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Flat knot 6.909

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,2,3,1,0,1,1,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.909']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 14*K1**2 - 10*K1*K2 - 4*K1*K3 - K1 + 7*K2 + 3*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.909']
Outer characteristic polynomial of the knot is: t^7+50t^5+33t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.909']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 960*K1**4*K2**2 + 3008*K1**4*K2 - 5088*K1**4 - 256*K1**3*K2**2*K3 + 1120*K1**3*K2*K3 + 96*K1**3*K3*K4 - 800*K1**3*K3 - 384*K1**2*K2**4 + 1408*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 8192*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 11632*K1**2*K2 - 736*K1**2*K3**2 - 64*K1**2*K3*K5 - 176*K1**2*K4**2 - 6920*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9584*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 + 2320*K1*K3*K4 + 560*K1*K4*K5 + 64*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1544*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 880*K2**2*K3**2 - 64*K2**2*K3*K7 - 376*K2**2*K4**2 - 32*K2**2*K4*K8 + 2904*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 6582*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1248*K2*K3*K5 + 328*K2*K4*K6 + 88*K2*K5*K7 + 16*K2*K6*K8 + 80*K3**2*K6 - 3352*K3**2 + 16*K3*K4*K7 - 1574*K4**2 - 480*K5**2 - 90*K6**2 - 16*K7**2 - 2*K8**2 + 6838
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.909']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4424', 'vk6.4519', 'vk6.5810', 'vk6.5937', 'vk6.7873', 'vk6.7982', 'vk6.9297', 'vk6.9416', 'vk6.10157', 'vk6.10228', 'vk6.10375', 'vk6.17897', 'vk6.17960', 'vk6.18275', 'vk6.18612', 'vk6.24404', 'vk6.25167', 'vk6.30044', 'vk6.30105', 'vk6.36893', 'vk6.37353', 'vk6.43831', 'vk6.44114', 'vk6.44439', 'vk6.48627', 'vk6.50532', 'vk6.50619', 'vk6.51143', 'vk6.51666', 'vk6.55846', 'vk6.56066', 'vk6.65508']
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 O1O2O3O4U2U5O6O5U3U6U1U4
R3 orbit {'O1O2O3O4U2U5O6O5U3U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U5U2O6O5U6U3
Gauss code of K* O1O2O3O4U3U5U1U4O5O6U2U6
Gauss code of -K* O1O2O3O4U5U3O5O6U1U4U6U2
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 -1 -2 -1 3 1 0],[ 1 0 -1 0 3 2 0],[ 2 1 0 1 2 2 1],[ 1 0 -1 0 2 1 0],[-3 -3 -2 -2 0 -2 -1],[-1 -2 -2 -1 2 0 0],[ 0 0 -1 0 1 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 -2 -1 -2 -3 -2],[-1 2 0 0 -1 -2 -2],[ 0 1 0 0 0 0 -1],[ 1 2 1 0 0 0 -1],[ 1 3 2 0 0 0 -1],[ 2 2 2 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,2,1,2,3,2,0,1,2,2,0,0,1,0,1,1]
Phi over symmetry [-3,-1,0,1,1,2,0,2,1,2,3,1,0,1,1,1,1,1,0,0,0]
Phi of -K [-2,-1,-1,0,1,3,0,0,1,1,3,0,1,0,1,1,1,2,1,2,0]
Phi of K* [-3,-1,0,1,1,2,0,2,1,2,3,1,0,1,1,1,1,1,0,0,0]
Phi of -K* [-2,-1,-1,0,1,3,1,1,1,2,2,0,0,1,2,0,2,3,0,1,2]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+34t^4+18t^2
Outer characteristic polynomial t^7+50t^5+33t^3+3t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 14*K1**2 - 10*K1*K2 - 4*K1*K3 - K1 + 7*K2 + 3*K3 + K4 + 7
2-strand cable arrow polynomial 256*K1**4*K2**3 - 960*K1**4*K2**2 + 3008*K1**4*K2 - 5088*K1**4 - 256*K1**3*K2**2*K3 + 1120*K1**3*K2*K3 + 96*K1**3*K3*K4 - 800*K1**3*K3 - 384*K1**2*K2**4 + 1408*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 8192*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 11632*K1**2*K2 - 736*K1**2*K3**2 - 64*K1**2*K3*K5 - 176*K1**2*K4**2 - 6920*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2112*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9584*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 + 2320*K1*K3*K4 + 560*K1*K4*K5 + 64*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1544*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 880*K2**2*K3**2 - 64*K2**2*K3*K7 - 376*K2**2*K4**2 - 32*K2**2*K4*K8 + 2904*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 6582*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1248*K2*K3*K5 + 328*K2*K4*K6 + 88*K2*K5*K7 + 16*K2*K6*K8 + 80*K3**2*K6 - 3352*K3**2 + 16*K3*K4*K7 - 1574*K4**2 - 480*K5**2 - 90*K6**2 - 16*K7**2 - 2*K8**2 + 6838
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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