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Flat knot 6.1141

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,0,1,0,1,2,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1141', '7.24265']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.404', '6.843', '6.1141']
Outer characteristic polynomial of the knot is: t^7+42t^5+82t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1141']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 416*K1**4*K2 - 704*K1**4 + 512*K1**3*K2**3*K3 + 1632*K1**3*K2*K3 - 128*K1**3*K3 - 2816*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3168*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7728*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 4216*K1**2*K2 - 1216*K1**2*K3**2 - 32*K1**2*K4**2 - 1700*K1**2 + 640*K1*K2**5*K3 - 384*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4992*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1920*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 96*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6312*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1128*K1*K3*K4 + 72*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 576*K2**6 - 768*K2**4*K3**2 - 32*K2**4*K4**2 + 544*K2**4*K4 - 3456*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 2496*K2**2*K3**2 - 64*K2**2*K3*K7 - 248*K2**2*K4**2 - 32*K2**2*K4*K8 + 2208*K2**2*K4 - 48*K2**2*K5**2 - 16*K2**2*K6**2 - 22*K2**2 - 32*K2*K3**2*K4 + 1064*K2*K3*K5 + 136*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1344*K3**2 - 384*K4**2 - 104*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1824
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1141']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.305', 'vk6.340', 'vk6.345', 'vk6.691', 'vk6.694', 'vk6.737', 'vk6.742', 'vk6.1481', 'vk6.1490', 'vk6.1939', 'vk6.1974', 'vk6.1979', 'vk6.2452', 'vk6.2631', 'vk6.2636', 'vk6.3106', 'vk6.18260', 'vk6.18263', 'vk6.18595', 'vk6.18600', 'vk6.24744', 'vk6.24747', 'vk6.25150', 'vk6.36873', 'vk6.36878', 'vk6.37334', 'vk6.44095', 'vk6.44098', 'vk6.56058', 'vk6.56063', 'vk6.60613', 'vk6.65729']
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 O1O2O3O4U1U2U4O5O6U5U6U3
R3 orbit {'O1O2O3O4U1U2U4O5O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U6O5O6U1U3U4
Gauss code of K* O1O2O3U4U5O4O5O6U1U2U6U3
Gauss code of -K* O1O2O3U2U3O4O5O6U4U1U5U6
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 2 -1 1],[ 3 0 1 3 2 0 0],[ 1 -1 0 2 1 0 0],[-2 -3 -2 0 0 -1 1],[-2 -2 -1 0 0 0 0],[ 1 0 0 1 0 0 1],[-1 0 0 -1 0 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 1 -1 -2 -3],[-2 0 0 0 0 -1 -2],[-1 -1 0 0 -1 0 0],[ 1 1 0 1 0 0 0],[ 1 2 1 0 0 0 -1],[ 3 3 2 0 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,-1,1,2,3,0,0,1,2,1,0,0,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,0,1,0,1,2,0,-1,0]
Phi of -K [-3,-1,-1,1,2,2,1,2,4,2,3,0,2,1,2,1,2,3,2,1,0]
Phi of K* [-2,-2,-1,1,1,3,0,1,2,3,3,2,1,2,2,2,1,4,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,0,2,3,0,1,0,1,0,1,2,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial t^6+22t^4+32t^2+1
Outer characteristic polynomial t^7+42t^5+82t^3+11t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 416*K1**4*K2 - 704*K1**4 + 512*K1**3*K2**3*K3 + 1632*K1**3*K2*K3 - 128*K1**3*K3 - 2816*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 3168*K1**2*K2**3 - 512*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7728*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 4216*K1**2*K2 - 1216*K1**2*K3**2 - 32*K1**2*K4**2 - 1700*K1**2 + 640*K1*K2**5*K3 - 384*K1*K2**4*K3 - 256*K1*K2**4*K5 + 4992*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1920*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 96*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6312*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1128*K1*K3*K4 + 72*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 576*K2**6 - 768*K2**4*K3**2 - 32*K2**4*K4**2 + 544*K2**4*K4 - 3456*K2**4 + 448*K2**3*K3*K5 + 64*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 2496*K2**2*K3**2 - 64*K2**2*K3*K7 - 248*K2**2*K4**2 - 32*K2**2*K4*K8 + 2208*K2**2*K4 - 48*K2**2*K5**2 - 16*K2**2*K6**2 - 22*K2**2 - 32*K2*K3**2*K4 + 1064*K2*K3*K5 + 136*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1344*K3**2 - 384*K4**2 - 104*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1824
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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