| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,1,1,2,0,0,1,1,-1,1,1,1,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1322'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+27t^5+40t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1322'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 608*K1**4*K2 - 720*K1**4 + 224*K1**3*K2*K3 - 96*K1**3*K3 + 1088*K1**2*K2**3 - 3424*K1**2*K2**2 - 576*K1**2*K2*K4 + 3616*K1**2*K2 - 80*K1**2*K3**2 - 2404*K1**2 + 192*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2624*K1*K2*K3 + 368*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 752*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 664*K2**2*K4 - 1382*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 544*K3**2 - 228*K4**2 - 12*K5**2 - 2*K6**2 + 1698 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1322'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11454', 'vk6.11753', 'vk6.12767', 'vk6.13109', 'vk6.20674', 'vk6.22113', 'vk6.28177', 'vk6.29601', 'vk6.31212', 'vk6.31557', 'vk6.32384', 'vk6.32791', 'vk6.39622', 'vk6.41861', 'vk6.46230', 'vk6.47835', 'vk6.52210', 'vk6.52475', 'vk6.53039', 'vk6.53361', 'vk6.57612', 'vk6.58771', 'vk6.62276', 'vk6.63215', 'vk6.63779', 'vk6.63892', 'vk6.64205', 'vk6.64391', 'vk6.67070', 'vk6.67936', 'vk6.69684', 'vk6.70366'] |
| 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 | O1O2O3U4O5O6U3U6O4U1U2U5 |
| R3 orbit | {'O1O2O3U4O5O6U3U6O4U1U2U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U3O5U6U1O6O4U5 |
| Gauss code of K* | O1O2O3U1U2U4O5U3U6O4O6U5 |
| Gauss code of -K* | O1O2O3U4O5O6U5U1O4U6U2U3 |
| 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 0 -1 0 2 1],[ 2 0 1 0 1 2 1],[ 0 -1 0 0 -1 1 1],[ 1 0 0 0 0 1 1],[ 0 -1 1 0 0 2 1],[-2 -2 -1 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -1 -2],[-1 0 0 -1 -1 -1 -1],[ 0 1 1 0 -1 0 -1],[ 0 2 1 1 0 0 -1],[ 1 1 1 0 0 0 0],[ 2 2 1 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,2,1,2,1,1,1,1,1,0,1,0,1,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,1,1,1,2,0,0,1,1,-1,1,1,1,2,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,1,1,2,2,1,1,1,2,-1,0,0,0,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,0,1,2,2,0,0,1,2,1,1,1,1,1,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,1,1,1,2,0,0,1,1,-1,1,1,1,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+22w^2z+25w |
| Inner characteristic polynomial | t^6+17t^4+12t^2 |
| Outer characteristic polynomial | t^7+27t^5+40t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -384*K1**4*K2**2 + 608*K1**4*K2 - 720*K1**4 + 224*K1**3*K2*K3 - 96*K1**3*K3 + 1088*K1**2*K2**3 - 3424*K1**2*K2**2 - 576*K1**2*K2*K4 + 3616*K1**2*K2 - 80*K1**2*K3**2 - 2404*K1**2 + 192*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2624*K1*K2*K3 + 368*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 752*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 664*K2**2*K4 - 1382*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 544*K3**2 - 228*K4**2 - 12*K5**2 - 2*K6**2 + 1698 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{6}, {3, 5}, {1, 4}, {2}]] |
| If K is slice | False |