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Flat knot 6.990

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,2,2,0,1,2,1,0,0,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.990']
Arrow polynomial of the knot is: 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.212', '6.358', '6.588', '6.589', '6.603', '6.990']
Outer characteristic polynomial of the knot is: t^7+48t^5+84t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.990']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 400*K1**4 - 256*K1**2*K2**4 + 1120*K1**2*K2**3 - 4384*K1**2*K2**2 - 96*K1**2*K2*K4 + 5424*K1**2*K2 - 16*K1**2*K3**2 - 16*K1**2*K4**2 - 4024*K1**2 + 480*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4072*K1*K2*K3 + 336*K1*K3*K4 + 64*K1*K4*K5 - 96*K2**6 + 160*K2**4*K4 - 1512*K2**4 - 160*K2**2*K3**2 - 72*K2**2*K4**2 + 1688*K2**2*K4 - 2668*K2**2 + 56*K2*K3*K5 + 32*K2*K4*K6 - 1076*K3**2 - 526*K4**2 - 36*K5**2 - 12*K6**2 + 3004
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.990']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71601', 'vk6.71612', 'vk6.71724', 'vk6.71735', 'vk6.72146', 'vk6.72157', 'vk6.72338', 'vk6.74045', 'vk6.74054', 'vk6.74613', 'vk6.76801', 'vk6.77224', 'vk6.77228', 'vk6.77536', 'vk6.77540', 'vk6.77675', 'vk6.79044', 'vk6.79052', 'vk6.79609', 'vk6.79620', 'vk6.80565', 'vk6.80576', 'vk6.81016', 'vk6.81028', 'vk6.81352', 'vk6.81366', 'vk6.81396', 'vk6.85418', 'vk6.85420', 'vk6.85495', 'vk6.87978', 'vk6.89319']
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 O1O2O3O4U1U5O6U2O5U4U6U3
R3 orbit {'O1O2O3O4U1U5O6U2O5U4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U1O6U3O5U6U4
Gauss code of K* O1O2O3U4U5U3U1O4O6U2O5U6
Gauss code of -K* O1O2O3U4O5U2O4O6U3U1U5U6
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 -3 -1 2 1 0 1],[ 3 0 1 3 2 2 2],[ 1 -1 0 2 0 1 1],[-2 -3 -2 0 -1 -1 0],[-1 -2 0 1 0 -1 0],[ 0 -2 -1 1 1 0 1],[-1 -2 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 -2 -3],[-1 0 0 0 -1 -1 -2],[-1 1 0 0 -1 0 -2],[ 0 1 1 1 0 -1 -2],[ 1 2 1 0 1 0 -1],[ 3 3 2 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,2,3,0,1,1,2,1,0,2,1,2,1]
Phi over symmetry [-3,-1,0,1,1,2,1,1,2,2,2,0,1,2,1,0,0,1,0,1,0]
Phi of -K [-3,-1,0,1,1,2,1,1,2,2,2,0,1,2,1,0,0,1,0,1,0]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,1,2,0,0,2,2,0,1,2,0,1,1]
Phi of -K* [-3,-1,0,1,1,2,1,2,2,2,3,1,0,1,2,1,1,1,0,1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+32t^4+51t^2
Outer characteristic polynomial t^7+48t^5+84t^3+4t
Flat arrow polynomial 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4
2-strand cable arrow polynomial 96*K1**4*K2 - 400*K1**4 - 256*K1**2*K2**4 + 1120*K1**2*K2**3 - 4384*K1**2*K2**2 - 96*K1**2*K2*K4 + 5424*K1**2*K2 - 16*K1**2*K3**2 - 16*K1**2*K4**2 - 4024*K1**2 + 480*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4072*K1*K2*K3 + 336*K1*K3*K4 + 64*K1*K4*K5 - 96*K2**6 + 160*K2**4*K4 - 1512*K2**4 - 160*K2**2*K3**2 - 72*K2**2*K4**2 + 1688*K2**2*K4 - 2668*K2**2 + 56*K2*K3*K5 + 32*K2*K4*K6 - 1076*K3**2 - 526*K4**2 - 36*K5**2 - 12*K6**2 + 3004
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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