| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,1,1,1,0,1,0,1,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.241', '7.23651'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874'] |
| Outer characteristic polynomial of the knot is: t^7+30t^5+42t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.241', '7.23651'] |
| 2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1536*K1**4*K2 - 2720*K1**4 + 256*K1**3*K2*K3 - 256*K1**3*K3 - 576*K1**2*K2**4 + 1856*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5312*K1**2*K2**2 - 352*K1**2*K2*K4 + 4440*K1**2*K2 - 256*K1**2*K3**2 - 684*K1**2 + 544*K1*K2**3*K3 - 768*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3288*K1*K2*K3 + 240*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1288*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 840*K2**2*K4 - 662*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 404*K3**2 - 94*K4**2 - 24*K5**2 - 2*K6**2 + 1204 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.241'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.47', 'vk6.94', 'vk6.189', 'vk6.246', 'vk6.286', 'vk6.661', 'vk6.668', 'vk6.786', 'vk6.1238', 'vk6.1327', 'vk6.1386', 'vk6.1432', 'vk6.1472', 'vk6.1601', 'vk6.1912', 'vk6.2078', 'vk6.2421', 'vk6.2492', 'vk6.2617', 'vk6.2965', 'vk6.3773', 'vk6.3964', 'vk6.7161', 'vk6.7336', 'vk6.14594', 'vk6.15814', 'vk6.16215', 'vk6.17770', 'vk6.24270', 'vk6.24922', 'vk6.25383', 'vk6.25935', 'vk6.29815', 'vk6.33406', 'vk6.33556', 'vk6.38010', 'vk6.38065', 'vk6.45040', 'vk6.53729', 'vk6.60755'] |
| 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 | O1O2O3O4O5U3O6U5U6U4U1U2 |
| R3 orbit | {'O1O2O3O4O5U3U4O6U5U6U1U2', 'O1O2O3O4O5U3O6U5U6U4U1U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U5U2U6U1O6U3 |
| Gauss code of K* | O1O2O3O4O5U4U5U6U3U1O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U5U3U6U1U2 |
| 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 1 -2 1 0 1],[ 1 0 1 -2 1 0 1],[-1 -1 0 -2 1 0 1],[ 2 2 2 0 2 1 1],[-1 -1 -1 -2 0 -1 1],[ 0 0 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -1 -2],[-1 -1 0 1 -1 -1 -2],[-1 -1 -1 0 -1 -1 -1],[ 0 0 1 1 0 0 -1],[ 1 1 1 1 0 0 -2],[ 2 2 2 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,1,2,-1,1,1,2,1,1,1,0,1,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,1,1,1,0,1,0,1,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,-1,1,1,1,2,1,1,1,1,0,1,0,1,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,0,1,2,-1,0,1,1,1,1,1,1,1,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,1,1,0,1,-1,-1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+22t^4+15t^2 |
| Outer characteristic polynomial | t^7+30t^5+42t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -640*K1**4*K2**2 + 1536*K1**4*K2 - 2720*K1**4 + 256*K1**3*K2*K3 - 256*K1**3*K3 - 576*K1**2*K2**4 + 1856*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5312*K1**2*K2**2 - 352*K1**2*K2*K4 + 4440*K1**2*K2 - 256*K1**2*K3**2 - 684*K1**2 + 544*K1*K2**3*K3 - 768*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3288*K1*K2*K3 + 240*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1288*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 840*K2**2*K4 - 662*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 404*K3**2 - 94*K4**2 - 24*K5**2 - 2*K6**2 + 1204 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]] |
| If K is slice | False |