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Flat knot 6.1906

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,0,1,0,0,2,0,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1906']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.133', '6.517', '6.545', '6.1198', '6.1251', '6.1906']
Outer characteristic polynomial of the knot is: t^7+25t^5+107t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1906']
2-strand cable arrow polynomial of the knot is: 768*K1**2*K2**5 - 2432*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2208*K1**2*K2**3 - 2880*K1**2*K2**2 - 192*K1**2*K2*K4 + 1744*K1**2*K2 - 1184*K1**2 + 896*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 - 384*K1*K2**3*K3*K4 + 2944*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2144*K1*K2*K3 + 192*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 1856*K2**6 - 1152*K2**4*K3**2 - 192*K2**4*K4**2 + 1536*K2**4*K4 - 1600*K2**4 + 672*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1408*K2**2*K3**2 - 288*K2**2*K4**2 + 1168*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 + 288*K2**2 + 544*K2*K3*K5 + 56*K2*K4*K6 - 464*K3**2 - 178*K4**2 - 32*K5**2 - 8*K6**2 + 920
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1906']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4761', 'vk6.5092', 'vk6.6318', 'vk6.6756', 'vk6.8280', 'vk6.8734', 'vk6.9656', 'vk6.9969', 'vk6.20718', 'vk6.22166', 'vk6.28278', 'vk6.29698', 'vk6.39733', 'vk6.41987', 'vk6.46295', 'vk6.47880', 'vk6.48795', 'vk6.49008', 'vk6.49622', 'vk6.49826', 'vk6.50825', 'vk6.51043', 'vk6.51298', 'vk6.51493', 'vk6.57651', 'vk6.58799', 'vk6.62326', 'vk6.63264', 'vk6.67117', 'vk6.67984', 'vk6.69711', 'vk6.70390']
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 O1O2O3U1U2U3O4O5O6U4U6U5
R3 orbit {'O1O2O3U1U2U3O4O5O6U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U6O4O5O6U2U1U3
Gauss code of K* O1O2O3U4U5U6O4O5O6U1U3U2
Gauss code of -K* O1O2O3U1U2U3O4O5O6U5U4U6
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 0 2 -2 1 1],[ 2 0 1 2 0 0 0],[ 0 -1 0 1 0 0 0],[-2 -2 -1 0 0 0 0],[ 2 0 0 0 0 2 1],[-1 0 0 0 -2 0 0],[-1 0 0 0 -1 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -1 0 -2],[-1 0 0 0 0 -1 0],[-1 0 0 0 0 -2 0],[ 0 1 0 0 0 0 -1],[ 2 0 1 2 0 0 0],[ 2 2 0 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,0,0,1,0,2,0,0,1,0,0,2,0,0,1,0]
Phi over symmetry [-2,-2,0,1,1,2,0,0,1,2,0,1,0,0,2,0,0,1,0,0,0]
Phi of -K [-2,-2,0,1,1,2,0,1,3,3,2,2,1,2,4,1,1,1,0,1,1]
Phi of K* [-2,-1,-1,0,2,2,1,1,1,2,4,0,1,3,1,1,3,2,1,2,0]
Phi of -K* [-2,-2,0,1,1,2,0,0,1,2,0,1,0,0,2,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+8z+5
Enhanced Jones-Krushkal polynomial -6w^4z^2+9w^3z^2-10w^3z+18w^2z+5w
Inner characteristic polynomial t^6+11t^4+30t^2
Outer characteristic polynomial t^7+25t^5+107t^3+6t
Flat arrow polynomial -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 768*K1**2*K2**5 - 2432*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2208*K1**2*K2**3 - 2880*K1**2*K2**2 - 192*K1**2*K2*K4 + 1744*K1**2*K2 - 1184*K1**2 + 896*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 - 384*K1*K2**3*K3*K4 + 2944*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2144*K1*K2*K3 + 192*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 1856*K2**6 - 1152*K2**4*K3**2 - 192*K2**4*K4**2 + 1536*K2**4*K4 - 1600*K2**4 + 672*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1408*K2**2*K3**2 - 288*K2**2*K4**2 + 1168*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 + 288*K2**2 + 544*K2*K3*K5 + 56*K2*K4*K6 - 464*K3**2 - 178*K4**2 - 32*K5**2 - 8*K6**2 + 920
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {1, 3}, {2}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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