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Flat knot 6.445

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,2,4,4,2,0,0,1,0,0,1,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.445']
Arrow polynomial of the knot is: -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.108', '6.157', '6.283', '6.399', '6.445', '6.510']
Outer characteristic polynomial of the knot is: t^7+71t^5+111t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.445']
2-strand cable arrow polynomial of the knot is: 416*K1**4*K2 - 2192*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1728*K1**3*K3 + 96*K1**2*K2**2*K4 - 1440*K1**2*K2**2 - 960*K1**2*K2*K4 + 6576*K1**2*K2 - 528*K1**2*K3**2 - 128*K1**2*K4**2 - 5176*K1**2 + 64*K1*K2**3*K3 - 448*K1*K2**2*K3 - 256*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 5624*K1*K2*K3 - 32*K1*K3**2*K5 + 1824*K1*K3*K4 + 472*K1*K4*K5 + 48*K1*K5*K6 - 64*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1136*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4184*K2**2 - 32*K2*K3**2*K4 + 720*K2*K3*K5 + 136*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 160*K3**2*K6 - 2404*K3**2 + 8*K3*K4*K7 - 1094*K4**2 - 392*K5**2 - 128*K6**2 - 12*K7**2 - 2*K8**2 + 4246
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.445']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16974', 'vk6.16976', 'vk6.17217', 'vk6.17219', 'vk6.20872', 'vk6.20885', 'vk6.22281', 'vk6.22292', 'vk6.23377', 'vk6.23679', 'vk6.23685', 'vk6.28347', 'vk6.35431', 'vk6.35865', 'vk6.35875', 'vk6.39975', 'vk6.39996', 'vk6.42048', 'vk6.43172', 'vk6.43178', 'vk6.46515', 'vk6.46536', 'vk6.55127', 'vk6.55137', 'vk6.55386', 'vk6.57679', 'vk6.57692', 'vk6.58876', 'vk6.59841', 'vk6.59863', 'vk6.68400', 'vk6.69743']
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 O1O2O3O4O5U6U2O6U4U3U1U5
R3 orbit {'O1O2O3O4O5U6U2O6U4U3U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U5U3U2O6U4U6
Gauss code of K* O1O2O3O4U3U5U2U1U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U4U3U6U2
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 0 0 4 -1],[ 1 0 -1 1 1 4 0],[ 2 1 0 1 0 2 2],[ 0 -1 -1 0 0 2 0],[ 0 -1 0 0 0 1 0],[-4 -4 -2 -2 -1 0 -4],[ 1 0 -2 0 0 4 0]]
Primitive based matrix [[ 0 4 0 0 -1 -1 -2],[-4 0 -1 -2 -4 -4 -2],[ 0 1 0 0 0 -1 0],[ 0 2 0 0 0 -1 -1],[ 1 4 0 0 0 0 -2],[ 1 4 1 1 0 0 -1],[ 2 2 0 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,0,0,1,1,2,1,2,4,4,2,0,0,1,0,0,1,1,0,2,1]
Phi over symmetry [-4,0,0,1,1,2,1,2,4,4,2,0,0,1,0,0,1,1,0,2,1]
Phi of -K [-2,-1,-1,0,0,4,-1,0,1,2,4,0,1,1,1,0,0,1,0,2,3]
Phi of K* [-4,0,0,1,1,2,2,3,1,1,4,0,0,1,1,0,1,2,0,0,-1]
Phi of -K* [-2,-1,-1,0,0,4,1,2,0,1,2,0,1,1,4,0,0,4,0,1,2]
Symmetry type of based matrix c
u-polynomial -t^4+t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial 5w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+49t^4+72t^2+4
Outer characteristic polynomial t^7+71t^5+111t^3+9t
Flat arrow polynomial -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial 416*K1**4*K2 - 2192*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1728*K1**3*K3 + 96*K1**2*K2**2*K4 - 1440*K1**2*K2**2 - 960*K1**2*K2*K4 + 6576*K1**2*K2 - 528*K1**2*K3**2 - 128*K1**2*K4**2 - 5176*K1**2 + 64*K1*K2**3*K3 - 448*K1*K2**2*K3 - 256*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 5624*K1*K2*K3 - 32*K1*K3**2*K5 + 1824*K1*K3*K4 + 472*K1*K4*K5 + 48*K1*K5*K6 - 64*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1136*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4184*K2**2 - 32*K2*K3**2*K4 + 720*K2*K3*K5 + 136*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 160*K3**2*K6 - 2404*K3**2 + 8*K3*K4*K7 - 1094*K4**2 - 392*K5**2 - 128*K6**2 - 12*K7**2 - 2*K8**2 + 4246
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
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