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Flat knot 6.1553

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,0,0,0,1,0,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1553']
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+19t^5+16t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1553']
2-strand cable arrow polynomial of the knot is: -1264K14 + 64K1*3K2K3 - 512K1*3K3 + 128K12K2*3 - 1744K1*2K2*2 - 352K1*2K2K4 + 4048K1*2K2 - 272K12K3*2 - 2736K1*2 + 96K1K23K3 - 32K1K2*2K3 - 192K1K2K3K4 + 3304K1K2K3 + 520K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 312K24 - 176K2*2K3*2 - 48K2*2K4*2 + 472K2*2K4 - 2062K22 + 200K2K3K5 + 16K2K4K6 - 1084K3*2 - 270K4*2 - 44K5*2 - 2K6**2 + 2172
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1553']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71382', 'vk6.71441', 'vk6.71908', 'vk6.71967', 'vk6.72448', 'vk6.72610', 'vk6.72727', 'vk6.72811', 'vk6.72874', 'vk6.73042', 'vk6.73348', 'vk6.73511', 'vk6.74249', 'vk6.74379', 'vk6.74430', 'vk6.75044', 'vk6.75520', 'vk6.75837', 'vk6.76426', 'vk6.76619', 'vk6.77039', 'vk6.77750', 'vk6.77801', 'vk6.78232', 'vk6.78477', 'vk6.78638', 'vk6.78833', 'vk6.79297', 'vk6.79417', 'vk6.79835', 'vk6.79879', 'vk6.80266', 'vk6.80762', 'vk6.80865', 'vk6.85163', 'vk6.86521', 'vk6.87203', 'vk6.87343', 'vk6.89251', 'vk6.89424']
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: [-2,0,0,0,1,1,0,1,1,1,2,0,0,0,1,0,0,1,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1553']

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+19t^5+16t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1553']

2-strand cable arrow polynomial of the knot is: -1264*K1**4 + 64*K1**3*K2*K3 - 512*K1**3*K3 + 128*K1**2*K2**3 - 1744*K1**2*K2**2 - 352*K1**2*K2*K4 + 4048*K1**2*K2 - 272*K1**2*K3**2 - 2736*K1**2 + 96*K1*K2**3*K3 - 32*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 3304*K1*K2*K3 + 520*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 312*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 472*K2**2*K4 - 2062*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1084*K3**2 - 270*K4**2 - 44*K5**2 - 2*K6**2 + 2172

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1553']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71382', 'vk6.71441', 'vk6.71908', 'vk6.71967', 'vk6.72448', 'vk6.72610', 'vk6.72727', 'vk6.72811', 'vk6.72874', 'vk6.73042', 'vk6.73348', 'vk6.73511', 'vk6.74249', 'vk6.74379', 'vk6.74430', 'vk6.75044', 'vk6.75520', 'vk6.75837', 'vk6.76426', 'vk6.76619', 'vk6.77039', 'vk6.77750', 'vk6.77801', 'vk6.78232', 'vk6.78477', 'vk6.78638', 'vk6.78833', 'vk6.79297', 'vk6.79417', 'vk6.79835', 'vk6.79879', 'vk6.80266', 'vk6.80762', 'vk6.80865', 'vk6.85163', 'vk6.86521', 'vk6.87203', 'vk6.87343', 'vk6.89251', 'vk6.89424']

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	O1O2O3U4O5U3U1O4O6U5U2U6
R3 orbit	{'O1O2O3U4U2O5U1O4O6U3U5U6', 'O1O2O3U4O5U3U1O4O6U5U2U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3U4U2U5O4O6U3U1O5U6
Gauss code of K*	O1O2O3U4U2U5O6U1O5O4U6U3
Gauss code of -K*	O1O2O3U1U4O5O6U3O4U6U2U5
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 -1 0 0 -1 0 2],[ 1 0 1 0 0 1 2],[ 0 -1 0 0 0 0 2],[ 0 0 0 0 0 0 1],[ 1 0 0 0 0 0 1],[ 0 -1 0 0 0 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 0 0 0 -1 -1],[-2 0 -1 -1 -2 -1 -2],[ 0 1 0 0 0 0 0],[ 0 1 0 0 0 0 -1],[ 0 2 0 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 2 0 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,0,0,0,1,1,1,1,2,1,2,0,0,0,0,0,0,1,0,1,0]
Phi over symmetry	[-2,0,0,0,1,1,0,1,1,1,2,0,0,0,1,0,0,1,1,1,0]
Phi of -K	[-1,-1,0,0,0,2,0,0,0,1,1,1,1,1,2,0,0,0,0,1,1]
Phi of K*	[-2,0,0,0,1,1,0,1,1,1,2,0,0,0,1,0,0,1,1,1,0]
Phi of -K*	[-1,-1,0,0,0,2,0,0,0,0,1,0,1,1,2,0,0,1,0,1,2]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	z^2+14z+25
Enhanced Jones-Krushkal polynomial	w^3z^2+14w^2z+25w
Inner characteristic polynomial	t^6+13t^4+5t^2
Outer characteristic polynomial	t^7+19t^5+16t^3+3t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-1264K14 + 64K1*3K2K3 - 512K1*3K3 + 128K12K2*3 - 1744K1*2K2*2 - 352K1*2K2K4 + 4048K1*2K2 - 272K12K3*2 - 2736K1*2 + 96K1K23K3 - 32K1K2*2K3 - 192K1K2K3K4 + 3304K1K2K3 + 520K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 312K24 - 176K2*2K3*2 - 48K2*2K4*2 + 472K2*2K4 - 2062K22 + 200K2K3K5 + 16K2K4K6 - 1084K3*2 - 270K4*2 - 44K5*2 - 2K6**2 + 2172
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}], [{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}], [{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}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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