| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,2,3,1,2,1,2,0,0,1,2,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.701'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+61t^5+87t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.701'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 1120*K1**4 + 160*K1**3*K2*K3 - 192*K1**3*K3 + 768*K1**2*K2**3 - 2416*K1**2*K2**2 - 160*K1**2*K2*K4 + 2728*K1**2*K2 - 1052*K1**2 + 160*K1*K2**3*K3 - 384*K1*K2**2*K3 + 1616*K1*K2*K3 + 16*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 536*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 312*K2**2*K4 - 640*K2**2 + 24*K2*K3*K5 - 244*K3**2 - 14*K4**2 + 884 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.701'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16335', 'vk6.16378', 'vk6.18060', 'vk6.18400', 'vk6.22670', 'vk6.22751', 'vk6.24503', 'vk6.24928', 'vk6.34616', 'vk6.34697', 'vk6.36640', 'vk6.37066', 'vk6.42309', 'vk6.42340', 'vk6.43926', 'vk6.44247', 'vk6.54606', 'vk6.54645', 'vk6.55888', 'vk6.56178', 'vk6.59095', 'vk6.59133', 'vk6.60412', 'vk6.60773', 'vk6.64641', 'vk6.64689', 'vk6.65522', 'vk6.65840', 'vk6.67996', 'vk6.68022', 'vk6.68608', 'vk6.68827', 'vk6.83062', 'vk6.83364', 'vk6.85820', 'vk6.86151', 'vk6.86838', 'vk6.89684', 'vk6.89929', 'vk6.90008'] |
| 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 | O1O2O3O4U5O6U2U1O5U3U4U6 |
| R3 orbit | {'O1O2O3O4U5U1O5U6U2U3O6U4', 'O1O2O3O4U5O6U2U1O5U3U4U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U1U2O6U4U3O5U6 |
| Gauss code of K* | O1O2O3U4U5U1U2O6U3O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U1O4U2U3U6U5 |
| 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 -2 0 2 -1 3],[ 2 0 0 1 2 1 3],[ 2 0 0 0 1 2 2],[ 0 -1 0 0 1 0 1],[-2 -2 -1 -1 0 -2 0],[ 1 -1 -2 0 2 0 3],[-3 -3 -2 -1 0 -3 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-2 0 0 -1 -2 -1 -2],[ 0 1 1 0 0 0 -1],[ 1 3 2 0 0 -2 -1],[ 2 2 1 0 2 0 0],[ 2 3 2 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,2,2,0,1,3,2,3,1,2,1,2,0,0,1,2,1,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,3,2,3,1,2,1,2,0,0,1,2,1,0] |
| Phi of -K | [-2,-2,-1,0,2,3,0,-1,2,3,3,0,1,2,2,1,1,1,1,2,1] |
| Phi of K* | [-3,-2,0,1,2,2,1,2,1,2,3,1,1,2,3,1,1,2,0,-1,0] |
| Phi of -K* | [-2,-2,-1,0,2,3,0,1,1,2,3,2,0,1,2,0,2,3,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+39t^4+52t^2+4 |
| Outer characteristic polynomial | t^7+61t^5+87t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 480*K1**4*K2 - 1120*K1**4 + 160*K1**3*K2*K3 - 192*K1**3*K3 + 768*K1**2*K2**3 - 2416*K1**2*K2**2 - 160*K1**2*K2*K4 + 2728*K1**2*K2 - 1052*K1**2 + 160*K1*K2**3*K3 - 384*K1*K2**2*K3 + 1616*K1*K2*K3 + 16*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 536*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 312*K2**2*K4 - 640*K2**2 + 24*K2*K3*K5 - 244*K3**2 - 14*K4**2 + 884 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {1, 5}, {4}, {2, 3}]] |
| If K is slice | False |