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Flat knot 6.162

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,1,2,1,3,4,0,0,1,1,0,2,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.162']
Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511']
Outer characteristic polynomial of the knot is: t^7+108t^5+92t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.162']
2-strand cable arrow polynomial of the knot is: -528*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1184*K1**3*K3 - 736*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 3104*K1**2*K2 - 2928*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 32*K1**2*K4*K6 - 4076*K1**2 + 64*K1*K2**3*K3 - 416*K1*K2**2*K3 + 128*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6952*K1*K2*K3 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2560*K1*K3*K4 + 112*K1*K4*K5 + 24*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**4 - 1296*K2**2*K3**2 - 8*K2**2*K4**2 + 232*K2**2*K4 - 8*K2**2*K6**2 - 2970*K2**2 - 96*K2*K3**2*K4 + 992*K2*K3*K5 + 104*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 + 112*K3**2*K6 - 2652*K3**2 + 16*K3*K4*K7 - 626*K4**2 - 176*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 3354
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.162']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16995', 'vk6.17236', 'vk6.20213', 'vk6.21501', 'vk6.23399', 'vk6.23706', 'vk6.27403', 'vk6.29021', 'vk6.35462', 'vk6.35904', 'vk6.38818', 'vk6.41001', 'vk6.42896', 'vk6.43195', 'vk6.45573', 'vk6.47348', 'vk6.55158', 'vk6.55403', 'vk6.57049', 'vk6.58154', 'vk6.59534', 'vk6.59877', 'vk6.61551', 'vk6.62729', 'vk6.64968', 'vk6.65173', 'vk6.66666', 'vk6.67497', 'vk6.68258', 'vk6.68412', 'vk6.69311', 'vk6.70071']
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 O1O2O3O4O5U1O6U4U3U5U6U2
R3 orbit {'O1O2O3O4O5U1O6U4U3U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U6U1U3U2O6U5
Gauss code of K* O1O2O3O4O5U6U5U2U1U3O6U4
Gauss code of -K* O1O2O3O4O5U2O6U3U5U4U1U6
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 1 -1 -1 2 3],[ 4 0 4 2 1 3 3],[-1 -4 0 -2 -2 1 3],[ 1 -2 2 0 0 2 3],[ 1 -1 2 0 0 1 2],[-2 -3 -1 -2 -1 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 -1 -3 -2 -3 -3],[-2 1 0 -1 -1 -2 -3],[-1 3 1 0 -2 -2 -4],[ 1 2 1 2 0 0 -1],[ 1 3 2 2 0 0 -2],[ 4 3 3 4 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,1,3,2,3,3,1,1,2,3,2,2,4,0,1,2]
Phi over symmetry [-4,-1,-1,1,2,3,1,2,1,3,4,0,0,1,1,0,2,2,0,-1,0]
Phi of -K [-4,-1,-1,1,2,3,1,2,1,3,4,0,0,1,1,0,2,2,0,-1,0]
Phi of K* [-3,-2,-1,1,1,4,0,-1,1,2,4,0,1,2,3,0,0,1,0,1,2]
Phi of -K* [-4,-1,-1,1,2,3,1,2,4,3,3,0,2,1,2,2,2,3,1,3,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+76t^4+35t^2+1
Outer characteristic polynomial t^7+108t^5+92t^3+10t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -528*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1184*K1**3*K3 - 736*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 3104*K1**2*K2 - 2928*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 32*K1**2*K4*K6 - 4076*K1**2 + 64*K1*K2**3*K3 - 416*K1*K2**2*K3 + 128*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6952*K1*K2*K3 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2560*K1*K3*K4 + 112*K1*K4*K5 + 24*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**4 - 1296*K2**2*K3**2 - 8*K2**2*K4**2 + 232*K2**2*K4 - 8*K2**2*K6**2 - 2970*K2**2 - 96*K2*K3**2*K4 + 992*K2*K3*K5 + 104*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 + 112*K3**2*K6 - 2652*K3**2 + 16*K3*K4*K7 - 626*K4**2 - 176*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 3354
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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