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Flat knot 6.1663

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1663']
Arrow polynomial of the knot is: -4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']
Outer characteristic polynomial of the knot is: t^7+32t^5+31t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1663']
2-strand cable arrow polynomial of the knot is: -2992K14 + 864K1*3K2K3 + 32K1*3K3K4 - 416K1*3K3 + 224K12K2*2K4 - 3440K12K2*2 - 736K1*2K2K4 + 6808K1*2K2 - 1360K12K3*2 - 96K1*2K3K5 - 128K1*2K4*2 - 3568K1*2 + 96K1K22K3K4 - 1312K1K22K3 - 128K1K2*2K5 - 256K1K2K3K4 + 5864K1K2K3 - 96K1K2K4K5 + 2432K1K3K4 + 312K1K4K5 + 24K1K5K6 - 112K24 - 144K2*2K3*2 - 112K2*2K4*2 + 1064K2*2K4 - 3572K22 + 360K2K3K5 + 112K2K4K6 - 2164K3*2 - 992K4*2 - 180K5*2 - 28K6**2 + 3638
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1663']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13903', 'vk6.14000', 'vk6.14175', 'vk6.14414', 'vk6.14978', 'vk6.15101', 'vk6.15643', 'vk6.16097', 'vk6.16721', 'vk6.16750', 'vk6.16837', 'vk6.18803', 'vk6.19288', 'vk6.19580', 'vk6.23159', 'vk6.23216', 'vk6.25401', 'vk6.26475', 'vk6.33722', 'vk6.33799', 'vk6.34282', 'vk6.35153', 'vk6.37530', 'vk6.42715', 'vk6.44707', 'vk6.54114', 'vk6.54926', 'vk6.54953', 'vk6.56405', 'vk6.56611', 'vk6.59350', 'vk6.64592']
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,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,2,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.1663']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']

Outer characteristic polynomial of the knot is: t^7+32t^5+31t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1663']

2-strand cable arrow polynomial of the knot is: -2992*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 416*K1**3*K3 + 224*K1**2*K2**2*K4 - 3440*K1**2*K2**2 - 736*K1**2*K2*K4 + 6808*K1**2*K2 - 1360*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 3568*K1**2 + 96*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5864*K1*K2*K3 - 96*K1*K2*K4*K5 + 2432*K1*K3*K4 + 312*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**4 - 144*K2**2*K3**2 - 112*K2**2*K4**2 + 1064*K2**2*K4 - 3572*K2**2 + 360*K2*K3*K5 + 112*K2*K4*K6 - 2164*K3**2 - 992*K4**2 - 180*K5**2 - 28*K6**2 + 3638

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1663']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13903', 'vk6.14000', 'vk6.14175', 'vk6.14414', 'vk6.14978', 'vk6.15101', 'vk6.15643', 'vk6.16097', 'vk6.16721', 'vk6.16750', 'vk6.16837', 'vk6.18803', 'vk6.19288', 'vk6.19580', 'vk6.23159', 'vk6.23216', 'vk6.25401', 'vk6.26475', 'vk6.33722', 'vk6.33799', 'vk6.34282', 'vk6.35153', 'vk6.37530', 'vk6.42715', 'vk6.44707', 'vk6.54114', 'vk6.54926', 'vk6.54953', 'vk6.56405', 'vk6.56611', 'vk6.59350', 'vk6.64592']

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	O1O2O3U2O4U1U5U3O5O6U4U6
R3 orbit	{'O1O2O3U2O4U1U5U3O5O6U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O4O6U1U6U3O5U2
Gauss code of K*	O1O2O3U2U4O5O4U1U6U3O6U5
Gauss code of -K*	O1O2O3U4O5U1U5U3O6O4U6U2
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 2 1 -1 1],[ 2 0 0 2 2 1 1],[ 1 0 0 1 1 0 0],[-2 -2 -1 0 0 -2 1],[-1 -2 -1 0 0 -1 1],[ 1 -1 0 2 1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 -1 -1 -2],[ 1 1 0 1 0 0 0],[ 1 2 1 1 0 0 -1],[ 2 2 1 2 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,2,0,0,1]
Phi over symmetry	[-2,-1,-1,1,1,2,-1,0,1,2,2,1,0,1,1,1,1,2,0,0,1]
Phi of -K	[-2,-1,-1,1,1,2,0,1,1,2,2,0,1,1,1,1,2,2,-1,1,2]
Phi of K*	[-2,-1,-1,1,1,2,1,2,1,2,2,1,1,1,1,1,2,2,0,0,1]
Phi of -K*	[-2,-1,-1,1,1,2,0,1,1,2,2,0,0,1,1,1,1,2,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	0
Normalized Jones-Krushkal polynomial	5z^2+24z+29
Enhanced Jones-Krushkal polynomial	5w^3z^2+24w^2z+29w
Inner characteristic polynomial	t^6+20t^4+13t^2+1
Outer characteristic polynomial	t^7+32t^5+31t^3+6t
Flat arrow polynomial	-4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
2-strand cable arrow polynomial	-2992K14 + 864K1*3K2K3 + 32K1*3K3K4 - 416K1*3K3 + 224K12K2*2K4 - 3440K12K2*2 - 736K1*2K2K4 + 6808K1*2K2 - 1360K12K3*2 - 96K1*2K3K5 - 128K1*2K4*2 - 3568K1*2 + 96K1K22K3K4 - 1312K1K22K3 - 128K1K2*2K5 - 256K1K2K3K4 + 5864K1K2K3 - 96K1K2K4K5 + 2432K1K3K4 + 312K1K4K5 + 24K1K5K6 - 112K24 - 144K2*2K3*2 - 112K2*2K4*2 + 1064K2*2K4 - 3572K22 + 360K2K3K5 + 112K2K4K6 - 2164K3*2 - 992K4*2 - 180K5*2 - 28K6**2 + 3638
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

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