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Flat knot 6.361

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-2,0,2,2,2,1,2,2,1,0,1,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.361']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']
Outer characteristic polynomial of the knot is: t^7+45t^5+66t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.361']
2-strand cable arrow polynomial of the knot is: -512K14K2*2 + 1056K1*4K2 - 1568K14 - 128K1*3K2*2K3 + 384K13K2K3 - 224K1*3K3 + 1888K12K2*3 - 7024K1*2K2*2 - 672K1*2K2K4 + 6536K1*2K2 - 128K12K3*2 - 4476K1*2 + 896K1K23K3 + 128K1K2*2K3K4 - 1024K1K22K3 - 160K1K2K3K4 + 7072K1K2K3 + 784K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 2088K24 - 1504K2*2K3*2 - 176K2*2K4*2 + 1848K2*2K4 - 2918K22 + 760K2K3K5 + 48K2K4K6 - 2012K3*2 - 638K4*2 - 96K5*2 - 2K6**2 + 3892
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.361']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3614', 'vk6.3685', 'vk6.3876', 'vk6.3997', 'vk6.7036', 'vk6.7073', 'vk6.7248', 'vk6.7367', 'vk6.17707', 'vk6.17754', 'vk6.24254', 'vk6.24313', 'vk6.36557', 'vk6.36632', 'vk6.43663', 'vk6.43768', 'vk6.48242', 'vk6.48309', 'vk6.48392', 'vk6.48421', 'vk6.49998', 'vk6.50027', 'vk6.50110', 'vk6.50141', 'vk6.55739', 'vk6.55794', 'vk6.60311', 'vk6.60392', 'vk6.65439', 'vk6.65466', 'vk6.68567', 'vk6.68594']
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,-2,0,2,2,2,1,2,2,1,0,1,1,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.361']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']

Outer characteristic polynomial of the knot is: t^7+45t^5+66t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.361']

2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 1056*K1**4*K2 - 1568*K1**4 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 224*K1**3*K3 + 1888*K1**2*K2**3 - 7024*K1**2*K2**2 - 672*K1**2*K2*K4 + 6536*K1**2*K2 - 128*K1**2*K3**2 - 4476*K1**2 + 896*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 7072*K1*K2*K3 + 784*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2088*K2**4 - 1504*K2**2*K3**2 - 176*K2**2*K4**2 + 1848*K2**2*K4 - 2918*K2**2 + 760*K2*K3*K5 + 48*K2*K4*K6 - 2012*K3**2 - 638*K4**2 - 96*K5**2 - 2*K6**2 + 3892

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.361']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3614', 'vk6.3685', 'vk6.3876', 'vk6.3997', 'vk6.7036', 'vk6.7073', 'vk6.7248', 'vk6.7367', 'vk6.17707', 'vk6.17754', 'vk6.24254', 'vk6.24313', 'vk6.36557', 'vk6.36632', 'vk6.43663', 'vk6.43768', 'vk6.48242', 'vk6.48309', 'vk6.48392', 'vk6.48421', 'vk6.49998', 'vk6.50027', 'vk6.50110', 'vk6.50141', 'vk6.55739', 'vk6.55794', 'vk6.60311', 'vk6.60392', 'vk6.65439', 'vk6.65466', 'vk6.68567', 'vk6.68594']

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	O1O2O3O4O5U3U4O6U5U1U6U2
R3 orbit	{'O1O2O3O4O5U3U4O6U5U1U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U6U5U1O6U2U3
Gauss code of K*	O1O2O3O4U2U4U5U6U1O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U4U5U6U1U3
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 -2 0 1 2],[ 2 0 2 -2 0 2 2],[-1 -2 0 -2 0 1 1],[ 2 2 2 0 1 2 1],[ 0 0 0 -1 0 1 1],[-1 -2 -1 -2 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 -1 -1 -1 -1 -2],[-1 1 0 1 0 -2 -2],[-1 1 -1 0 -1 -2 -2],[ 0 1 0 1 0 -1 0],[ 2 1 2 2 1 0 2],[ 2 2 2 2 0 -2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,1,1,1,1,2,-1,0,2,2,1,2,2,1,0,-2]
Phi over symmetry	[-2,-2,0,1,1,2,-2,0,2,2,2,1,2,2,1,0,1,1,1,1,1]
Phi of -K	[-2,-2,0,1,1,2,-2,1,1,1,3,2,1,1,2,0,1,1,1,0,0]
Phi of K*	[-2,-1,-1,0,2,2,0,0,1,2,3,-1,0,1,1,1,1,1,2,1,-2]
Phi of -K*	[-2,-2,0,1,1,2,-2,0,2,2,2,1,2,2,1,0,1,1,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+31t^4+23t^2
Outer characteristic polynomial	t^7+45t^5+66t^3+13t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-512K14K2*2 + 1056K1*4K2 - 1568K14 - 128K1*3K2*2K3 + 384K13K2K3 - 224K1*3K3 + 1888K12K2*3 - 7024K1*2K2*2 - 672K1*2K2K4 + 6536K1*2K2 - 128K12K3*2 - 4476K1*2 + 896K1K23K3 + 128K1K2*2K3K4 - 1024K1K22K3 - 160K1K2K3K4 + 7072K1K2K3 + 784K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 2088K24 - 1504K2*2K3*2 - 176K2*2K4*2 + 1848K2*2K4 - 2918K22 + 760K2K3K5 + 48K2K4K6 - 2012K3*2 - 638K4*2 - 96K5*2 - 2K6**2 + 3892
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}], [{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}], [{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}, {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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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