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Flat knot 6.821

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,-1,1,4,3,4,0,3,1,2,-1,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.821']
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+88t^5+128t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.821']
2-strand cable arrow polynomial of the knot is: -1824K14 + 1536K1*3K2K3 + 64K1*3K3K4 - 896K1*3K3 - 128K12K2*2K3*2 + 192K1*2K2*2K4 - 4064K12K2*2 - 1216K1*2K2K4 + 6544K1*2K2 - 960K12K3*2 - 128K1*2K4*2 - 4608K1*2 + 576K1K23K3 - 896K1K2*2K3 - 320K1K2*2K5 + 128K1K2K33 + 7136K1K2K3 + 1280K1K3K4 + 32K1K4K5 - 432K24 - 288K2*2K3*2 - 16K2*2K4*2 + 1088K2*2K4 - 3964K22 + 144K2K3K5 + 16K2K4K6 - 32K3*4 - 2264K3*2 - 508K4*2 - 8K5*2 - 4K6**2 + 3818
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.821']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11656', 'vk6.12009', 'vk6.12998', 'vk6.13269', 'vk6.20439', 'vk6.21798', 'vk6.27811', 'vk6.29319', 'vk6.31459', 'vk6.32641', 'vk6.32985', 'vk6.39235', 'vk6.47562', 'vk6.52376', 'vk6.53263', 'vk6.57308', 'vk6.61997', 'vk6.64328', 'vk6.64508', 'vk6.66898']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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: [-3,-3,1,1,2,2,-1,1,4,3,4,0,3,1,2,-1,0,0,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.821']

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+88t^5+128t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.821']

2-strand cable arrow polynomial of the knot is: -1824*K1**4 + 1536*K1**3*K2*K3 + 64*K1**3*K3*K4 - 896*K1**3*K3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 4064*K1**2*K2**2 - 1216*K1**2*K2*K4 + 6544*K1**2*K2 - 960*K1**2*K3**2 - 128*K1**2*K4**2 - 4608*K1**2 + 576*K1*K2**3*K3 - 896*K1*K2**2*K3 - 320*K1*K2**2*K5 + 128*K1*K2*K3**3 + 7136*K1*K2*K3 + 1280*K1*K3*K4 + 32*K1*K4*K5 - 432*K2**4 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 1088*K2**2*K4 - 3964*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 - 2264*K3**2 - 508*K4**2 - 8*K5**2 - 4*K6**2 + 3818

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.821']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11656', 'vk6.12009', 'vk6.12998', 'vk6.13269', 'vk6.20439', 'vk6.21798', 'vk6.27811', 'vk6.29319', 'vk6.31459', 'vk6.32641', 'vk6.32985', 'vk6.39235', 'vk6.47562', 'vk6.52376', 'vk6.53263', 'vk6.57308', 'vk6.61997', 'vk6.64328', 'vk6.64508', 'vk6.66898']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3O4U1O5U6U5U4U3O6U2
R3 orbit	{'O1O2O3O4U1O5U6U5U4U3O6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3O5U2U1U6U5O6U4
Gauss code of K*	Same
Gauss code of -K*	O1O2O3O4U3O5U2U1U6U5O6U4
Diagrammatic symmetry type	+
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -3 1 2 2 1 -3],[ 3 0 3 2 1 0 1],[-1 -3 0 1 1 1 -4],[-2 -2 -1 0 0 0 -4],[-2 -1 -1 0 0 0 -3],[-1 0 -1 0 0 0 -1],[ 3 -1 4 4 3 1 0]]
Primitive based matrix	[[ 0 2 2 1 1 -3 -3],[-2 0 0 0 -1 -1 -3],[-2 0 0 0 -1 -2 -4],[-1 0 0 0 -1 0 -1],[-1 1 1 1 0 -3 -4],[ 3 1 2 0 3 0 1],[ 3 3 4 1 4 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,-1,3,3,0,0,1,1,3,0,1,2,4,1,0,1,3,4,-1]
Phi over symmetry	[-3,-3,1,1,2,2,-1,1,4,3,4,0,3,1,2,-1,0,0,1,1,0]
Phi of -K	[-3,-3,1,1,2,2,-1,1,4,3,4,0,3,1,2,-1,0,0,1,1,0]
Phi of K*	[-2,-2,-1,-1,3,3,0,0,1,1,3,0,1,2,4,1,0,1,3,4,-1]
Phi of -K*	[-3,-3,1,1,2,2,-1,1,4,3,4,0,3,1,2,-1,0,0,1,1,0]
Symmetry type of based matrix	+
u-polynomial	2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+60t^4+58t^2+1
Outer characteristic polynomial	t^7+88t^5+128t^3+11t
Flat arrow polynomial	-4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
2-strand cable arrow polynomial	-1824K14 + 1536K1*3K2K3 + 64K1*3K3K4 - 896K1*3K3 - 128K12K2*2K3*2 + 192K1*2K2*2K4 - 4064K12K2*2 - 1216K1*2K2K4 + 6544K1*2K2 - 960K12K3*2 - 128K1*2K4*2 - 4608K1*2 + 576K1K23K3 - 896K1K2*2K3 - 320K1K2*2K5 + 128K1K2K33 + 7136K1K2K3 + 1280K1K3K4 + 32K1K4K5 - 432K24 - 288K2*2K3*2 - 16K2*2K4*2 + 1088K2*2K4 - 3964K22 + 144K2K3K5 + 16K2K4K6 - 32K3*4 - 2264K3*2 - 508K4*2 - 8K5*2 - 4K6**2 + 3818
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}]]
If K is slice	False

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