| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,3,1,0,2,2,0,1,0,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1831'] |
| 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+27t^5+56t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1831'] |
| 2-strand cable arrow polynomial of the knot is: -32*K1**4 + 64*K1**3*K2*K3 - 640*K1**2*K2**2 + 352*K1**2*K2 - 224*K1**2*K3**2 - 32*K1**2*K5**2 - 984*K1**2 + 192*K1*K2**3*K3 + 2048*K1*K2*K3 + 224*K1*K3*K4 + 96*K1*K4*K5 + 80*K1*K5*K6 - 144*K2**4 - 352*K2**2*K3**2 - 16*K2**2*K4**2 + 48*K2**2*K4 - 852*K2**2 + 352*K2*K3*K5 + 32*K2*K4*K6 - 960*K3**2 - 124*K4**2 - 184*K5**2 - 44*K6**2 + 1098 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1831'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71457', 'vk6.71494', 'vk6.71510', 'vk6.71551', 'vk6.71983', 'vk6.72040', 'vk6.72539', 'vk6.72648', 'vk6.72929', 'vk6.72966', 'vk6.73125', 'vk6.77078', 'vk6.77114', 'vk6.77128', 'vk6.77167', 'vk6.77464', 'vk6.81288', 'vk6.81438', 'vk6.81535', 'vk6.85485', 'vk6.86885', 'vk6.86912', 'vk6.87257', 'vk6.87725', 'vk6.89347', 'vk6.89504'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3U1U4O5U3O4O6U5U2U6 |
| R3 orbit | {'O1O2O3U1U4O5U3O4O6U5U2U6', 'O1O2O3U1U4U2O5O4O6U3U5U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U2U5O4O6U1O5U6U3 |
| Gauss code of K* | O1O2O3U4U2U5O4O6U1O5U6U3 |
| Gauss code of -K* | Same |
| Diagrammatic symmetry type | - |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 0 1 0 -1 2],[ 2 0 2 1 1 1 1],[ 0 -2 0 0 0 0 2],[-1 -1 0 0 -1 0 1],[ 0 -1 0 1 0 -1 1],[ 1 -1 0 0 1 0 1],[-2 -1 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -1 -1],[-1 1 0 -1 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 0 2 0 0 0 0 -2],[ 1 1 0 1 0 0 -1],[ 2 1 1 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,1,2,1,1,1,0,0,1,0,1,1,0,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,2,3,1,0,2,2,0,1,0,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,2,3,1,0,2,2,0,1,0,0,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,0,1,2,3,1,0,2,2,0,1,0,0,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,1,1,1,0,0,1,0,1,1,0,2,1] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z+13 |
| Enhanced Jones-Krushkal polynomial | -6w^3z+12w^2z+13w |
| Inner characteristic polynomial | t^6+17t^4+26t^2 |
| Outer characteristic polynomial | t^7+27t^5+56t^3 |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -32*K1**4 + 64*K1**3*K2*K3 - 640*K1**2*K2**2 + 352*K1**2*K2 - 224*K1**2*K3**2 - 32*K1**2*K5**2 - 984*K1**2 + 192*K1*K2**3*K3 + 2048*K1*K2*K3 + 224*K1*K3*K4 + 96*K1*K4*K5 + 80*K1*K5*K6 - 144*K2**4 - 352*K2**2*K3**2 - 16*K2**2*K4**2 + 48*K2**2*K4 - 852*K2**2 + 352*K2*K3*K5 + 32*K2*K4*K6 - 960*K3**2 - 124*K4**2 - 184*K5**2 - 44*K6**2 + 1098 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}]] |
| If K is slice | True |