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Flat knot 6.1798

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,0,1,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1798', '7.42770']
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+22t^5+42t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1798']
2-strand cable arrow polynomial of the knot is: -896K14K2*2 + 1920K1*4K2 - 4544K14 + 2176K1*3K2K3 - 1312K1*3K3 - 832K12K2*4 + 2048K1*2K2*3 + 384K1*2K2*2K4 - 9184K12K2*2 - 1216K1*2K2K4 + 8832K1*2K2 - 1504K12K3*2 - 96K1*2K3K5 - 2420K1*2 + 2144K1K23K3 + 96K1K2*2K3K4 - 1472K1K22K3 - 608K1K2*2K5 - 448K1K2K3K4 - 96K1K2K3K6 + 7824K1K2K3 + 1208K1K3K4 + 72K1K4K5 - 32K2*6 + 96K2*4K4 - 1912K24 - 32K2*3K6 - 1328K22K3*2 - 112K2*2K4*2 + 1464K2*2K4 - 2094K22 + 776K2K3K5 + 88K2K4K6 + 24K3*2K6 - 1492K32 - 266K4*2 - 56K5*2 - 18K6**2 + 2816
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1798']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.330', 'vk6.369', 'vk6.446', 'vk6.725', 'vk6.774', 'vk6.893', 'vk6.1468', 'vk6.1525', 'vk6.1594', 'vk6.1964', 'vk6.2003', 'vk6.2071', 'vk6.2498', 'vk6.2751', 'vk6.3015', 'vk6.3136', 'vk6.3777', 'vk6.3968', 'vk6.7169', 'vk6.7344', 'vk6.18784', 'vk6.19857', 'vk6.24910', 'vk6.25371', 'vk6.25928', 'vk6.26296', 'vk6.26741', 'vk6.38003', 'vk6.38058', 'vk6.45031', 'vk6.50091', 'vk6.60759']
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,-1,0,1,2,2,0,1,0,1,0,0,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1798', '7.42770']

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+22t^5+42t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1798']

2-strand cable arrow polynomial of the knot is: -896*K1**4*K2**2 + 1920*K1**4*K2 - 4544*K1**4 + 2176*K1**3*K2*K3 - 1312*K1**3*K3 - 832*K1**2*K2**4 + 2048*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9184*K1**2*K2**2 - 1216*K1**2*K2*K4 + 8832*K1**2*K2 - 1504*K1**2*K3**2 - 96*K1**2*K3*K5 - 2420*K1**2 + 2144*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1472*K1*K2**2*K3 - 608*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7824*K1*K2*K3 + 1208*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1912*K2**4 - 32*K2**3*K6 - 1328*K2**2*K3**2 - 112*K2**2*K4**2 + 1464*K2**2*K4 - 2094*K2**2 + 776*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1492*K3**2 - 266*K4**2 - 56*K5**2 - 18*K6**2 + 2816

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1798']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.330', 'vk6.369', 'vk6.446', 'vk6.725', 'vk6.774', 'vk6.893', 'vk6.1468', 'vk6.1525', 'vk6.1594', 'vk6.1964', 'vk6.2003', 'vk6.2071', 'vk6.2498', 'vk6.2751', 'vk6.3015', 'vk6.3136', 'vk6.3777', 'vk6.3968', 'vk6.7169', 'vk6.7344', 'vk6.18784', 'vk6.19857', 'vk6.24910', 'vk6.25371', 'vk6.25928', 'vk6.26296', 'vk6.26741', 'vk6.38003', 'vk6.38058', 'vk6.45031', 'vk6.50091', 'vk6.60759']

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	O1O2O3U4U2O5O6U5U6O4U1U3
R3 orbit	{'O1O2O3U4U2O5O6U5U6O4U1U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3U1U3O4U5U6O5O6U2U4
Gauss code of K*	O1O2U3O4O5U4U6U5O3O6U1U2
Gauss code of -K*	O1O2U3O4O5U4U5O6O3U1U6U2
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 2 -1 -1 1],[ 1 0 1 2 0 -1 1],[ 0 -1 0 0 0 0 0],[-2 -2 0 0 -2 -1 1],[ 1 0 0 2 0 0 0],[ 1 1 0 1 0 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 1 0 -1 -2 -2],[-1 -1 0 0 -1 0 -1],[ 0 0 0 0 0 0 -1],[ 1 1 1 0 0 0 1],[ 1 2 0 0 0 0 0],[ 1 2 1 1 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,-1,0,1,2,2,0,1,0,1,0,0,1,0,-1,0]
Phi over symmetry	[-2,-1,0,1,1,1,-1,0,1,2,2,0,1,0,1,0,0,1,0,-1,0]
Phi of -K	[-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,1,2,1,1,2,2]
Phi of K*	[-2,-1,0,1,1,1,2,2,1,1,2,1,1,2,1,0,1,1,0,-1,0]
Phi of -K*	[-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,1,1,0,0,2,0,0,-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+14t^4+23t^2+4
Outer characteristic polynomial	t^7+22t^5+42t^3+11t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-896K14K2*2 + 1920K1*4K2 - 4544K14 + 2176K1*3K2K3 - 1312K1*3K3 - 832K12K2*4 + 2048K1*2K2*3 + 384K1*2K2*2K4 - 9184K12K2*2 - 1216K1*2K2K4 + 8832K1*2K2 - 1504K12K3*2 - 96K1*2K3K5 - 2420K1*2 + 2144K1K23K3 + 96K1K2*2K3K4 - 1472K1K22K3 - 608K1K2*2K5 - 448K1K2K3K4 - 96K1K2K3K6 + 7824K1K2K3 + 1208K1K3K4 + 72K1K4K5 - 32K2*6 + 96K2*4K4 - 1912K24 - 32K2*3K6 - 1328K22K3*2 - 112K2*2K4*2 + 1464K2*2K4 - 2094K22 + 776K2K3K5 + 88K2K4K6 + 24K3*2K6 - 1492K32 - 266K4*2 - 56K5*2 - 18K6**2 + 2816
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice	False

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