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Flat knot 6.1586

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,1,-1,0,-1,1,1,1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1586']
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+17t^5+49t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1586']
2-strand cable arrow polynomial of the knot is: 128K14K2*3 - 576K1*4K2*2 + 768K1*4K2 - 1632K14 - 128K1*3K2*2K3 + 416K13K2K3 - 480K1*3K3 - 768K12K2*4 + 3296K1*2K2*3 - 8128K1*2K2*2 + 32K1*2K2K32 - 672K1*2K2K4 + 8392K1*2K2 - 160K12K3*2 - 5340K1*2 + 1088K1K23K3 - 512K1K2*2K3 - 96K1K2*2K5 - 32K1K2K3K4 + 5576K1K2K3 + 264K1K3K4 - 32K26 + 64K2*4K4 - 1848K24 - 432K2*2K3*2 - 48K2*2K4*2 + 752K2*2K4 - 2334K22 + 128K2K3K5 + 16K2K4K6 - 1120K3*2 - 150K4*2 - 12K5*2 - 2K6**2 + 3580
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1586']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4745', 'vk6.5074', 'vk6.6283', 'vk6.6724', 'vk6.8244', 'vk6.8695', 'vk6.9634', 'vk6.9951', 'vk6.20401', 'vk6.21752', 'vk6.27739', 'vk6.29275', 'vk6.39179', 'vk6.41413', 'vk6.45911', 'vk6.47546', 'vk6.48785', 'vk6.48998', 'vk6.49601', 'vk6.49806', 'vk6.50801', 'vk6.51018', 'vk6.51284', 'vk6.51481', 'vk6.57270', 'vk6.58497', 'vk6.61922', 'vk6.63021', 'vk6.66887', 'vk6.67771', 'vk6.69523', 'vk6.70231']
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: [-2,0,0,0,1,1,0,1,1,1,1,-1,0,-1,1,1,1,1,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1586']

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+17t^5+49t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1586']

2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 576*K1**4*K2**2 + 768*K1**4*K2 - 1632*K1**4 - 128*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 480*K1**3*K3 - 768*K1**2*K2**4 + 3296*K1**2*K2**3 - 8128*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8392*K1**2*K2 - 160*K1**2*K3**2 - 5340*K1**2 + 1088*K1*K2**3*K3 - 512*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5576*K1*K2*K3 + 264*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 1848*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 752*K2**2*K4 - 2334*K2**2 + 128*K2*K3*K5 + 16*K2*K4*K6 - 1120*K3**2 - 150*K4**2 - 12*K5**2 - 2*K6**2 + 3580

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1586']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4745', 'vk6.5074', 'vk6.6283', 'vk6.6724', 'vk6.8244', 'vk6.8695', 'vk6.9634', 'vk6.9951', 'vk6.20401', 'vk6.21752', 'vk6.27739', 'vk6.29275', 'vk6.39179', 'vk6.41413', 'vk6.45911', 'vk6.47546', 'vk6.48785', 'vk6.48998', 'vk6.49601', 'vk6.49806', 'vk6.50801', 'vk6.51018', 'vk6.51284', 'vk6.51481', 'vk6.57270', 'vk6.58497', 'vk6.61922', 'vk6.63021', 'vk6.66887', 'vk6.67771', 'vk6.69523', 'vk6.70231']

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	O1O2O3U4O5U3U5O6O4U1U2U6
R3 orbit	{'O1O2O3U4O5U3U5O6O4U1U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U2U3O5O4U6U1O6U5
Gauss code of K*	O1O2O3U1U2U4O5U6O4O6U3U5
Gauss code of -K*	O1O2O3U4U1O5O6U5O4U6U2U3
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 0 0 1 1],[ 2 0 1 0 1 1 1],[ 0 -1 0 0 -1 1 0],[ 0 0 0 0 -1 1 -1],[ 0 -1 1 1 0 1 1],[-1 -1 -1 -1 -1 0 0],[-1 -1 0 1 -1 0 0]]
Primitive based matrix	[[ 0 1 1 0 0 0 -2],[-1 0 0 1 0 -1 -1],[-1 0 0 -1 -1 -1 -1],[ 0 -1 1 0 0 -1 0],[ 0 0 1 0 0 -1 -1],[ 0 1 1 1 1 0 -1],[ 2 1 1 0 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,0,0,0,2,0,-1,0,1,1,1,1,1,1,0,1,0,1,1,1]
Phi over symmetry	[-2,0,0,0,1,1,0,1,1,1,1,-1,0,-1,1,1,1,1,0,1,0]
Phi of -K	[-2,0,0,0,1,1,1,1,2,2,2,-1,-1,0,0,0,0,1,0,2,0]
Phi of K*	[-1,-1,0,0,0,2,0,0,0,0,2,0,1,2,2,1,1,1,0,1,2]
Phi of -K*	[-2,0,0,0,1,1,0,1,1,1,1,-1,0,-1,1,1,1,1,0,1,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	3z^2+20z+29
Enhanced Jones-Krushkal polynomial	3w^3z^2-4w^3z+24w^2z+29w
Inner characteristic polynomial	t^6+11t^4+20t^2
Outer characteristic polynomial	t^7+17t^5+49t^3+8t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	128K14K2*3 - 576K1*4K2*2 + 768K1*4K2 - 1632K14 - 128K1*3K2*2K3 + 416K13K2K3 - 480K1*3K3 - 768K12K2*4 + 3296K1*2K2*3 - 8128K1*2K2*2 + 32K1*2K2K32 - 672K1*2K2K4 + 8392K1*2K2 - 160K12K3*2 - 5340K1*2 + 1088K1K23K3 - 512K1K2*2K3 - 96K1K2*2K5 - 32K1K2K3K4 + 5576K1K2K3 + 264K1K3K4 - 32K26 + 64K2*4K4 - 1848K24 - 432K2*2K3*2 - 48K2*2K4*2 + 752K2*2K4 - 2334K22 + 128K2K3K5 + 16K2K4K6 - 1120K3*2 - 150K4*2 - 12K5*2 - 2K6**2 + 3580
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}], [{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}], [{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}, {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}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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