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Flat knot 6.435

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,1,2,1,4,3,0,1,2,1,1,3,2,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.435']
Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511']
Outer characteristic polynomial of the knot is: t^7+98t^5+301t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.435']
2-strand cable arrow polynomial of the knot is: 128*K1**2*K2*K3**2 + 744*K1**2*K2 - 1664*K1**2*K3**2 - 2108*K1**2 - 608*K1*K2**2*K3 - 480*K1*K2*K3*K4 + 3680*K1*K2*K3 - 32*K1*K3**2*K5 + 2216*K1*K3*K4 + 104*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 816*K2**2*K3**2 - 8*K2**2*K4**2 + 696*K2**2*K4 - 8*K2**2*K6**2 - 2218*K2**2 - 32*K2*K3*K4*K5 + 1224*K2*K3*K5 + 24*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 + 16*K3**2*K6 - 2104*K3**2 + 40*K3*K4*K7 + 24*K3*K5*K8 - 870*K4**2 - 404*K5**2 - 46*K6**2 - 40*K7**2 - 18*K8**2 + 2430
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.435']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16926', 'vk6.17169', 'vk6.20220', 'vk6.21516', 'vk6.23320', 'vk6.23615', 'vk6.27420', 'vk6.29032', 'vk6.35358', 'vk6.35782', 'vk6.38833', 'vk6.41028', 'vk6.42839', 'vk6.43119', 'vk6.45598', 'vk6.47359', 'vk6.55086', 'vk6.55338', 'vk6.57059', 'vk6.58185', 'vk6.59485', 'vk6.59775', 'vk6.61580', 'vk6.62754', 'vk6.64932', 'vk6.65140', 'vk6.66681', 'vk6.67521', 'vk6.68225', 'vk6.68368', 'vk6.69332', 'vk6.70085']
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 O1O2O3O4O5U6U2O6U1U4U3U5
R3 orbit {'O1O2O3O4O5U6U2O6U1U4U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U3U2U5O6U4U6
Gauss code of K* O1O2O3O4U1U5U3U2U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U3U2U6U4
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 -2 1 1 4 -1],[ 3 0 1 3 2 4 2],[ 2 -1 0 1 0 2 2],[-1 -3 -1 0 0 2 -1],[-1 -2 0 0 0 1 -1],[-4 -4 -2 -2 -1 0 -4],[ 1 -2 -2 1 1 4 0]]
Primitive based matrix [[ 0 4 1 1 -1 -2 -3],[-4 0 -1 -2 -4 -2 -4],[-1 1 0 0 -1 0 -2],[-1 2 0 0 -1 -1 -3],[ 1 4 1 1 0 -2 -2],[ 2 2 0 1 2 0 -1],[ 3 4 2 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,-1,1,2,3,1,2,4,2,4,0,1,0,2,1,1,3,2,2,1]
Phi over symmetry [-4,-1,-1,1,2,3,1,2,1,4,3,0,1,2,1,1,3,2,-1,0,0]
Phi of -K [-3,-2,-1,1,1,4,0,0,1,2,3,-1,2,3,4,1,1,1,0,1,2]
Phi of K* [-4,-1,-1,1,2,3,1,2,1,4,3,0,1,2,1,1,3,2,-1,0,0]
Phi of -K* [-3,-2,-1,1,1,4,1,2,2,3,4,2,0,1,2,1,1,4,0,1,2]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial t^6+66t^4+172t^2+1
Outer characteristic polynomial t^7+98t^5+301t^3+8t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial 128*K1**2*K2*K3**2 + 744*K1**2*K2 - 1664*K1**2*K3**2 - 2108*K1**2 - 608*K1*K2**2*K3 - 480*K1*K2*K3*K4 + 3680*K1*K2*K3 - 32*K1*K3**2*K5 + 2216*K1*K3*K4 + 104*K1*K4*K5 + 56*K1*K5*K6 + 8*K1*K6*K7 - 816*K2**2*K3**2 - 8*K2**2*K4**2 + 696*K2**2*K4 - 8*K2**2*K6**2 - 2218*K2**2 - 32*K2*K3*K4*K5 + 1224*K2*K3*K5 + 24*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 + 16*K3**2*K6 - 2104*K3**2 + 40*K3*K4*K7 + 24*K3*K5*K8 - 870*K4**2 - 404*K5**2 - 46*K6**2 - 40*K7**2 - 18*K8**2 + 2430
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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