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Flat knot 6.1206

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1206']
Arrow polynomial of the knot is: -4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']
Outer characteristic polynomial of the knot is: t^7+36t^5+27t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1206']
2-strand cable arrow polynomial of the knot is: -64K16 + 224K1*4K2 - 800K14 + 32K1*3K2K3 - 432K1*2K2*2 + 1176K1*2K2 - 288K12K3*2 - 112K1*2K4*2 - 660K1*2 + 720K1K2K3 + 440K1K3K4 + 152K1K4K5 - 16K24 - 16K2*2K3*2 - 16K2*2K4*2 + 88K2*2K4 - 676K22 + 88K2K3K5 + 32K2K4K6 - 372K3*2 - 220K4*2 - 88K5*2 - 12K6**2 + 818
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1206']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4068', 'vk6.4099', 'vk6.5310', 'vk6.5341', 'vk6.7434', 'vk6.7459', 'vk6.8933', 'vk6.8964', 'vk6.10114', 'vk6.10281', 'vk6.10304', 'vk6.14558', 'vk6.15278', 'vk6.15407', 'vk6.15778', 'vk6.16193', 'vk6.29864', 'vk6.29895', 'vk6.33912', 'vk6.33997', 'vk6.34233', 'vk6.34382', 'vk6.48454', 'vk6.49156', 'vk6.50202', 'vk6.50225', 'vk6.51606', 'vk6.53955', 'vk6.54020', 'vk6.54182', 'vk6.54460', 'vk6.63325']
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,-1,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1206']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']

Outer characteristic polynomial of the knot is: t^7+36t^5+27t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1206']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 224*K1**4*K2 - 800*K1**4 + 32*K1**3*K2*K3 - 432*K1**2*K2**2 + 1176*K1**2*K2 - 288*K1**2*K3**2 - 112*K1**2*K4**2 - 660*K1**2 + 720*K1*K2*K3 + 440*K1*K3*K4 + 152*K1*K4*K5 - 16*K2**4 - 16*K2**2*K3**2 - 16*K2**2*K4**2 + 88*K2**2*K4 - 676*K2**2 + 88*K2*K3*K5 + 32*K2*K4*K6 - 372*K3**2 - 220*K4**2 - 88*K5**2 - 12*K6**2 + 818

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1206']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4068', 'vk6.4099', 'vk6.5310', 'vk6.5341', 'vk6.7434', 'vk6.7459', 'vk6.8933', 'vk6.8964', 'vk6.10114', 'vk6.10281', 'vk6.10304', 'vk6.14558', 'vk6.15278', 'vk6.15407', 'vk6.15778', 'vk6.16193', 'vk6.29864', 'vk6.29895', 'vk6.33912', 'vk6.33997', 'vk6.34233', 'vk6.34382', 'vk6.48454', 'vk6.49156', 'vk6.50202', 'vk6.50225', 'vk6.51606', 'vk6.53955', 'vk6.54020', 'vk6.54182', 'vk6.54460', 'vk6.63325']

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	O1O2O3O4U2U4U5O6O5U1U3U6
R3 orbit	{'O1O2O3O4U2U4U5O6O5U1U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U2U4O6O5U6U1U3
Gauss code of K*	O1O2O3U4U3O5O6O4U5U1U6U2
Gauss code of -K*	O1O2O3U4U1O4O5O6U5U2U6U3
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 -2 -2 1 1 1 1],[ 2 0 -1 2 1 2 1],[ 2 1 0 2 1 2 1],[-1 -2 -2 0 0 -1 0],[-1 -1 -1 0 0 -1 0],[-1 -2 -2 1 1 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 1 1 1 -2 -2],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 0 -2 -2],[ 2 2 1 1 2 0 1],[ 2 2 1 1 2 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,-1,-1,-1,2,2,0,0,1,1,0,1,1,2,2,-1]
Phi over symmetry	[-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,0]
Phi of -K	[-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,0]
Phi of K*	[-1,-1,-1,-1,2,2,-1,0,0,1,1,1,1,1,1,0,2,2,2,2,-1]
Phi of -K*	[-2,-2,1,1,1,1,-1,1,1,2,2,1,1,2,2,0,-1,0,-1,0,1]
Symmetry type of based matrix	c
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	9z+19
Enhanced Jones-Krushkal polynomial	9w^2z+19w
Inner characteristic polynomial	t^6+24t^4+7t^2
Outer characteristic polynomial	t^7+36t^5+27t^3
Flat arrow polynomial	-4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
2-strand cable arrow polynomial	-64K16 + 224K1*4K2 - 800K14 + 32K1*3K2K3 - 432K1*2K2*2 + 1176K1*2K2 - 288K12K3*2 - 112K1*2K4*2 - 660K1*2 + 720K1K2K3 + 440K1K3K4 + 152K1K4K5 - 16K24 - 16K2*2K3*2 - 16K2*2K4*2 + 88K2*2K4 - 676K22 + 88K2K3K5 + 32K2K4K6 - 372K3*2 - 220K4*2 - 88K5*2 - 12K6**2 + 818
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {3, 5}, {4}, {1, 2}]]
If K is slice	False

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