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Flat knot 6.1301

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,1,3,3,0,0,2,2,0,0,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1301']
Arrow polynomial of the knot is: -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.665', '6.1301', '6.1514', '6.1646', '6.1669', '6.1709', '6.1710', '6.1744', '6.1776']
Outer characteristic polynomial of the knot is: t^7+45t^5+49t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1301']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 192*K1**4*K2**2 + 2080*K1**4*K2 - 6640*K1**4 + 928*K1**3*K2*K3 + 160*K1**3*K3*K4 - 1856*K1**3*K3 - 4976*K1**2*K2**2 - 896*K1**2*K2*K4 + 12600*K1**2*K2 - 1584*K1**2*K3**2 - 128*K1**2*K3*K5 - 176*K1**2*K4**2 - 6876*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 9136*K1*K2*K3 + 2400*K1*K3*K4 + 240*K1*K4*K5 - 376*K2**4 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 1064*K2**2*K4 - 6204*K2**2 + 216*K2*K3*K5 + 32*K2*K4*K6 - 3232*K3**2 - 1014*K4**2 - 116*K5**2 - 12*K6**2 + 6524
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1301']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3665', 'vk6.3762', 'vk6.3951', 'vk6.4048', 'vk6.4491', 'vk6.4588', 'vk6.5877', 'vk6.6006', 'vk6.7158', 'vk6.7331', 'vk6.7424', 'vk6.7922', 'vk6.8043', 'vk6.9356', 'vk6.17903', 'vk6.17998', 'vk6.18739', 'vk6.24438', 'vk6.24864', 'vk6.25325', 'vk6.37486', 'vk6.43873', 'vk6.44219', 'vk6.44522', 'vk6.48289', 'vk6.48354', 'vk6.50080', 'vk6.50190', 'vk6.50583', 'vk6.50648', 'vk6.55862', 'vk6.60719']
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 O1O2O3U2O4O5U6U4O6U1U5U3
R3 orbit {'O1O2O3U2O4O5U6U4O6U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O5U6U5O4O6U2
Gauss code of K* O1O2O3U1U4U3O4U5U2O6O5U6
Gauss code of -K* O1O2O3U4O5O4U2U5O6U1U6U3
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 -1 2 0 2 -1],[ 2 0 0 3 1 2 1],[ 1 0 0 1 0 0 1],[-2 -3 -1 0 0 1 -3],[ 0 -1 0 0 0 0 0],[-2 -2 0 -1 0 0 -2],[ 1 -1 -1 3 0 2 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 1 0 -1 -3 -3],[-2 -1 0 0 0 -2 -2],[ 0 0 0 0 0 0 -1],[ 1 1 0 0 0 1 0],[ 1 3 2 0 -1 0 -1],[ 2 3 2 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-1,0,1,3,3,0,0,2,2,0,0,1,-1,0,1]
Phi over symmetry [-2,-2,0,1,1,2,-1,0,1,3,3,0,0,2,2,0,0,1,-1,0,1]
Phi of -K [-2,-1,-1,0,2,2,0,1,1,1,2,1,1,0,1,1,2,3,2,2,-1]
Phi of K* [-2,-2,0,1,1,2,-1,2,1,3,2,2,0,2,1,1,1,1,-1,0,1]
Phi of -K* [-2,-1,-1,0,2,2,0,1,1,2,3,1,0,0,1,0,2,3,0,0,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+31t^4+30t^2+1
Outer characteristic polynomial t^7+45t^5+49t^3+4t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -384*K1**6 - 192*K1**4*K2**2 + 2080*K1**4*K2 - 6640*K1**4 + 928*K1**3*K2*K3 + 160*K1**3*K3*K4 - 1856*K1**3*K3 - 4976*K1**2*K2**2 - 896*K1**2*K2*K4 + 12600*K1**2*K2 - 1584*K1**2*K3**2 - 128*K1**2*K3*K5 - 176*K1**2*K4**2 - 6876*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 9136*K1*K2*K3 + 2400*K1*K3*K4 + 240*K1*K4*K5 - 376*K2**4 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 1064*K2**2*K4 - 6204*K2**2 + 216*K2*K3*K5 + 32*K2*K4*K6 - 3232*K3**2 - 1014*K4**2 - 116*K5**2 - 12*K6**2 + 6524
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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