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Flat knot 6.1360

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,-1,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1360']
Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384']
Outer characteristic polynomial of the knot is: t^7+43t^5+63t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1360']
2-strand cable arrow polynomial of the knot is: -1280*K1**4*K2**2 + 2752*K1**4*K2 - 4416*K1**4 - 128*K1**3*K3 + 2048*K1**2*K2**3 - 8992*K1**2*K2**2 - 192*K1**2*K2*K4 + 10000*K1**2*K2 - 64*K1**2*K3**2 - 4584*K1**2 + 256*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 7152*K1*K2*K3 + 752*K1*K3*K4 + 64*K1*K4*K5 - 1248*K2**4 - 160*K2**2*K3**2 - 8*K2**2*K4**2 + 1512*K2**2*K4 - 4166*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 - 1808*K3**2 - 612*K4**2 - 56*K5**2 - 2*K6**2 + 4514
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1360']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11597', 'vk6.11948', 'vk6.12939', 'vk6.13250', 'vk6.20433', 'vk6.21794', 'vk6.27801', 'vk6.29312', 'vk6.31392', 'vk6.32566', 'vk6.32950', 'vk6.39229', 'vk6.41441', 'vk6.47561', 'vk6.53192', 'vk6.53505', 'vk6.57294', 'vk6.61972', 'vk6.64285', 'vk6.64495']
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 O1O2O3U2O4O5U3U6U1O6U4U5
R3 orbit {'O1O2O3U2O4O5U3U6U1O6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U3U6U1O4O5U2
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U5O6U3U6U1O4O5U2
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 -1 -1 -1 1 3 -1],[ 1 0 -1 1 1 2 1],[ 1 1 0 1 1 1 1],[ 1 -1 -1 0 1 2 1],[-1 -1 -1 -1 0 1 -1],[-3 -2 -1 -2 -1 0 -3],[ 1 -1 -1 -1 1 3 0]]
Primitive based matrix [[ 0 3 1 -1 -1 -1 -1],[-3 0 -1 -1 -2 -2 -3],[-1 1 0 -1 -1 -1 -1],[ 1 1 1 0 1 1 1],[ 1 2 1 -1 0 1 1],[ 1 2 1 -1 -1 0 1],[ 1 3 1 -1 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,-1,-1,-1,-1]
Phi over symmetry [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,-1,-1,-1,-1]
Phi of -K [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,-1,-1,1,2,-1,1,2,1,1,1]
Phi of K* [-3,-1,1,1,1,1,1,1,2,2,3,1,1,1,1,-1,-1,-1,-1,-1,-1]
Phi of -K* [-1,-1,-1,-1,1,3,-1,-1,-1,1,3,-1,-1,1,2,-1,1,2,1,1,1]
Symmetry type of based matrix +
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+29t^4+25t^2+1
Outer characteristic polynomial t^7+43t^5+63t^3+5t
Flat arrow polynomial -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -1280*K1**4*K2**2 + 2752*K1**4*K2 - 4416*K1**4 - 128*K1**3*K3 + 2048*K1**2*K2**3 - 8992*K1**2*K2**2 - 192*K1**2*K2*K4 + 10000*K1**2*K2 - 64*K1**2*K3**2 - 4584*K1**2 + 256*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 7152*K1*K2*K3 + 752*K1*K3*K4 + 64*K1*K4*K5 - 1248*K2**4 - 160*K2**2*K3**2 - 8*K2**2*K4**2 + 1512*K2**2*K4 - 4166*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 - 1808*K3**2 - 612*K4**2 - 56*K5**2 - 2*K6**2 + 4514
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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