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Flat knot 6.387

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,1,3,3,5,0,1,1,2,0,0,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.387']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 5*K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.277', '6.387']
Outer characteristic polynomial of the knot is: t^7+83t^5+67t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.387']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 544*K1**4*K2 - 1040*K1**4 + 96*K1**3*K2*K3 - 384*K1**3*K3 + 576*K1**2*K2**3 - 3872*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 6280*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K3*K5 - 5320*K1**2 + 608*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 224*K1*K2**2*K5 + 64*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 6216*K1*K2*K3 + 1088*K1*K3*K4 + 232*K1*K4*K5 + 24*K1*K5*K6 - 64*K2**6 + 192*K2**4*K4 - 1200*K2**4 + 32*K2**3*K3*K5 - 64*K2**3*K6 - 800*K2**2*K3**2 - 128*K2**2*K4**2 + 1680*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4060*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 856*K2*K3*K5 + 104*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 72*K3**2*K6 - 2320*K3**2 + 16*K3*K4*K7 - 758*K4**2 - 320*K5**2 - 60*K6**2 - 16*K7**2 - 2*K8**2 + 4358
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.387']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17115', 'vk6.17357', 'vk6.20578', 'vk6.21985', 'vk6.23506', 'vk6.23845', 'vk6.28040', 'vk6.29497', 'vk6.35651', 'vk6.36089', 'vk6.39454', 'vk6.41653', 'vk6.43012', 'vk6.43323', 'vk6.46038', 'vk6.47704', 'vk6.55254', 'vk6.55505', 'vk6.57444', 'vk6.58613', 'vk6.59656', 'vk6.60004', 'vk6.62115', 'vk6.63083', 'vk6.65052', 'vk6.65248', 'vk6.66980', 'vk6.67843', 'vk6.68314', 'vk6.68463', 'vk6.69595', 'vk6.70286']
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 O1O2O3O4O5U4U2O6U3U1U6U5
R3 orbit {'O1O2O3O4O5U4U2O6U3U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U5U3O6U4U2
Gauss code of K* O1O2O3O4U2U5U1U6U4O6O5U3
Gauss code of -K* O1O2O3O4U2O5O6U1U6U4U5U3
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 -2 -1 -1 4 2],[ 2 0 -1 1 0 5 2],[ 2 1 0 1 0 3 1],[ 1 -1 -1 0 0 3 1],[ 1 0 0 0 0 1 0],[-4 -5 -3 -3 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 4 2 -1 -1 -2 -2],[-4 0 0 -1 -3 -3 -5],[-2 0 0 0 -1 -1 -2],[ 1 1 0 0 0 0 0],[ 1 3 1 0 0 -1 -1],[ 2 3 1 0 1 0 1],[ 2 5 2 0 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,1,1,2,2,0,1,3,3,5,0,1,1,2,0,0,0,1,1,-1]
Phi over symmetry [-4,-2,1,1,2,2,0,1,3,3,5,0,1,1,2,0,0,0,1,1,-1]
Phi of -K [-2,-2,-1,-1,2,4,-1,0,1,3,3,0,1,2,1,0,2,2,3,4,2]
Phi of K* [-4,-2,1,1,2,2,2,2,4,1,3,2,3,2,3,0,0,0,1,1,-1]
Phi of -K* [-2,-2,-1,-1,2,4,-1,0,1,2,5,0,1,1,3,0,0,1,1,3,0]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+53t^4+12t^2
Outer characteristic polynomial t^7+83t^5+67t^3+4t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 5*K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial -128*K1**4*K2**2 + 544*K1**4*K2 - 1040*K1**4 + 96*K1**3*K2*K3 - 384*K1**3*K3 + 576*K1**2*K2**3 - 3872*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 6280*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K3*K5 - 5320*K1**2 + 608*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 224*K1*K2**2*K5 + 64*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 6216*K1*K2*K3 + 1088*K1*K3*K4 + 232*K1*K4*K5 + 24*K1*K5*K6 - 64*K2**6 + 192*K2**4*K4 - 1200*K2**4 + 32*K2**3*K3*K5 - 64*K2**3*K6 - 800*K2**2*K3**2 - 128*K2**2*K4**2 + 1680*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 4060*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 856*K2*K3*K5 + 104*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 72*K3**2*K6 - 2320*K3**2 + 16*K3*K4*K7 - 758*K4**2 - 320*K5**2 - 60*K6**2 - 16*K7**2 - 2*K8**2 + 4358
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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