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Flat knot 6.1065

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,1,2,3,0,0,0,0,1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1065']
Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']
Outer characteristic polynomial of the knot is: t^7+30t^5+34t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1065']
2-strand cable arrow polynomial of the knot is: 288*K1**4*K2 - 3152*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 896*K1**3*K3 - 2720*K1**2*K2**2 - 1568*K1**2*K2*K4 + 7104*K1**2*K2 - 800*K1**2*K3**2 - 32*K1**2*K3*K5 - 272*K1**2*K4**2 - 5088*K1**2 + 96*K1*K2**3*K3 - 320*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6648*K1*K2*K3 + 2936*K1*K3*K4 + 392*K1*K4*K5 + 8*K1*K5*K6 - 80*K2**4 - 160*K2**2*K3**2 - 56*K2**2*K4**2 + 1032*K2**2*K4 - 4166*K2**2 + 280*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 2756*K3**2 - 1352*K4**2 - 188*K5**2 - 18*K6**2 + 4598
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1065']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11011', 'vk6.11090', 'vk6.12181', 'vk6.12288', 'vk6.18207', 'vk6.18543', 'vk6.24666', 'vk6.25089', 'vk6.30576', 'vk6.30671', 'vk6.31850', 'vk6.31897', 'vk6.36801', 'vk6.37256', 'vk6.44041', 'vk6.44382', 'vk6.51810', 'vk6.51877', 'vk6.52678', 'vk6.52772', 'vk6.56001', 'vk6.56275', 'vk6.60537', 'vk6.60877', 'vk6.63490', 'vk6.63534', 'vk6.63972', 'vk6.64016', 'vk6.65666', 'vk6.65949', 'vk6.68711', 'vk6.68920']
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 O1O2O3O4U5U2O6U3O5U6U1U4
R3 orbit {'O1O2O3O4U5U2O6U3O5U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U5O6U2O5U3U6
Gauss code of K* O1O2O3U2U4U5U3O6O4U1O5U6
Gauss code of -K* O1O2O3U4O5U3O6O4U1U5U6U2
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 0 3 -1 0],[ 1 0 0 1 3 0 0],[ 1 0 0 1 1 0 0],[ 0 -1 -1 0 1 0 0],[-3 -3 -1 -1 0 -2 -1],[ 1 0 0 0 2 0 0],[ 0 0 0 0 1 0 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -1 -1 -2 -3],[ 0 1 0 0 0 0 0],[ 0 1 0 0 -1 0 -1],[ 1 1 0 1 0 0 0],[ 1 2 0 0 0 0 0],[ 1 3 0 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,1,1,1,2,3,0,0,0,0,1,0,1,0,0,0]
Phi over symmetry [-3,0,0,1,1,1,1,1,1,2,3,0,0,0,0,1,0,1,0,0,0]
Phi of -K [-1,-1,-1,0,0,3,0,0,0,1,1,0,0,1,3,1,1,2,0,2,2]
Phi of K* [-3,0,0,1,1,1,2,2,1,2,3,0,0,1,0,1,1,1,0,0,0]
Phi of -K* [-1,-1,-1,0,0,3,0,0,0,0,2,0,0,1,1,0,1,3,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+18t^4+14t^2
Outer characteristic polynomial t^7+30t^5+34t^3+6t
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 288*K1**4*K2 - 3152*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 896*K1**3*K3 - 2720*K1**2*K2**2 - 1568*K1**2*K2*K4 + 7104*K1**2*K2 - 800*K1**2*K3**2 - 32*K1**2*K3*K5 - 272*K1**2*K4**2 - 5088*K1**2 + 96*K1*K2**3*K3 - 320*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 6648*K1*K2*K3 + 2936*K1*K3*K4 + 392*K1*K4*K5 + 8*K1*K5*K6 - 80*K2**4 - 160*K2**2*K3**2 - 56*K2**2*K4**2 + 1032*K2**2*K4 - 4166*K2**2 + 280*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 24*K3**2*K6 - 2756*K3**2 - 1352*K4**2 - 188*K5**2 - 18*K6**2 + 4598
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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