| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,2,2,3,0,0,2,1,1,2,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1203'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+30t^5+57t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1203'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 352*K1**4*K2 - 1472*K1**4 + 192*K1**3*K2*K3 - 352*K1**3*K3 - 1104*K1**2*K2**2 - 128*K1**2*K2*K4 + 3096*K1**2*K2 - 544*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 2052*K1**2 - 96*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2360*K1*K2*K3 - 32*K1*K3**2*K5 + 736*K1*K3*K4 + 184*K1*K4*K5 + 40*K1*K5*K6 - 16*K2**4 - 16*K2**2*K3**2 - 16*K2**2*K4**2 + 216*K2**2*K4 - 1764*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 + 40*K3**2*K6 - 996*K3**2 - 332*K4**2 - 144*K5**2 - 44*K6**2 + 1922 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1203'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4063', 'vk6.4096', 'vk6.4248', 'vk6.4329', 'vk6.5305', 'vk6.5338', 'vk6.5521', 'vk6.5525', 'vk6.5642', 'vk6.5646', 'vk6.7464', 'vk6.7713', 'vk6.8936', 'vk6.8969', 'vk6.9113', 'vk6.9194', 'vk6.14548', 'vk6.15280', 'vk6.15409', 'vk6.15772', 'vk6.16187', 'vk6.26279', 'vk6.26722', 'vk6.29841', 'vk6.29874', 'vk6.33926', 'vk6.34223', 'vk6.38229', 'vk6.38237', 'vk6.45006', 'vk6.45014', 'vk6.48558', 'vk6.49154', 'vk6.49263', 'vk6.49267', 'vk6.50230', 'vk6.51593', 'vk6.53977', 'vk6.54478', 'vk6.63306'] |
| 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 | O1O2O3O4U2U4U1O5O6U5U3U6 |
| R3 orbit | {'O1O2O3U1O4U2U4O5O6U5U3U6', 'O1O2O3O4U2U4U1O5O6U5U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U2U6O5O6U4U1U3 |
| Gauss code of K* | O1O2O3U4U5O4O6O5U3U1U6U2 |
| Gauss code of -K* | O1O2O3U1U3O4O5O6U5U2U6U4 |
| 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 1 -1 2],[ 1 0 -1 2 1 0 1],[ 2 1 0 2 1 0 1],[-1 -2 -2 0 0 0 2],[-1 -1 -1 0 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -1 -2 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -1 -1],[-1 0 0 0 0 -1 -1],[-1 2 0 0 0 -2 -2],[ 1 1 0 0 0 0 0],[ 1 1 1 2 0 0 -1],[ 2 1 1 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,2,1,1,1,0,0,1,1,0,2,2,0,0,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,1,2,2,3,0,0,2,1,1,2,2,0,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,0,1,1,2,3,0,0,1,2,2,2,2,0,-1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,-1,1,2,2,3,0,0,2,1,1,2,2,0,0,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,1,1,2,1,0,0,0,1,1,2,1,0,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+18t^4+23t^2 |
| Outer characteristic polynomial | t^7+30t^5+57t^3+3t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -64*K1**6 + 352*K1**4*K2 - 1472*K1**4 + 192*K1**3*K2*K3 - 352*K1**3*K3 - 1104*K1**2*K2**2 - 128*K1**2*K2*K4 + 3096*K1**2*K2 - 544*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 2052*K1**2 - 96*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2360*K1*K2*K3 - 32*K1*K3**2*K5 + 736*K1*K3*K4 + 184*K1*K4*K5 + 40*K1*K5*K6 - 16*K2**4 - 16*K2**2*K3**2 - 16*K2**2*K4**2 + 216*K2**2*K4 - 1764*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 + 40*K3**2*K6 - 996*K3**2 - 332*K4**2 - 144*K5**2 - 44*K6**2 + 1922 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |