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Flat knot 6.1531

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,0,0,1,1,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1531']
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+24t^5+28t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1531']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 672K1*4K2 - 1456K14 + 96K1*3K2K3 + 32K1*3K3K4 + 256K1*2K2*3 - 1920K1*2K2*2 + 2864K1*2K2 - 176K12K3*2 - 48K1*2K4*2 - 1640K1*2 + 64K1K23K3 + 1728K1K2K3 + 392K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 328K24 - 128K2*2K3*2 - 48K2*2K4*2 + 344K2*2K4 - 1390K22 + 112K2K3K5 + 16K2K4K6 - 644K3*2 - 286K4*2 - 36K5*2 - 2K6**2 + 1660
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1531']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4432', 'vk6.4527', 'vk6.5818', 'vk6.5945', 'vk6.7879', 'vk6.7990', 'vk6.9305', 'vk6.9424', 'vk6.10153', 'vk6.10224', 'vk6.10371', 'vk6.17893', 'vk6.17956', 'vk6.18277', 'vk6.18614', 'vk6.24400', 'vk6.24700', 'vk6.25165', 'vk6.30048', 'vk6.30109', 'vk6.30890', 'vk6.31013', 'vk6.32078', 'vk6.32197', 'vk6.36895', 'vk6.37286', 'vk6.37355', 'vk6.43835', 'vk6.44112', 'vk6.44437', 'vk6.50526', 'vk6.50611', 'vk6.51135', 'vk6.51982', 'vk6.52077', 'vk6.55844', 'vk6.56068', 'vk6.60570', 'vk6.60909', 'vk6.65979']
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,-1,1,1,2,2,0,0,1,1,1,1,1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1531']

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+24t^5+28t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1531']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 672*K1**4*K2 - 1456*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 256*K1**2*K2**3 - 1920*K1**2*K2**2 + 2864*K1**2*K2 - 176*K1**2*K3**2 - 48*K1**2*K4**2 - 1640*K1**2 + 64*K1*K2**3*K3 + 1728*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 328*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 344*K2**2*K4 - 1390*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 644*K3**2 - 286*K4**2 - 36*K5**2 - 2*K6**2 + 1660

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1531']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4432', 'vk6.4527', 'vk6.5818', 'vk6.5945', 'vk6.7879', 'vk6.7990', 'vk6.9305', 'vk6.9424', 'vk6.10153', 'vk6.10224', 'vk6.10371', 'vk6.17893', 'vk6.17956', 'vk6.18277', 'vk6.18614', 'vk6.24400', 'vk6.24700', 'vk6.25165', 'vk6.30048', 'vk6.30109', 'vk6.30890', 'vk6.31013', 'vk6.32078', 'vk6.32197', 'vk6.36895', 'vk6.37286', 'vk6.37355', 'vk6.43835', 'vk6.44112', 'vk6.44437', 'vk6.50526', 'vk6.50611', 'vk6.51135', 'vk6.51982', 'vk6.52077', 'vk6.55844', 'vk6.56068', 'vk6.60570', 'vk6.60909', 'vk6.65979']

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	O1O2O3U2O4U5U3O6O5U1U4U6
R3 orbit	{'O1O2O3U2U4O5U3O4O6U1U6U5', 'O1O2O3U2O4U5U3O6O5U1U4U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3U4U5U3O6O4U1U6O5U2
Gauss code of K*	O1O2O3U1U4U5O4U2O6O5U3U6
Gauss code of -K*	O1O2O3U4U1O5O4U2O6U5U6U3
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 -1 1 1 0 1],[ 2 0 -1 2 2 1 1],[ 1 1 0 1 1 0 0],[-1 -2 -1 0 0 -1 0],[-1 -2 -1 0 0 -1 0],[ 0 -1 0 1 1 0 1],[-1 -1 0 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -2],[-1 0 0 0 -1 -1 -2],[ 0 1 1 1 0 0 -1],[ 1 0 1 1 0 0 1],[ 2 1 2 2 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,0,0,1,0,1,0,1,1,2,1,1,2,0,1,-1]
Phi over symmetry	[-2,-1,0,1,1,1,-1,1,1,2,2,0,0,1,1,1,1,1,0,0,0]
Phi of -K	[-2,-1,0,1,1,1,2,1,1,1,2,1,1,1,2,0,0,0,0,0,0]
Phi of K*	[-1,-1,-1,0,1,2,0,0,0,1,1,0,0,1,1,0,2,2,1,1,2]
Phi of -K*	[-2,-1,0,1,1,1,-1,1,1,2,2,0,0,1,1,1,1,1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	11z+23
Enhanced Jones-Krushkal polynomial	-2w^3z+13w^2z+23w
Inner characteristic polynomial	t^6+16t^4+15t^2
Outer characteristic polynomial	t^7+24t^5+28t^3
Flat arrow polynomial	4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	-192K14K2*2 + 672K1*4K2 - 1456K14 + 96K1*3K2K3 + 32K1*3K3K4 + 256K1*2K2*3 - 1920K1*2K2*2 + 2864K1*2K2 - 176K12K3*2 - 48K1*2K4*2 - 1640K1*2 + 64K1K23K3 + 1728K1K2K3 + 392K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 328K24 - 128K2*2K3*2 - 48K2*2K4*2 + 344K2*2K4 - 1390K22 + 112K2K3K5 + 16K2K4K6 - 644K3*2 - 286K4*2 - 36K5*2 - 2K6**2 + 1660
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}], [{6}, {5}, {4}, {3}, {2}, {1}]]
If K is slice	False

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