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Flat knot 6.1498

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,1,0,1,1,1,1,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1498']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926']
Outer characteristic polynomial of the knot is: t^7+24t^5+27t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1498']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 448*K1**4*K2**2 + 2208*K1**4*K2 - 4880*K1**4 + 1312*K1**3*K2*K3 - 1088*K1**3*K3 - 192*K1**2*K2**4 + 704*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6416*K1**2*K2**2 - 832*K1**2*K2*K4 + 10128*K1**2*K2 - 1072*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 4692*K1**2 + 480*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 224*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 6936*K1*K2*K3 + 1720*K1*K3*K4 + 384*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 664*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 128*K2**2*K4**2 + 1208*K2**2*K4 - 4322*K2**2 + 520*K2*K3*K5 + 104*K2*K4*K6 - 2072*K3**2 - 802*K4**2 - 236*K5**2 - 22*K6**2 + 4592
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1498']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4148', 'vk6.4179', 'vk6.5390', 'vk6.5421', 'vk6.7508', 'vk6.7533', 'vk6.9013', 'vk6.9044', 'vk6.12441', 'vk6.12472', 'vk6.13342', 'vk6.13567', 'vk6.13598', 'vk6.14253', 'vk6.14700', 'vk6.14736', 'vk6.15207', 'vk6.15860', 'vk6.15896', 'vk6.30842', 'vk6.30873', 'vk6.32030', 'vk6.32061', 'vk6.33068', 'vk6.33099', 'vk6.33862', 'vk6.34325', 'vk6.48490', 'vk6.50269', 'vk6.53514', 'vk6.53935', 'vk6.54257']
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 O1O2O3U1O4U5U3O5O6U4U6U2
R3 orbit {'O1O2O3U1O4U5U3O5O6U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O4O6U1U6O5U3
Gauss code of K* O1O2O3U4U3U5O4U1O6O5U6U2
Gauss code of -K* O1O2O3U2U4O5O4U3O6U5U1U6
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 0 -1 1],[ 2 0 2 1 1 1 0],[-1 -2 0 0 0 -2 1],[-1 -1 0 0 0 -1 1],[ 0 -1 0 0 0 0 1],[ 1 -1 2 1 0 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 0 -2 -2],[ 0 0 1 0 0 0 -1],[ 1 1 1 2 0 0 -1],[ 2 1 0 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,1,1,1,1,1,0,0,2,2,0,1,1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,3,1,0,1,1,1,1,0,0,-1,-1]
Phi of -K [-2,-1,0,1,1,1,0,1,1,2,3,1,0,1,1,1,1,0,0,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,0,1,3,0,1,0,1,1,1,2,1,1,0]
Phi of -K* [-2,-1,0,1,1,1,1,1,0,1,2,0,1,1,2,1,0,0,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2+21w^2z+35w
Inner characteristic polynomial t^6+16t^4+14t^2+1
Outer characteristic polynomial t^7+24t^5+27t^3+6t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -448*K1**6 - 448*K1**4*K2**2 + 2208*K1**4*K2 - 4880*K1**4 + 1312*K1**3*K2*K3 - 1088*K1**3*K3 - 192*K1**2*K2**4 + 704*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6416*K1**2*K2**2 - 832*K1**2*K2*K4 + 10128*K1**2*K2 - 1072*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 4692*K1**2 + 480*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 224*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 6936*K1*K2*K3 + 1720*K1*K3*K4 + 384*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 664*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 128*K2**2*K4**2 + 1208*K2**2*K4 - 4322*K2**2 + 520*K2*K3*K5 + 104*K2*K4*K6 - 2072*K3**2 - 802*K4**2 - 236*K5**2 - 22*K6**2 + 4592
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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