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Flat knot 6.1369

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1369']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+45t^5+130t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1369']
2-strand cable arrow polynomial of the knot is: -768*K1**4*K2**2 + 1152*K1**4*K2 - 1664*K1**4 + 576*K1**3*K2*K3 - 320*K1**3*K3 - 256*K1**2*K2**4 + 2688*K1**2*K2**3 - 8224*K1**2*K2**2 - 1024*K1**2*K2*K4 + 7488*K1**2*K2 - 128*K1**2*K3**2 - 4216*K1**2 + 704*K1*K2**3*K3 - 320*K1*K2**2*K3 - 256*K1*K2**2*K5 + 5424*K1*K2*K3 + 80*K1*K3*K4 - 1584*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 808*K2**2*K4 - 2054*K2**2 + 144*K2*K3*K5 + 8*K2*K4*K6 - 784*K3**2 - 32*K4**2 - 8*K5**2 - 2*K6**2 + 2862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1369']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11604', 'vk6.11957', 'vk6.12946', 'vk6.13257', 'vk6.20421', 'vk6.21786', 'vk6.27777', 'vk6.29297', 'vk6.31407', 'vk6.32581', 'vk6.32960', 'vk6.39205', 'vk6.41427', 'vk6.47554', 'vk6.53203', 'vk6.53512', 'vk6.57290', 'vk6.61964', 'vk6.64292', 'vk6.64500']
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 O1O2O3U4O5O4U1U2U5O6U3U6
R3 orbit {'O1O2O3U4O5O4U1U2U5O6U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O4U5U2U3O6O5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U1O4U5U2U3O6O5U6
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 1 1 1 1],[ 3 0 1 3 3 1 1],[ 1 -1 0 2 1 0 1],[-1 -3 -2 0 0 -1 1],[-1 -3 -1 0 0 1 1],[-1 -1 0 1 -1 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 1 -1 -3],[-1 0 1 1 0 -1 -3],[-1 -1 0 0 1 0 -1],[-1 -1 0 0 -1 -1 -1],[-1 0 -1 1 0 -2 -3],[ 1 1 0 1 2 0 -1],[ 3 3 1 1 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,-1,0,1,1,1,1,2,3,1]
Phi over symmetry [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,1,-1,-1,0]
Phi of -K [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,1,-1,-1,0]
Phi of K* [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,-1,0,1,1,1,1,2,3,1]
Phi of -K* [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,2,0,-1,1,-1,-1,0]
Symmetry type of based matrix +
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+31t^4+72t^2+4
Outer characteristic polynomial t^7+45t^5+130t^3+16t
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -768*K1**4*K2**2 + 1152*K1**4*K2 - 1664*K1**4 + 576*K1**3*K2*K3 - 320*K1**3*K3 - 256*K1**2*K2**4 + 2688*K1**2*K2**3 - 8224*K1**2*K2**2 - 1024*K1**2*K2*K4 + 7488*K1**2*K2 - 128*K1**2*K3**2 - 4216*K1**2 + 704*K1*K2**3*K3 - 320*K1*K2**2*K3 - 256*K1*K2**2*K5 + 5424*K1*K2*K3 + 80*K1*K3*K4 - 1584*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 808*K2**2*K4 - 2054*K2**2 + 144*K2*K3*K5 + 8*K2*K4*K6 - 784*K3**2 - 32*K4**2 - 8*K5**2 - 2*K6**2 + 2862
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}]]
If K is slice False
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