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Flat knot 6.950

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,-1,0,0,-1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.950']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.950', '6.1142']
Outer characteristic polynomial of the knot is: t^7+44t^5+76t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.950']
2-strand cable arrow polynomial of the knot is: -848*K1**4 - 160*K1**3*K3 - 768*K1**2*K2**6 + 1920*K1**2*K2**5 - 5184*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3424*K1**2*K2**3 - 4368*K1**2*K2**2 - 256*K1**2*K2*K4 + 4072*K1**2*K2 - 112*K1**2*K3**2 - 16*K1**2*K4**2 - 2692*K1**2 + 1280*K1*K2**5*K3 - 640*K1*K2**4*K3 - 384*K1*K2**4*K5 + 3616*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3192*K1*K2*K3 + 384*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 2016*K2**6 - 128*K2**5*K6 - 448*K2**4*K3**2 - 64*K2**4*K4**2 + 1216*K2**4*K4 - 944*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 768*K2**2*K3**2 - 152*K2**2*K4**2 + 808*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 684*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 864*K3**2 - 250*K4**2 - 20*K5**2 - 4*K6**2 + 2032
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.950']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4636', 'vk6.4905', 'vk6.6060', 'vk6.6569', 'vk6.8091', 'vk6.8471', 'vk6.9467', 'vk6.9842', 'vk6.20282', 'vk6.21613', 'vk6.27558', 'vk6.29122', 'vk6.38967', 'vk6.41214', 'vk6.45742', 'vk6.47437', 'vk6.48674', 'vk6.48861', 'vk6.49410', 'vk6.49647', 'vk6.50684', 'vk6.50865', 'vk6.51157', 'vk6.51376', 'vk6.57123', 'vk6.58315', 'vk6.61721', 'vk6.62863', 'vk6.66748', 'vk6.67632', 'vk6.69406', 'vk6.70130']
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 O1O2O3O4U5U1O6O5U6U2U3U4
R3 orbit {'O1O2O3O4U5U1O6O5U6U2U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U3U5O6O5U4U6
Gauss code of K* O1O2O3O4U5U2U3U4O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U1U2U3U6
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 -2 -1 1 3 0 -1],[ 2 0 0 1 2 2 -1],[ 1 0 0 1 2 2 -1],[-1 -1 -1 0 1 0 -1],[-3 -2 -2 -1 0 -2 -1],[ 0 -2 -2 0 2 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -1 -2 -2],[-1 1 0 0 -1 -1 -1],[ 0 2 0 0 -1 -2 -2],[ 1 1 1 1 0 1 1],[ 1 2 1 2 -1 0 0],[ 2 2 1 2 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,1,2,1,2,2,0,1,1,1,1,2,2,-1,-1,0]
Phi over symmetry [-3,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,-1,0,0,-1,1,2]
Phi of -K [-2,-1,-1,0,1,3,1,2,0,2,3,1,-1,1,2,0,1,3,1,1,1]
Phi of K* [-3,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,-1,0,0,-1,1,2]
Phi of -K* [-2,-1,-1,0,1,3,-1,0,2,1,2,1,1,1,1,2,1,2,0,2,1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 5z+11
Enhanced Jones-Krushkal polynomial -6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w
Inner characteristic polynomial t^6+28t^4+11t^2
Outer characteristic polynomial t^7+44t^5+76t^3+6t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2
2-strand cable arrow polynomial -848*K1**4 - 160*K1**3*K3 - 768*K1**2*K2**6 + 1920*K1**2*K2**5 - 5184*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3424*K1**2*K2**3 - 4368*K1**2*K2**2 - 256*K1**2*K2*K4 + 4072*K1**2*K2 - 112*K1**2*K3**2 - 16*K1**2*K4**2 - 2692*K1**2 + 1280*K1*K2**5*K3 - 640*K1*K2**4*K3 - 384*K1*K2**4*K5 + 3616*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3192*K1*K2*K3 + 384*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 2016*K2**6 - 128*K2**5*K6 - 448*K2**4*K3**2 - 64*K2**4*K4**2 + 1216*K2**4*K4 - 944*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 768*K2**2*K3**2 - 152*K2**2*K4**2 + 808*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 684*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 864*K3**2 - 250*K4**2 - 20*K5**2 - 4*K6**2 + 2032
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]]
If K is slice False
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