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Flat knot 6.1655

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,0,2,0,0,-1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1655']
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+50t^5+71t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1655']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 6144*K1**4*K2 - 6368*K1**4 - 384*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 1984*K1**3*K3 - 128*K1**2*K2**4 + 3776*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10608*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 8040*K1**2*K2 - 1056*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 1640*K1**2 + 704*K1*K2**3*K3 - 2176*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6816*K1*K2*K3 + 920*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1792*K2**4 - 592*K2**2*K3**2 - 48*K2**2*K4**2 + 1312*K2**2*K4 - 1774*K2**2 + 280*K2*K3*K5 + 16*K2*K4*K6 - 1036*K3**2 - 212*K4**2 - 28*K5**2 - 2*K6**2 + 2466
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1655']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3251', 'vk6.3277', 'vk6.3302', 'vk6.3379', 'vk6.3412', 'vk6.3437', 'vk6.3477', 'vk6.3512', 'vk6.4628', 'vk6.5917', 'vk6.6038', 'vk6.7956', 'vk6.8075', 'vk6.9388', 'vk6.17839', 'vk6.17854', 'vk6.19062', 'vk6.19864', 'vk6.24352', 'vk6.25676', 'vk6.25691', 'vk6.26309', 'vk6.26752', 'vk6.37784', 'vk6.43777', 'vk6.43792', 'vk6.45054', 'vk6.48107', 'vk6.48123', 'vk6.48148', 'vk6.48198', 'vk6.50664']
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 O1O2O3U1O4U5U4U6O5O6U3U2
R3 orbit {'O1O2O3U1O4U5U4U6O5O6U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U1O4O5U4U6U5O6U3
Gauss code of K* O1O2O3U1U3O4O5U6U5U4O6U2
Gauss code of -K* O1O2O3U2O4U5U6U4O6O5U1U3
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 -2 1],[ 2 0 2 1 0 2 2],[-1 -2 0 0 1 -3 0],[-1 -1 0 0 1 -3 0],[-1 0 -1 -1 0 -1 0],[ 2 -2 3 3 1 0 2],[-1 -2 0 0 0 -2 0]]
Primitive based matrix [[ 0 1 1 1 1 -2 -2],[-1 0 1 0 0 -1 -3],[-1 -1 0 0 -1 0 -1],[-1 0 0 0 0 -2 -2],[-1 0 1 0 0 -2 -3],[ 2 1 0 2 2 0 2],[ 2 3 1 2 3 -2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,2,2,-1,0,0,1,3,0,1,0,1,0,2,2,2,3,-2]
Phi over symmetry [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,0,2,0,0,-1,0,0,-1]
Phi of -K [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,0,2,0,0,-1,0,0,-1]
Phi of K* [-1,-1,-1,-1,2,2,-1,-1,0,2,3,0,0,0,1,0,0,2,1,1,-2]
Phi of -K* [-2,-2,1,1,1,1,-2,1,2,3,3,0,2,1,2,0,-1,-1,0,0,0]
Symmetry type of based matrix c
u-polynomial 2t^2-4t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+38t^4+51t^2+4
Outer characteristic polynomial t^7+50t^5+71t^3+7t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 6144*K1**4*K2 - 6368*K1**4 - 384*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 1984*K1**3*K3 - 128*K1**2*K2**4 + 3776*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10608*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 8040*K1**2*K2 - 1056*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 1640*K1**2 + 704*K1*K2**3*K3 - 2176*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6816*K1*K2*K3 + 920*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1792*K2**4 - 592*K2**2*K3**2 - 48*K2**2*K4**2 + 1312*K2**2*K4 - 1774*K2**2 + 280*K2*K3*K5 + 16*K2*K4*K6 - 1036*K3**2 - 212*K4**2 - 28*K5**2 - 2*K6**2 + 2466
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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