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Flat knot 6.413

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,0,1,2,3,1,1,1,1,-1,-1,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.413', '7.19583']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+35t^5+54t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.413', '7.19583']
2-strand cable arrow polynomial of the knot is: -576*K1**4 + 768*K1**3*K2*K3 - 64*K1**3*K3 - 2384*K1**2*K2**2 - 544*K1**2*K2*K4 + 2136*K1**2*K2 - 896*K1**2*K3**2 - 96*K1**2*K4**2 - 1324*K1**2 + 960*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 3520*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1184*K1*K3*K4 + 224*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 192*K2**4*K4 - 960*K2**4 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1056*K2**2*K3**2 - 296*K2**2*K4**2 - 32*K2**2*K4*K8 + 1208*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 1198*K2**2 - 96*K2*K3**2*K4 + 792*K2*K3*K5 + 280*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1048*K3**2 - 488*K4**2 - 152*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 1448
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.413']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.490', 'vk6.560', 'vk6.633', 'vk6.969', 'vk6.1062', 'vk6.1143', 'vk6.1656', 'vk6.1769', 'vk6.1860', 'vk6.2147', 'vk6.2242', 'vk6.2325', 'vk6.2588', 'vk6.2907', 'vk6.3074', 'vk6.3182', 'vk6.12072', 'vk6.13063', 'vk6.20509', 'vk6.21095', 'vk6.21884', 'vk6.22527', 'vk6.27937', 'vk6.28539', 'vk6.29421', 'vk6.32726', 'vk6.39352', 'vk6.41523', 'vk6.46817', 'vk6.46888', 'vk6.53300', 'vk6.57376']
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 O1O2O3O4O5U4U5O6U3U2U1U6
R3 orbit {'O1O2O3O4O5U4U5O6U3U2U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U4U3O6U1U2
Gauss code of K* O1O2O3O4U3U2U1U5U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U6U4U3U2
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 -1 -1 -1 -1 1 3],[ 1 0 0 0 -1 1 3],[ 1 0 0 0 -1 1 2],[ 1 0 0 0 -1 1 1],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 -1 -1 -1 -1],[-3 0 0 0 -1 -2 -3],[-1 0 0 -1 -1 -1 -1],[ 1 0 1 0 1 1 1],[ 1 1 1 -1 0 0 0],[ 1 2 1 -1 0 0 0],[ 1 3 1 -1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,1,1,1,1,0,0,1,2,3,1,1,1,1,-1,-1,-1,0,0,0]
Phi over symmetry [-3,-1,1,1,1,1,0,0,1,2,3,1,1,1,1,-1,-1,-1,0,0,0]
Phi of -K [-1,-1,-1,-1,1,3,-1,-1,-1,1,4,0,0,1,1,0,1,2,1,3,2]
Phi of K* [-3,-1,1,1,1,1,2,1,2,3,4,1,1,1,1,0,0,-1,0,-1,-1]
Phi of -K* [-1,-1,-1,-1,1,3,-1,0,0,1,1,1,1,1,0,0,1,2,1,3,0]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial t^6+21t^4+26t^2
Outer characteristic polynomial t^7+35t^5+54t^3+6t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -576*K1**4 + 768*K1**3*K2*K3 - 64*K1**3*K3 - 2384*K1**2*K2**2 - 544*K1**2*K2*K4 + 2136*K1**2*K2 - 896*K1**2*K3**2 - 96*K1**2*K4**2 - 1324*K1**2 + 960*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 3520*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1184*K1*K3*K4 + 224*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 192*K2**4*K4 - 960*K2**4 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1056*K2**2*K3**2 - 296*K2**2*K4**2 - 32*K2**2*K4*K8 + 1208*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 1198*K2**2 - 96*K2*K3**2*K4 + 792*K2*K3*K5 + 280*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1048*K3**2 - 488*K4**2 - 152*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 1448
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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