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Flat knot 6.438

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,0,1,2,3,-1,2,3,4,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.438']
Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147']
Outer characteristic polynomial of the knot is: t^7+81t^5+324t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.438']
2-strand cable arrow polynomial of the knot is: -144*K1**4 + 192*K1**3*K2*K3 - 288*K1**2*K2**2 + 200*K1**2*K2 - 944*K1**2*K3**2 - 860*K1**2 + 2040*K1*K2*K3 + 880*K1*K3*K4 + 104*K1*K4*K5 + 88*K1*K5*K6 + 8*K1*K6*K7 - 752*K2**2*K3**2 - 8*K2**2*K4**2 + 208*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 1214*K2**2 + 1096*K2*K3*K5 + 56*K2*K4*K6 + 24*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 + 144*K3**2*K6 - 1236*K3**2 - 404*K4**2 - 460*K5**2 - 138*K6**2 - 12*K7**2 - 12*K8**2 + 1494
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.438']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17120', 'vk6.17363', 'vk6.20256', 'vk6.21563', 'vk6.23520', 'vk6.23855', 'vk6.27492', 'vk6.29086', 'vk6.35677', 'vk6.36106', 'vk6.38911', 'vk6.41114', 'vk6.43028', 'vk6.43336', 'vk6.45662', 'vk6.47393', 'vk6.55273', 'vk6.55521', 'vk6.57085', 'vk6.58243', 'vk6.59694', 'vk6.60033', 'vk6.61641', 'vk6.62819', 'vk6.65078', 'vk6.65267', 'vk6.66724', 'vk6.67588', 'vk6.68332', 'vk6.68480', 'vk6.69370', 'vk6.70110']
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 O1O2O3O4O5U6U2O6U1U5U4U3
R3 orbit {'O1O2O3O4O5U6U2O6U1U5U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U1U5O6U4U6
Gauss code of K* O1O2O3O4U1U5U4U3U2O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U3U2U1U6U4
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 2 2 2 -1],[ 3 0 1 4 3 2 2],[ 2 -1 0 2 1 0 2],[-2 -4 -2 0 0 0 -2],[-2 -3 -1 0 0 0 -2],[-2 -2 0 0 0 0 -2],[ 1 -2 -2 2 2 2 0]]
Primitive based matrix [[ 0 2 2 2 -1 -2 -3],[-2 0 0 0 -2 0 -2],[-2 0 0 0 -2 -1 -3],[-2 0 0 0 -2 -2 -4],[ 1 2 2 2 0 -2 -2],[ 2 0 1 2 2 0 -1],[ 3 2 3 4 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,1,2,3,0,0,2,0,2,0,2,1,3,2,2,4,2,2,1]
Phi over symmetry [-3,-2,-1,2,2,2,0,0,1,2,3,-1,2,3,4,1,1,1,0,0,0]
Phi of -K [-3,-2,-1,2,2,2,0,0,1,2,3,-1,2,3,4,1,1,1,0,0,0]
Phi of K* [-2,-2,-2,1,2,3,0,0,1,2,1,0,1,3,2,1,4,3,-1,0,0]
Phi of -K* [-3,-2,-1,2,2,2,1,2,2,3,4,2,0,1,2,2,2,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 4z+9
Enhanced Jones-Krushkal polynomial 8w^4z-14w^3z+10w^2z+9w
Inner characteristic polynomial t^6+55t^4+180t^2
Outer characteristic polynomial t^7+81t^5+324t^3
Flat arrow polynomial -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1
2-strand cable arrow polynomial -144*K1**4 + 192*K1**3*K2*K3 - 288*K1**2*K2**2 + 200*K1**2*K2 - 944*K1**2*K3**2 - 860*K1**2 + 2040*K1*K2*K3 + 880*K1*K3*K4 + 104*K1*K4*K5 + 88*K1*K5*K6 + 8*K1*K6*K7 - 752*K2**2*K3**2 - 8*K2**2*K4**2 + 208*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 1214*K2**2 + 1096*K2*K3*K5 + 56*K2*K4*K6 + 24*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 + 144*K3**2*K6 - 1236*K3**2 - 404*K4**2 - 460*K5**2 - 138*K6**2 - 12*K7**2 - 12*K8**2 + 1494
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}]]
If K is slice False
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