K14 + 128K1*3K2K3 - 2272K1*2K2*2 + 3112K1*2K2 - 816K12K3*2 - 112K1*2K4*2 - 604K1*2 + 2624K1K2K3 + 784K1K3K4 + 88K1K4K5 - 640K24 - 416K2*2K3*2 - 48K2*2K4*2 + 536K2*2K4 - 940K22 + 296K2K3K5 + 32K2K4K6 - 688K3*2 - 280K4*2 - 60K5*2 - 4K6**2 + 1342
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}]]
If K is slice	False

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