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Flat knot 6.970

Min(phi) over symmetries of the knot is: [-2,-1,1,1,1,-1,1,1,2,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['5.23', '6.970']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^6+20t^4+24t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.970']
2-strand cable arrow polynomial of the knot is: 2304*K1**4*K2 - 5184*K1**4 + 2304*K1**3*K2*K3 - 1152*K1**3*K3 - 384*K1**2*K2**4 + 480*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 6640*K1**2*K2**2 - 640*K1**2*K2*K4 + 6104*K1**2*K2 - 2880*K1**2*K3**2 - 96*K1**2*K4**2 + 160*K1**2 + 512*K1*K2**3*K3 - 896*K1*K2**2*K3 - 128*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5280*K1*K2*K3 + 1608*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 552*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 512*K2**2*K4 - 1214*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 780*K3**2 - 166*K4**2 - 20*K5**2 - 2*K6**2 + 1420
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.970']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3256', 'vk6.3284', 'vk6.3297', 'vk6.3386', 'vk6.3415', 'vk6.3430', 'vk6.3470', 'vk6.3515', 'vk6.4611', 'vk6.5902', 'vk6.6029', 'vk6.7945', 'vk6.8066', 'vk6.9381', 'vk6.17842', 'vk6.17859', 'vk6.19059', 'vk6.19883', 'vk6.24359', 'vk6.25671', 'vk6.25688', 'vk6.26329', 'vk6.26772', 'vk6.37779', 'vk6.43780', 'vk6.43797', 'vk6.45066', 'vk6.48116', 'vk6.48128', 'vk6.48151', 'vk6.48205', 'vk6.50655']
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 O1O2O3O4U5U4O6O5U6U1U3U2
R3 orbit {'O1O2O3O4U5U4O6O5U6U1U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U2U4U5O6O5U1U6
Gauss code of K* O1O2O3O4U2U4U3U5O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U6U2U1U3
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 0 -1],[ 2 0 2 1 1 2 -1],[-1 -2 0 0 1 -1 -1],[-1 -1 0 0 1 -1 -1],[-1 -1 -1 -1 0 -1 -1],[ 0 -2 1 1 1 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 1 1 1 -1 -2],[-1 0 1 0 -1 -1],[-1 -1 0 -1 -1 -1],[-1 0 1 0 -1 -2],[ 1 1 1 1 0 1],[ 2 1 1 2 -1 0]]
If based matrix primitive False
Phi of primitive based matrix [-1,-1,-1,1,2,-1,0,1,1,1,1,1,1,2,-1]
Phi over symmetry [-2,-1,1,1,1,-1,1,1,2,1,1,1,-1,-1,0]
Phi of -K [-2,-1,1,1,1,2,1,2,2,1,1,1,-1,0,1]
Phi of K* [-1,-1,-1,1,2,-1,-1,1,2,0,1,1,1,2,2]
Phi of -K* [-2,-1,1,1,1,-1,1,1,2,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^5+12t^3+7t
Outer characteristic polynomial t^6+20t^4+24t^2+1
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 2304*K1**4*K2 - 5184*K1**4 + 2304*K1**3*K2*K3 - 1152*K1**3*K3 - 384*K1**2*K2**4 + 480*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 6640*K1**2*K2**2 - 640*K1**2*K2*K4 + 6104*K1**2*K2 - 2880*K1**2*K3**2 - 96*K1**2*K4**2 + 160*K1**2 + 512*K1*K2**3*K3 - 896*K1*K2**2*K3 - 128*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 5280*K1*K2*K3 + 1608*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 552*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 512*K2**2*K4 - 1214*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 780*K3**2 - 166*K4**2 - 20*K5**2 - 2*K6**2 + 1420
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}]]
If K is slice False
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