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Flat knot 6.1720

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,2,0,1,1,1,0,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1720', '7.38551']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^7+24t^5+51t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1720', '7.38551']
2-strand cable arrow polynomial of the knot is: 3840*K1**4*K2 - 6144*K1**4 + 1536*K1**3*K2*K3 - 1408*K1**3*K3 - 384*K1**2*K2**4 + 3008*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 10096*K1**2*K2**2 - 1408*K1**2*K2*K4 + 5816*K1**2*K2 - 1024*K1**2*K3**2 - 96*K1**2*K4**2 + 1448*K1**2 + 1376*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 352*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 5088*K1*K2*K3 + 640*K1*K3*K4 + 80*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2568*K2**4 - 32*K2**3*K6 - 720*K2**2*K3**2 - 208*K2**2*K4**2 + 1472*K2**2*K4 + 546*K2**2 + 344*K2*K3*K5 + 48*K2*K4*K6 - 340*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 700
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1720']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.511', 'vk6.603', 'vk6.637', 'vk6.1009', 'vk6.1106', 'vk6.1150', 'vk6.1866', 'vk6.2290', 'vk6.2520', 'vk6.2568', 'vk6.2600', 'vk6.2806', 'vk6.2899', 'vk6.2925', 'vk6.3086', 'vk6.3207', 'vk6.5290', 'vk6.6552', 'vk6.8920', 'vk6.9836', 'vk6.20842', 'vk6.21119', 'vk6.22240', 'vk6.22551', 'vk6.28570', 'vk6.29801', 'vk6.39902', 'vk6.42220', 'vk6.46459', 'vk6.46841', 'vk6.48007', 'vk6.48080']
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 O1O2O3U1U4O5O6U5U6O4U3U2
R3 orbit {'O1O2O3U1U4O5O6U5U6O4U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U1O4U5U6O5O6U4U3
Gauss code of K* O1O2U3O4O5U6U5U4O6O3U1U2
Gauss code of -K* O1O2U3O4O5U4U5O3O6U2U1U6
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 1 0 -1 1],[ 2 0 2 1 2 0 0],[-1 -2 0 0 -1 -1 1],[-1 -1 0 0 -1 -1 1],[ 0 -2 1 1 0 0 0],[ 1 0 1 1 0 0 1],[-1 0 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 0 -1 0],[-1 0 1 0 -1 -1 -2],[ 0 1 0 1 0 0 -2],[ 1 1 1 1 0 0 0],[ 2 1 0 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,1,1,1,0,1,0,1,1,2,0,2,0]
Phi over symmetry [-2,-1,0,1,1,1,0,2,0,1,2,0,1,1,1,0,1,1,-1,-1,0]
Phi of -K [-2,-1,0,1,1,1,1,0,1,2,3,1,1,1,1,0,0,1,0,-1,-1]
Phi of K* [-1,-1,-1,0,1,2,-1,-1,1,1,3,0,0,1,1,0,1,2,1,0,1]
Phi of -K* [-2,-1,0,1,1,1,0,2,0,1,2,0,1,1,1,0,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+16t^4+30t^2+1
Outer characteristic polynomial t^7+24t^5+51t^3+4t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 3840*K1**4*K2 - 6144*K1**4 + 1536*K1**3*K2*K3 - 1408*K1**3*K3 - 384*K1**2*K2**4 + 3008*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 10096*K1**2*K2**2 - 1408*K1**2*K2*K4 + 5816*K1**2*K2 - 1024*K1**2*K3**2 - 96*K1**2*K4**2 + 1448*K1**2 + 1376*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 352*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 5088*K1*K2*K3 + 640*K1*K3*K4 + 80*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2568*K2**4 - 32*K2**3*K6 - 720*K2**2*K3**2 - 208*K2**2*K4**2 + 1472*K2**2*K4 + 546*K2**2 + 344*K2*K3*K5 + 48*K2*K4*K6 - 340*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 700
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{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}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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