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Flat knot 6.174

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,1,1,3,4,3,0,1,1,2,1,0,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.174']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.174']
Outer characteristic polynomial of the knot is: t^7+85t^5+96t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.174']
2-strand cable arrow polynomial of the knot is: 288*K1**4*K2 - 928*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 + 960*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5424*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 7528*K1**2*K2 - 496*K1**2*K3**2 - 32*K1**2*K4**2 - 6476*K1**2 + 864*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 288*K1*K2**2*K5 + 64*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7552*K1*K2*K3 - 32*K1*K3**2*K5 + 1256*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 + 96*K2**4*K4 - 1952*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 1312*K2**2*K3**2 - 72*K2**2*K4**2 + 2088*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4366*K2**2 + 1104*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 56*K3**2*K6 - 2604*K3**2 + 40*K3*K4*K7 - 918*K4**2 - 320*K5**2 - 66*K6**2 - 24*K7**2 - 2*K8**2 + 5174
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.174']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71376', 'vk6.71435', 'vk6.71898', 'vk6.71957', 'vk6.72438', 'vk6.72608', 'vk6.72725', 'vk6.72797', 'vk6.72860', 'vk6.73036', 'vk6.74240', 'vk6.74374', 'vk6.74447', 'vk6.74868', 'vk6.75062', 'vk6.76632', 'vk6.76916', 'vk6.77033', 'vk6.77401', 'vk6.77748', 'vk6.77799', 'vk6.79284', 'vk6.79412', 'vk6.79757', 'vk6.79828', 'vk6.79897', 'vk6.80858', 'vk6.80919', 'vk6.81368', 'vk6.85502', 'vk6.87205', 'vk6.89264']
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 O1O2O3O4O5U1O6U5U3U6U2U4
R3 orbit {'O1O2O3O4O5U1O6U5U3U6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U4U6U3U1O6U5
Gauss code of K* O1O2O3O4O5U6U4U2U5U1O6U3
Gauss code of -K* O1O2O3O4O5U3O6U5U1U4U2U6
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 -4 0 -1 3 0 2],[ 4 0 3 2 4 1 2],[ 0 -3 0 -1 2 -1 2],[ 1 -2 1 0 2 0 2],[-3 -4 -2 -2 0 -1 1],[ 0 -1 1 0 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 0 -1 -4],[-3 0 1 -1 -2 -2 -4],[-2 -1 0 -1 -2 -2 -2],[ 0 1 1 0 1 0 -1],[ 0 2 2 -1 0 -1 -3],[ 1 2 2 0 1 0 -2],[ 4 4 2 1 3 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,0,1,4,-1,1,2,2,4,1,2,2,2,-1,0,1,1,3,2]
Phi over symmetry [-4,-1,0,0,2,3,1,1,3,4,3,0,1,1,2,1,0,1,1,2,2]
Phi of -K [-4,-1,0,0,2,3,1,1,3,4,3,0,1,1,2,1,0,1,1,2,2]
Phi of K* [-3,-2,0,0,1,4,2,1,2,2,3,0,1,1,4,-1,0,1,1,3,1]
Phi of -K* [-4,-1,0,0,2,3,2,1,3,2,4,0,1,2,2,1,1,1,2,2,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2-4w^3z+22w^2z+33w
Inner characteristic polynomial t^6+55t^4+25t^2+1
Outer characteristic polynomial t^7+85t^5+96t^3+13t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + K4 + 5
2-strand cable arrow polynomial 288*K1**4*K2 - 928*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 + 960*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5424*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 7528*K1**2*K2 - 496*K1**2*K3**2 - 32*K1**2*K4**2 - 6476*K1**2 + 864*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 288*K1*K2**2*K5 + 64*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7552*K1*K2*K3 - 32*K1*K3**2*K5 + 1256*K1*K3*K4 + 272*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 + 96*K2**4*K4 - 1952*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 1312*K2**2*K3**2 - 72*K2**2*K4**2 + 2088*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4366*K2**2 + 1104*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 56*K3**2*K6 - 2604*K3**2 + 40*K3*K4*K7 - 918*K4**2 - 320*K5**2 - 66*K6**2 - 24*K7**2 - 2*K8**2 + 5174
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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