| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,1,1,1,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1343', '7.37468'] |
| 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+20t^5+35t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1343', '7.37468'] |
| 2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1888*K1**4*K2 - 5344*K1**4 + 1056*K1**3*K2*K3 - 928*K1**3*K3 - 640*K1**2*K2**4 + 2144*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 8960*K1**2*K2**2 - 1312*K1**2*K2*K4 + 9384*K1**2*K2 - 544*K1**2*K3**2 - 96*K1**2*K4**2 - 2132*K1**2 + 1280*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 512*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6808*K1*K2*K3 - 96*K1*K2*K4*K5 + 896*K1*K3*K4 + 176*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1912*K2**4 - 32*K2**3*K6 - 592*K2**2*K3**2 - 112*K2**2*K4**2 + 1632*K2**2*K4 - 2166*K2**2 + 504*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 :['6.1343'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.333', 'vk6.373', 'vk6.450', 'vk6.727', 'vk6.779', 'vk6.897', 'vk6.1463', 'vk6.1523', 'vk6.1606', 'vk6.1960', 'vk6.2000', 'vk6.2083', 'vk6.2490', 'vk6.2745', 'vk6.3013', 'vk6.3133', 'vk6.3784', 'vk6.3977', 'vk6.7176', 'vk6.7353', 'vk6.18788', 'vk6.19861', 'vk6.24913', 'vk6.25376', 'vk6.25909', 'vk6.26300', 'vk6.26745', 'vk6.37988', 'vk6.38045', 'vk6.45043', 'vk6.50106', 'vk6.60762'] |
| 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 | O1O2O3U1O4O5U4U5U3O6U2U6 |
| R3 orbit | {'O1O2O3U1O4O5U4U5U3O6U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O4U1U5U6O5O6U3 |
| Gauss code of K* | O1O2O3U4O5O4U6U5U3O6U1U2 |
| Gauss code of -K* | O1O2O3U2U3O4U1U5U4O6O5U6 |
| 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 -1 1 1],[ 2 0 2 1 0 0 1],[ 0 -2 0 0 -1 1 1],[-1 -1 0 0 -1 1 0],[ 1 0 1 1 0 1 0],[-1 0 -1 -1 -1 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 -1 0 -1],[ 0 0 1 1 0 -1 -2],[ 1 1 1 0 1 0 0],[ 2 1 0 1 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,1,0,1,1,0,1,0,1,1,2,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,1,1,1,0,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,0,2,2,3,0,1,2,1,1,0,0,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,1,1,2,0,2,2,0,0,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,2,0,1,1,1,1,0,1,1,1,0,0,-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+12t^4+16t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+35t^3+8t |
| 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 + 1888*K1**4*K2 - 5344*K1**4 + 1056*K1**3*K2*K3 - 928*K1**3*K3 - 640*K1**2*K2**4 + 2144*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 8960*K1**2*K2**2 - 1312*K1**2*K2*K4 + 9384*K1**2*K2 - 544*K1**2*K3**2 - 96*K1**2*K4**2 - 2132*K1**2 + 1280*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 512*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6808*K1*K2*K3 - 96*K1*K2*K4*K5 + 896*K1*K3*K4 + 176*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1912*K2**4 - 32*K2**3*K6 - 592*K2**2*K3**2 - 112*K2**2*K4**2 + 1632*K2**2*K4 - 2166*K2**2 + 504*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}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {2, 3}, {1}]] |
| If K is slice | False |