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Flat knot 6.1672

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1672']
Arrow polynomial of the knot is: -6K12 - 4K1K2 + 2K1 + 3K2 + 2K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866']
Outer characteristic polynomial of the knot is: t^7+20t^5+20t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1672']
2-strand cable arrow polynomial of the knot is: -320K16 - 192K1*4K2*2 + 544K1*4K2 - 1552K14 + 128K1*3K2K3 - 1232K1*2K2*2 + 2096K1*2K2 - 496K12K3*2 - 112K1*2K4*2 - 512K1*2 + 1592K1K2K3 + 568K1K3K4 + 88K1K4K5 - 184K24 - 160K2*2K3*2 - 48K2*2K4*2 + 216K2*2K4 - 828K22 + 120K2K3K5 + 32K2K4K6 - 524K3*2 - 206K4*2 - 36K5*2 - 4K6**2 + 1020
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1672']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11249', 'vk6.11329', 'vk6.12510', 'vk6.12623', 'vk6.13880', 'vk6.13975', 'vk6.14134', 'vk6.14359', 'vk6.14955', 'vk6.15076', 'vk6.15586', 'vk6.16058', 'vk6.17419', 'vk6.22588', 'vk6.22621', 'vk6.23931', 'vk6.24070', 'vk6.24164', 'vk6.26141', 'vk6.26558', 'vk6.30923', 'vk6.31048', 'vk6.33699', 'vk6.33774', 'vk6.34582', 'vk6.36231', 'vk6.37659', 'vk6.37706', 'vk6.42274', 'vk6.44802', 'vk6.52015', 'vk6.52108', 'vk6.54107', 'vk6.54415', 'vk6.54578', 'vk6.56497', 'vk6.56663', 'vk6.59048', 'vk6.60057', 'vk6.64571']
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,-1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,-1,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1672']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866']

Outer characteristic polynomial of the knot is: t^7+20t^5+20t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1672']

2-strand cable arrow polynomial of the knot is: -320*K1**6 - 192*K1**4*K2**2 + 544*K1**4*K2 - 1552*K1**4 + 128*K1**3*K2*K3 - 1232*K1**2*K2**2 + 2096*K1**2*K2 - 496*K1**2*K3**2 - 112*K1**2*K4**2 - 512*K1**2 + 1592*K1*K2*K3 + 568*K1*K3*K4 + 88*K1*K4*K5 - 184*K2**4 - 160*K2**2*K3**2 - 48*K2**2*K4**2 + 216*K2**2*K4 - 828*K2**2 + 120*K2*K3*K5 + 32*K2*K4*K6 - 524*K3**2 - 206*K4**2 - 36*K5**2 - 4*K6**2 + 1020

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1672']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11249', 'vk6.11329', 'vk6.12510', 'vk6.12623', 'vk6.13880', 'vk6.13975', 'vk6.14134', 'vk6.14359', 'vk6.14955', 'vk6.15076', 'vk6.15586', 'vk6.16058', 'vk6.17419', 'vk6.22588', 'vk6.22621', 'vk6.23931', 'vk6.24070', 'vk6.24164', 'vk6.26141', 'vk6.26558', 'vk6.30923', 'vk6.31048', 'vk6.33699', 'vk6.33774', 'vk6.34582', 'vk6.36231', 'vk6.37659', 'vk6.37706', 'vk6.42274', 'vk6.44802', 'vk6.52015', 'vk6.52108', 'vk6.54107', 'vk6.54415', 'vk6.54578', 'vk6.56497', 'vk6.56663', 'vk6.59048', 'vk6.60057', 'vk6.64571']

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	O1O2O3U2O4U5U3U1O5O6U4U6
R3 orbit	{'O1O2U1O3O4U5U2U3O5O6U4U6', 'O1O2O3U2O4U5U3U1O5O6U4U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3U4U5O4O6U3U1U6O5U2
Gauss code of K*	O1O2O3U1U4O5O4U3U6U2O6U5
Gauss code of -K*	O1O2O3U4O5U2U5U1O6O4U6U3
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 0 -1 1 1 -2 1],[ 0 0 -1 1 1 -1 1],[ 1 1 0 1 1 0 0],[-1 -1 -1 0 0 -1 1],[-1 -1 -1 0 0 -1 1],[ 2 1 0 1 1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 -1 0 -1],[-1 0 1 0 -1 -1 -1],[ 0 1 1 1 0 -1 -1],[ 1 1 0 1 1 0 0],[ 2 1 1 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,0,1,1,1,1,1,0,1,1,1,1,1,1,0]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,-1,-1,0]
Phi of -K	[-2,-1,0,1,1,1,1,1,2,2,2,0,1,1,2,0,0,0,0,-1,-1]
Phi of K*	[-1,-1,-1,0,1,2,-1,-1,0,2,2,0,0,1,2,0,1,2,0,1,1]
Phi of -K*	[-2,-1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	11z+23
Enhanced Jones-Krushkal polynomial	11w^2z+23w
Inner characteristic polynomial	t^6+12t^4+5t^2
Outer characteristic polynomial	t^7+20t^5+20t^3
Flat arrow polynomial	-6K12 - 4K1K2 + 2K1 + 3K2 + 2K3 + 4
2-strand cable arrow polynomial	-320K16 - 192K1*4K2*2 + 544K1*4K2 - 1552K14 + 128K1*3K2K3 - 1232K1*2K2*2 + 2096K1*2K2 - 496K12K3*2 - 112K1*2K4*2 - 512K1*2 + 1592K1K2K3 + 568K1K3K4 + 88K1K4K5 - 184K24 - 160K2*2K3*2 - 48K2*2K4*2 + 216K2*2K4 - 828K22 + 120K2K3K5 + 32K2K4K6 - 524K3*2 - 206K4*2 - 36K5*2 - 4K6**2 + 1020
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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