| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,0,1,0,1,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1845', '7.41825'] |
| 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+22t^5+44t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1845'] |
| 2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1632*K1**4*K2 - 4704*K1**4 + 800*K1**3*K2*K3 - 256*K1**3*K3 - 640*K1**2*K2**4 + 2656*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 9920*K1**2*K2**2 - 544*K1**2*K2*K4 + 9448*K1**2*K2 - 480*K1**2*K3**2 - 96*K1**2*K3*K5 - 2228*K1**2 + 1536*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 - 384*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6904*K1*K2*K3 + 928*K1*K3*K4 + 176*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 2424*K2**4 - 32*K2**3*K6 - 1008*K2**2*K3**2 - 112*K2**2*K4**2 + 1888*K2**2*K4 - 1910*K2**2 + 536*K2*K3*K5 + 88*K2*K4*K6 - 1168*K3**2 - 378*K4**2 - 108*K5**2 - 18*K6**2 + 2832 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1845'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.334', 'vk6.374', 'vk6.443', 'vk6.732', 'vk6.784', 'vk6.894', 'vk6.1474', 'vk6.1530', 'vk6.1591', 'vk6.1971', 'vk6.2011', 'vk6.2072', 'vk6.2495', 'vk6.2752', 'vk6.3008', 'vk6.3132', 'vk6.3785', 'vk6.3976', 'vk6.7177', 'vk6.7352', 'vk6.18789', 'vk6.19854', 'vk6.24920', 'vk6.25381', 'vk6.25911', 'vk6.26299', 'vk6.26742', 'vk6.37986', 'vk6.38043', 'vk6.45034', 'vk6.50107', 'vk6.60753'] |
| 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 | O1O2O3U2U3O4U5O6O5U1U6U4 |
| R3 orbit | {'O1O2O3U2U3O4U5O6O5U1U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U3O6O5U6O4U1U2 |
| Gauss code of K* | O1O2O3U1U4U5O4O5U3O6U2U6 |
| Gauss code of -K* | O1O2O3U4U2O4U1O5O6U5U6U3 |
| 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 -1 1 1 1 0],[ 2 0 -1 1 2 2 0],[ 1 1 0 1 0 1 0],[-1 -1 -1 0 0 -1 0],[-1 -2 0 0 0 0 -1],[-1 -2 -1 1 0 0 0],[ 0 0 0 0 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 0 -2],[ 0 0 0 1 0 0 0],[ 1 1 1 0 0 0 1],[ 2 2 1 2 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,1,1,1,0,2,0,0,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,0,1,0,1,0,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,2,2,1,1,2,1,1,2,1,1,0,1,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,1,1,0,2,1,1,2,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,0,1,0,1,0,0,-1,0] |
| 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+14t^4+25t^2+4 |
| Outer characteristic polynomial | t^7+22t^5+44t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -640*K1**4*K2**2 + 1632*K1**4*K2 - 4704*K1**4 + 800*K1**3*K2*K3 - 256*K1**3*K3 - 640*K1**2*K2**4 + 2656*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 9920*K1**2*K2**2 - 544*K1**2*K2*K4 + 9448*K1**2*K2 - 480*K1**2*K3**2 - 96*K1**2*K3*K5 - 2228*K1**2 + 1536*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 - 384*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6904*K1*K2*K3 + 928*K1*K3*K4 + 176*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 2424*K2**4 - 32*K2**3*K6 - 1008*K2**2*K3**2 - 112*K2**2*K4**2 + 1888*K2**2*K4 - 1910*K2**2 + 536*K2*K3*K5 + 88*K2*K4*K6 - 1168*K3**2 - 378*K4**2 - 108*K5**2 - 18*K6**2 + 2832 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |