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Flat knot 6.1121

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,1,1,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1121', '7.40804']
Arrow polynomial of the knot is: 4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']
Outer characteristic polynomial of the knot is: t^7+22t^5+30t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1121', '7.40804']
2-strand cable arrow polynomial of the knot is: -512K16 - 2816K1*4K2*2 + 4864K1*4K2 - 6048K14 + 1984K1*3K2K3 - 1152K1*3K3 - 2112K12K2*4 + 6016K1*2K2*3 + 256K1*2K2*2K4 - 13056K12K2*2 - 1344K1*2K2K4 + 9256K1*2K2 - 320K12K3*2 - 708K1*2 + 2432K1K23K3 - 2272K1K2*2K3 - 480K1K2*2K5 - 160K1K2K3K4 + 6840K1K2K3 + 344K1K3K4 + 16K1K4K5 - 32K2*6 + 64K2*4K4 - 3560K24 - 592K2*2K3*2 - 48K2*2K4*2 + 2160K2*2K4 - 478K22 + 240K2K3K5 + 16K2K4K6 - 652K3*2 - 158K4*2 - 24K5*2 - 2K6**2 + 2036
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1121']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.484', 'vk6.554', 'vk6.615', 'vk6.954', 'vk6.1051', 'vk6.1121', 'vk6.1641', 'vk6.1754', 'vk6.1832', 'vk6.2138', 'vk6.2235', 'vk6.2304', 'vk6.2576', 'vk6.2850', 'vk6.3053', 'vk6.3178', 'vk6.12043', 'vk6.13036', 'vk6.20483', 'vk6.20998', 'vk6.21838', 'vk6.22421', 'vk6.27876', 'vk6.28453', 'vk6.29386', 'vk6.32687', 'vk6.39314', 'vk6.40218', 'vk6.41494', 'vk6.46721', 'vk6.46870', 'vk6.57351']
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,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,1,1,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1121', '7.40804']

Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']

Outer characteristic polynomial of the knot is: t^7+22t^5+30t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1121', '7.40804']

2-strand cable arrow polynomial of the knot is: -512*K1**6 - 2816*K1**4*K2**2 + 4864*K1**4*K2 - 6048*K1**4 + 1984*K1**3*K2*K3 - 1152*K1**3*K3 - 2112*K1**2*K2**4 + 6016*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 13056*K1**2*K2**2 - 1344*K1**2*K2*K4 + 9256*K1**2*K2 - 320*K1**2*K3**2 - 708*K1**2 + 2432*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 480*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6840*K1*K2*K3 + 344*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3560*K2**4 - 592*K2**2*K3**2 - 48*K2**2*K4**2 + 2160*K2**2*K4 - 478*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 652*K3**2 - 158*K4**2 - 24*K5**2 - 2*K6**2 + 2036

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1121']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.484', 'vk6.554', 'vk6.615', 'vk6.954', 'vk6.1051', 'vk6.1121', 'vk6.1641', 'vk6.1754', 'vk6.1832', 'vk6.2138', 'vk6.2235', 'vk6.2304', 'vk6.2576', 'vk6.2850', 'vk6.3053', 'vk6.3178', 'vk6.12043', 'vk6.13036', 'vk6.20483', 'vk6.20998', 'vk6.21838', 'vk6.22421', 'vk6.27876', 'vk6.28453', 'vk6.29386', 'vk6.32687', 'vk6.39314', 'vk6.40218', 'vk6.41494', 'vk6.46721', 'vk6.46870', 'vk6.57351']

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	O1O2O3O4U3U4O5U1U2O6U5U6
R3 orbit	{'O1O2O3O4U3U4O5U1U2O6U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U6O5U3U4O6U1U2
Gauss code of K*	O1O2U3O4O5U6O3O6U4U5U1U2
Gauss code of -K*	O1O2U1O3O4U2O5O6U5U6U3U4
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 0 -1 1 1 1],[ 2 0 1 -1 1 2 1],[ 0 -1 0 -1 1 1 1],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[-1 -2 -1 0 0 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 0 -2],[-1 -1 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 1 1 0 -1 -1],[ 1 0 0 1 1 0 1],[ 2 2 1 1 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,0,1,0,2,0,1,0,1,1,1,1,1,1,-1]
Phi over symmetry	[-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,1,1,1,0,-1,0]
Phi of -K	[-2,-1,0,1,1,1,2,1,1,2,2,0,2,1,2,0,0,0,0,-1,0]
Phi of K*	[-1,-1,-1,0,1,2,-1,0,0,2,2,0,0,2,1,0,1,2,0,1,2]
Phi of -K*	[-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,1,1,1,0,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	6w^3z^2+25w^2z+27w
Inner characteristic polynomial	t^6+14t^4+15t^2
Outer characteristic polynomial	t^7+22t^5+30t^3+5t
Flat arrow polynomial	4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	-512K16 - 2816K1*4K2*2 + 4864K1*4K2 - 6048K14 + 1984K1*3K2K3 - 1152K1*3K3 - 2112K12K2*4 + 6016K1*2K2*3 + 256K1*2K2*2K4 - 13056K12K2*2 - 1344K1*2K2K4 + 9256K1*2K2 - 320K12K3*2 - 708K1*2 + 2432K1K23K3 - 2272K1K2*2K3 - 480K1K2*2K5 - 160K1K2K3K4 + 6840K1K2K3 + 344K1K3K4 + 16K1K4K5 - 32K2*6 + 64K2*4K4 - 3560K24 - 592K2*2K3*2 - 48K2*2K4*2 + 2160K2*2K4 - 478K22 + 240K2K3K5 + 16K2K4K6 - 652K3*2 - 158K4*2 - 24K5*2 - 2K6**2 + 2036
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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