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Flat knot 6.551

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,0,-1,1,2,3,-1,1,2,3,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.551']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+55t^5+42t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.551']
2-strand cable arrow polynomial of the knot is: 2784*K1**4*K2 - 4640*K1**4 + 1184*K1**3*K2*K3 - 1216*K1**3*K3 - 384*K1**2*K2**4 + 640*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 6896*K1**2*K2**2 - 512*K1**2*K2*K4 + 6952*K1**2*K2 - 1120*K1**2*K3**2 - 32*K1**2*K4**2 - 1528*K1**2 + 512*K1*K2**3*K3 - 704*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5336*K1*K2*K3 + 664*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 640*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 576*K2**2*K4 - 1982*K2**2 + 128*K2*K3*K5 + 16*K2*K4*K6 - 956*K3**2 - 124*K4**2 - 4*K5**2 - 2*K6**2 + 2170
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.551']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16801', 'vk6.16805', 'vk6.16856', 'vk6.16860', 'vk6.18183', 'vk6.18185', 'vk6.18520', 'vk6.18522', 'vk6.23241', 'vk6.23245', 'vk6.24640', 'vk6.25069', 'vk6.25071', 'vk6.35233', 'vk6.35258', 'vk6.36776', 'vk6.37221', 'vk6.37223', 'vk6.42752', 'vk6.42756', 'vk6.44359', 'vk6.44361', 'vk6.54992', 'vk6.55025', 'vk6.55980', 'vk6.55982', 'vk6.59392', 'vk6.59396', 'vk6.60517', 'vk6.65646', 'vk6.68182', 'vk6.68186']
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 O1O2O3O4O5U3U6U1O6U5U4U2
R3 orbit {'O1O2O3O4O5U3U6U1O6U5U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U1O6U5U6U3
Gauss code of K* O1O2O3U2O4O5O6U3U6U1U5U4
Gauss code of -K* O1O2O3U4O5O4O6U3U2U6U1U5
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 2 2 -1],[ 2 0 2 0 2 1 2],[-1 -2 0 -2 1 1 -1],[ 2 0 2 0 2 1 2],[-2 -2 -1 -2 0 0 -2],[-2 -1 -1 -1 0 0 -2],[ 1 -2 1 -2 2 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 0 -1 -2 -1 -1],[-2 0 0 -1 -2 -2 -2],[-1 1 1 0 -1 -2 -2],[ 1 2 2 1 0 -2 -2],[ 2 1 2 2 2 0 0],[ 2 1 2 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,0,1,2,1,1,1,2,2,2,1,2,2,2,2,0]
Phi over symmetry [-2,-2,-1,1,2,2,0,-1,1,2,3,-1,1,2,3,1,1,1,0,0,0]
Phi of -K [-2,-2,-1,1,2,2,0,-1,1,2,3,-1,1,2,3,1,1,1,0,0,0]
Phi of K* [-2,-2,-1,1,2,2,0,0,1,2,2,0,1,3,3,1,1,1,-1,-1,0]
Phi of -K* [-2,-2,-1,1,2,2,0,2,2,1,2,2,2,1,2,1,2,2,1,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+37t^4+12t^2
Outer characteristic polynomial t^7+55t^5+42t^3+2t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial 2784*K1**4*K2 - 4640*K1**4 + 1184*K1**3*K2*K3 - 1216*K1**3*K3 - 384*K1**2*K2**4 + 640*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 6896*K1**2*K2**2 - 512*K1**2*K2*K4 + 6952*K1**2*K2 - 1120*K1**2*K3**2 - 32*K1**2*K4**2 - 1528*K1**2 + 512*K1*K2**3*K3 - 704*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5336*K1*K2*K3 + 664*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 640*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 576*K2**2*K4 - 1982*K2**2 + 128*K2*K3*K5 + 16*K2*K4*K6 - 956*K3**2 - 124*K4**2 - 4*K5**2 - 2*K6**2 + 2170
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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