| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,2,1,3,4,2,1,2,2,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.555'] |
| 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+74t^5+68t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.555'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 608*K1**4*K2 - 1328*K1**4 + 288*K1**3*K2*K3 - 192*K1**3*K3 - 384*K1**2*K2**4 + 1920*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 5760*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 5560*K1**2*K2 - 400*K1**2*K3**2 - 2644*K1**2 + 960*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 4696*K1*K2*K3 + 480*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1704*K2**4 - 1040*K2**2*K3**2 - 16*K2**2*K4**2 + 1184*K2**2*K4 - 1398*K2**2 - 32*K2*K3**2*K4 + 432*K2*K3*K5 + 8*K2*K4*K6 - 904*K3**2 - 150*K4**2 - 28*K5**2 - 2*K6**2 + 2076 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.555'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19951', 'vk6.20049', 'vk6.21198', 'vk6.21333', 'vk6.26924', 'vk6.27114', 'vk6.28680', 'vk6.28805', 'vk6.38344', 'vk6.38499', 'vk6.40486', 'vk6.40702', 'vk6.45209', 'vk6.45399', 'vk6.47034', 'vk6.47149', 'vk6.56739', 'vk6.56849', 'vk6.57842', 'vk6.57990', 'vk6.61172', 'vk6.61376', 'vk6.62414', 'vk6.62541', 'vk6.66435', 'vk6.66562', 'vk6.67207', 'vk6.67355', 'vk6.69087', 'vk6.69214', 'vk6.69870', 'vk6.69957'] |
| 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 | O1O2O3O4O5U4U1U2O6U3U5U6 |
| R3 orbit | {'O1O2O3O4O5U4U1U2O6U3U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U1U3O6U4U5U2 |
| Gauss code of K* | O1O2O3U4O5O6O4U2U3U5U1U6 |
| Gauss code of -K* | O1O2O3U1O4O5O6U2U6U3U4U5 |
| 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 -1 0 -1 3 2],[ 3 0 1 2 0 4 2],[ 1 -1 0 1 0 3 2],[ 0 -2 -1 0 0 2 2],[ 1 0 0 0 0 1 1],[-3 -4 -3 -2 -1 0 1],[-2 -2 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -1 -3],[-3 0 1 -2 -1 -3 -4],[-2 -1 0 -2 -1 -2 -2],[ 0 2 2 0 0 -1 -2],[ 1 1 1 0 0 0 0],[ 1 3 2 1 0 0 -1],[ 3 4 2 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,1,3,-1,2,1,3,4,2,1,2,2,0,1,2,0,0,1] |
| Phi over symmetry | [-3,-2,0,1,1,3,-1,2,1,3,4,2,1,2,2,0,1,2,0,0,1] |
| Phi of -K | [-3,-1,-1,0,2,3,1,2,1,3,2,0,0,1,1,1,2,3,0,1,2] |
| Phi of K* | [-3,-2,0,1,1,3,2,1,1,3,2,0,1,2,3,0,1,1,0,1,2] |
| Phi of -K* | [-3,-1,-1,0,2,3,0,1,2,2,4,0,0,1,1,1,2,3,2,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+22w^2z+25w |
| Inner characteristic polynomial | t^6+50t^4+23t^2 |
| Outer characteristic polynomial | t^7+74t^5+68t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -320*K1**4*K2**2 + 608*K1**4*K2 - 1328*K1**4 + 288*K1**3*K2*K3 - 192*K1**3*K3 - 384*K1**2*K2**4 + 1920*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 32*K1**2*K2**2*K4 - 5760*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 5560*K1**2*K2 - 400*K1**2*K3**2 - 2644*K1**2 + 960*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 4696*K1*K2*K3 + 480*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1704*K2**4 - 1040*K2**2*K3**2 - 16*K2**2*K4**2 + 1184*K2**2*K4 - 1398*K2**2 - 32*K2*K3**2*K4 + 432*K2*K3*K5 + 8*K2*K4*K6 - 904*K3**2 - 150*K4**2 - 28*K5**2 - 2*K6**2 + 2076 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |