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Flat knot 6.408

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,0,2,3,1,1,1,1,1,1,2,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.408', '7.19580']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+46t^5+82t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.408', '7.19580']
2-strand cable arrow polynomial of the knot is: 512*K1**4*K2 - 2624*K1**4 - 256*K1**3*K2**2*K3 + 2688*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1504*K1**3*K3 - 512*K1**2*K2**4 + 320*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 5808*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 5408*K1**2*K2 - 2912*K1**2*K3**2 - 272*K1**2*K4**2 - 1836*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2880*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 704*K1*K2**2*K5 + 384*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 1024*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6792*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2016*K1*K3*K4 + 256*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1144*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 + 256*K2**2*K3**2*K4 - 2256*K2**2*K3**2 - 64*K2**2*K3*K7 + 32*K2**2*K4**3 - 264*K2**2*K4**2 - 32*K2**2*K4*K8 + 872*K2**2*K4 - 8*K2**2*K6**2 - 1330*K2**2 - 64*K2*K3**2*K4 + 1160*K2*K3*K5 + 168*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 - 80*K3**2*K4**2 + 32*K3**2*K6 - 1392*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 380*K4**2 - 64*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2060
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.408']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.475', 'vk6.538', 'vk6.541', 'vk6.938', 'vk6.1034', 'vk6.1039', 'vk6.1622', 'vk6.1721', 'vk6.1730', 'vk6.2117', 'vk6.2218', 'vk6.2223', 'vk6.2534', 'vk6.2832', 'vk6.3032', 'vk6.3165', 'vk6.20388', 'vk6.20390', 'vk6.21730', 'vk6.21733', 'vk6.27714', 'vk6.27718', 'vk6.29259', 'vk6.29264', 'vk6.39155', 'vk6.39162', 'vk6.45883', 'vk6.45886', 'vk6.57252', 'vk6.57256', 'vk6.61887', 'vk6.61895']
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 O1O2O3O4O5U4U5O6U2U1U3U6
R3 orbit {'O1O2O3O4O5U4U5O6U2U1U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U3U5U4O6U1U2
Gauss code of K* O1O2O3O4U2U1U3U5U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U6U2U4U3
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 1 3],[ 2 0 0 2 -1 1 3],[ 2 0 0 1 -1 1 2],[-1 -2 -1 0 -1 1 1],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 0 -1 0 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -2],[ 1 0 1 1 0 1 1],[ 2 2 1 1 -1 0 0],[ 2 3 1 2 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,0,1,0,2,3,1,1,1,1,1,1,2,-1,-1,0]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,0,2,3,1,1,1,1,1,1,2,-1,-1,0]
Phi of -K [-2,-2,-1,1,1,3,0,2,1,2,2,2,2,2,3,1,1,4,-1,1,2]
Phi of K* [-3,-1,-1,1,2,2,1,2,4,2,3,1,1,1,2,1,2,2,2,2,0]
Phi of -K* [-2,-2,-1,1,1,3,0,-1,1,1,2,-1,1,2,3,1,1,0,-1,0,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+26t^4+34t^2+1
Outer characteristic polynomial t^7+46t^5+82t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial 512*K1**4*K2 - 2624*K1**4 - 256*K1**3*K2**2*K3 + 2688*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1504*K1**3*K3 - 512*K1**2*K2**4 + 320*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 5808*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 5408*K1**2*K2 - 2912*K1**2*K3**2 - 272*K1**2*K4**2 - 1836*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 2880*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 704*K1*K2**2*K5 + 384*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 1024*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6792*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2016*K1*K3*K4 + 256*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1144*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 + 256*K2**2*K3**2*K4 - 2256*K2**2*K3**2 - 64*K2**2*K3*K7 + 32*K2**2*K4**3 - 264*K2**2*K4**2 - 32*K2**2*K4*K8 + 872*K2**2*K4 - 8*K2**2*K6**2 - 1330*K2**2 - 64*K2*K3**2*K4 + 1160*K2*K3*K5 + 168*K2*K4*K6 + 8*K2*K6*K8 - 128*K3**4 - 80*K3**2*K4**2 + 32*K3**2*K6 - 1392*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 380*K4**2 - 64*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2060
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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