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Flat knot 6.1839

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,1,0,1,2,1,0,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1839']
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+26t^5+59t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1839']
2-strand cable arrow polynomial of the knot is: 160K14K2 - 464K14 - 224K1*3K3 + 480K12K2*3 - 3536K1*2K2*2 + 64K1*2K2K32 - 32K1*2K2K4 + 5344K1*2K2 - 112K12K3*2 - 4776K1*2 + 384K1K23K3 - 704K1K2*2K3 - 128K1K2*2K5 - 320K1K2K3K4 + 5280K1K2K3 + 408K1K3K4 + 160K1K4K5 - 288K2*6 + 448K2*4K4 - 2136K24 - 128K2*3K6 - 592K22K3*2 - 112K2*2K4*2 + 2200K2*2K4 - 3222K22 + 608K2K3K5 + 48K2K4K6 - 1820K3*2 - 638K4*2 - 180K5*2 - 2K6**2 + 3828
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1839']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73765', 'vk6.73769', 'vk6.73794', 'vk6.73804', 'vk6.73904', 'vk6.73908', 'vk6.73929', 'vk6.73938', 'vk6.75740', 'vk6.75748', 'vk6.75908', 'vk6.75910', 'vk6.78698', 'vk6.78708', 'vk6.78747', 'vk6.78762', 'vk6.78897', 'vk6.78910', 'vk6.80322', 'vk6.80328', 'vk6.80353', 'vk6.80362', 'vk6.80446', 'vk6.80452', 'vk6.81716', 'vk6.81724', 'vk6.82489', 'vk6.82497', 'vk6.84456', 'vk6.84468', 'vk6.88360', 'vk6.88363']
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,-1,1,1,2,2,1,0,1,2,1,0,1,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.1839']

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+26t^5+59t^3+15t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1839']

2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 464*K1**4 - 224*K1**3*K3 + 480*K1**2*K2**3 - 3536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 5344*K1**2*K2 - 112*K1**2*K3**2 - 4776*K1**2 + 384*K1*K2**3*K3 - 704*K1*K2**2*K3 - 128*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 5280*K1*K2*K3 + 408*K1*K3*K4 + 160*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2136*K2**4 - 128*K2**3*K6 - 592*K2**2*K3**2 - 112*K2**2*K4**2 + 2200*K2**2*K4 - 3222*K2**2 + 608*K2*K3*K5 + 48*K2*K4*K6 - 1820*K3**2 - 638*K4**2 - 180*K5**2 - 2*K6**2 + 3828

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1839']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73765', 'vk6.73769', 'vk6.73794', 'vk6.73804', 'vk6.73904', 'vk6.73908', 'vk6.73929', 'vk6.73938', 'vk6.75740', 'vk6.75748', 'vk6.75908', 'vk6.75910', 'vk6.78698', 'vk6.78708', 'vk6.78747', 'vk6.78762', 'vk6.78897', 'vk6.78910', 'vk6.80322', 'vk6.80328', 'vk6.80353', 'vk6.80362', 'vk6.80446', 'vk6.80452', 'vk6.81716', 'vk6.81724', 'vk6.82489', 'vk6.82497', 'vk6.84456', 'vk6.84468', 'vk6.88360', 'vk6.88363']

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	O1O2O3U1U4O5U3O6O4U2U6U5
R3 orbit	{'O1O2O3U1U4O5U3O6O4U2U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5U2O6O5U1O4U6U3
Gauss code of K*	O1O2O3U4U1U5O4O6U3O5U2U6
Gauss code of -K*	O1O2O3U4U2O5U1O4O6U5U3U6
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 1 1 1 0],[ 2 0 2 1 1 2 1],[ 1 -2 0 1 0 2 0],[-1 -1 -1 0 -1 0 -1],[-1 -1 0 1 0 0 0],[-1 -2 -2 0 0 0 0],[ 0 -1 0 1 0 0 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 0 -1],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 -2 -2],[ 0 0 1 0 0 0 -1],[ 1 0 1 2 0 0 -2],[ 2 1 1 2 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,0,0,0,1,0,1,1,1,0,2,2,0,1,2]
Phi over symmetry	[-2,-1,0,1,1,1,-1,1,1,2,2,1,0,1,2,1,0,1,0,0,1]
Phi of -K	[-2,-1,0,1,1,1,-1,1,1,2,2,1,0,1,2,1,0,1,0,0,1]
Phi of K*	[-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,2,2,1,0,1,1,1,-1]
Phi of -K*	[-2,-1,0,1,1,1,2,1,1,1,2,0,0,1,2,0,1,0,1,0,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	-2w^4z^2+4w^3z^2-8w^3z+23w^2z+23w
Inner characteristic polynomial	t^6+18t^4+26t^2
Outer characteristic polynomial	t^7+26t^5+59t^3+15t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	160K14K2 - 464K14 - 224K1*3K3 + 480K12K2*3 - 3536K1*2K2*2 + 64K1*2K2K32 - 32K1*2K2K4 + 5344K1*2K2 - 112K12K3*2 - 4776K1*2 + 384K1K23K3 - 704K1K2*2K3 - 128K1K2*2K5 - 320K1K2K3K4 + 5280K1K2K3 + 408K1K3K4 + 160K1K4K5 - 288K2*6 + 448K2*4K4 - 2136K24 - 128K2*3K6 - 592K22K3*2 - 112K2*2K4*2 + 2200K2*2K4 - 3222K22 + 608K2K3K5 + 48K2K4K6 - 1820K3*2 - 638K4*2 - 180K5*2 - 2K6**2 + 3828
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}], [{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}], [{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}, {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}, {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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