| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,2,1,2,4,1,0,1,2,1,1,2,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.651'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+64t^5+44t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.651'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1312*K1**4*K2 - 2016*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 896*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1024*K1**3*K3 - 320*K1**2*K2**4 + 352*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 5584*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 8904*K1**2*K2 - 1024*K1**2*K3**2 - 112*K1**2*K4**2 - 6900*K1**2 + 576*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 8304*K1*K2*K3 + 1424*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 824*K2**4 - 512*K2**2*K3**2 - 16*K2**2*K4**2 + 1480*K2**2*K4 - 5286*K2**2 + 440*K2*K3*K5 + 8*K2*K4*K6 - 2732*K3**2 - 738*K4**2 - 96*K5**2 - 2*K6**2 + 5376 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.651'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11427', 'vk6.11722', 'vk6.12741', 'vk6.13084', 'vk6.20334', 'vk6.21676', 'vk6.27637', 'vk6.29182', 'vk6.31178', 'vk6.31519', 'vk6.32346', 'vk6.32763', 'vk6.39068', 'vk6.41327', 'vk6.45824', 'vk6.47495', 'vk6.52192', 'vk6.52449', 'vk6.53023', 'vk6.53339', 'vk6.57193', 'vk6.58408', 'vk6.61806', 'vk6.62932', 'vk6.63762', 'vk6.63872', 'vk6.64190', 'vk6.64376', 'vk6.66808', 'vk6.67677', 'vk6.69447', 'vk6.70170'] |
| 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 | O1O2O3O4U2O5U1U3O6U5U6U4 |
| R3 orbit | {'O1O2O3O4U2O5U1U3O6U5U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U6O5U2U4O6U3 |
| Gauss code of K* | O1O2O3U4U5U6U3O5U1O4O6U2 |
| Gauss code of -K* | O1O2O3U2O4O5U3O6U1U4U6U5 |
| 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 -3 -2 0 3 1 1],[ 3 0 0 2 4 2 1],[ 2 0 0 1 2 1 0],[ 0 -2 -1 0 2 1 1],[-3 -4 -2 -2 0 -1 1],[-1 -2 -1 -1 1 0 1],[-1 -1 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 0 -2 -3],[-3 0 1 -1 -2 -2 -4],[-1 -1 0 -1 -1 0 -1],[-1 1 1 0 -1 -1 -2],[ 0 2 1 1 0 -1 -2],[ 2 2 0 1 1 0 0],[ 3 4 1 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,0,2,3,-1,1,2,2,4,1,1,0,1,1,1,2,1,2,0] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,2,1,2,4,1,0,1,2,1,1,2,-1,-1,1] |
| Phi of -K | [-3,-2,0,1,1,3,1,1,2,3,2,1,2,3,3,0,0,1,-1,1,3] |
| Phi of K* | [-3,-1,-1,0,2,3,1,3,1,3,2,1,0,2,2,0,3,3,1,1,1] |
| Phi of -K* | [-3,-2,0,1,1,3,0,2,1,2,4,1,0,1,2,1,1,2,-1,-1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+40t^4+15t^2+1 |
| Outer characteristic polynomial | t^7+64t^5+44t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 1312*K1**4*K2 - 2016*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 896*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1024*K1**3*K3 - 320*K1**2*K2**4 + 352*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 5584*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 8904*K1**2*K2 - 1024*K1**2*K3**2 - 112*K1**2*K4**2 - 6900*K1**2 + 576*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 8304*K1*K2*K3 + 1424*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 824*K2**4 - 512*K2**2*K3**2 - 16*K2**2*K4**2 + 1480*K2**2*K4 - 5286*K2**2 + 440*K2*K3*K5 + 8*K2*K4*K6 - 2732*K3**2 - 738*K4**2 - 96*K5**2 - 2*K6**2 + 5376 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}]] |
| If K is slice | False |