| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,0,2,2,1,0,0,1,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1557'] |
| 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+34t^5+107t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1557'] |
| 2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 720*K1**4 - 512*K1**3*K3 + 192*K1**2*K2**3 - 1104*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 3424*K1**2*K2 - 368*K1**2*K3**2 - 64*K1**2*K3*K5 - 2872*K1**2 + 96*K1*K2**3*K3 - 640*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 2920*K1*K2*K3 + 544*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 280*K2**4 - 32*K2**3*K6 - 208*K2**2*K3**2 - 16*K2**2*K4**2 + 440*K2**2*K4 - 2006*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1052*K3**2 - 198*K4**2 - 52*K5**2 - 2*K6**2 + 2044 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1557'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4416', 'vk6.4511', 'vk6.5802', 'vk6.5929', 'vk6.7867', 'vk6.7974', 'vk6.9289', 'vk6.9408', 'vk6.10152', 'vk6.10225', 'vk6.10370', 'vk6.17885', 'vk6.17948', 'vk6.18293', 'vk6.18630', 'vk6.24392', 'vk6.24692', 'vk6.25180', 'vk6.30047', 'vk6.30110', 'vk6.30889', 'vk6.31014', 'vk6.32077', 'vk6.32198', 'vk6.36911', 'vk6.37278', 'vk6.37371', 'vk6.43827', 'vk6.44128', 'vk6.44452', 'vk6.50514', 'vk6.50595', 'vk6.51120', 'vk6.51981', 'vk6.52078', 'vk6.55848', 'vk6.56080', 'vk6.60578', 'vk6.60916', 'vk6.65987'] |
| 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 | O1O2O3U4O5U3U6O4O6U1U5U2 |
| R3 orbit | {'O1O2O3U4O5U3U6O4O6U1U5U2', 'O1O2O3U4U2O5U6O4O6U1U3U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U2U4U3O5O6U5U1O4U6 |
| Gauss code of K* | O1O2O3U1U3U4O5U2O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U2O5U6U1U3 |
| 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 0 -1 1 1],[ 2 0 2 1 0 1 3],[-1 -2 0 0 -2 0 0],[ 0 -1 0 0 0 0 1],[ 1 0 2 0 0 2 1],[-1 -1 0 0 -2 0 -1],[-1 -3 0 -1 -1 1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -3],[-1 -1 0 0 0 -2 -1],[-1 0 0 0 0 -2 -2],[ 0 1 0 0 0 0 -1],[ 1 1 2 2 0 0 0],[ 2 3 1 2 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,3,0,0,2,1,0,2,2,0,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,2,3,0,2,2,1,0,0,1,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,0,1,2,1,1,0,0,0,1,1,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,0,2,0,0,1,0,1,0,1,1,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,1,2,3,0,2,2,1,0,0,1,0,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+26t^4+78t^2 |
| Outer characteristic polynomial | t^7+34t^5+107t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | 160*K1**4*K2 - 720*K1**4 - 512*K1**3*K3 + 192*K1**2*K2**3 - 1104*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 3424*K1**2*K2 - 368*K1**2*K3**2 - 64*K1**2*K3*K5 - 2872*K1**2 + 96*K1*K2**3*K3 - 640*K1*K2**2*K3 - 64*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 2920*K1*K2*K3 + 544*K1*K3*K4 + 56*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 280*K2**4 - 32*K2**3*K6 - 208*K2**2*K3**2 - 16*K2**2*K4**2 + 440*K2**2*K4 - 2006*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1052*K3**2 - 198*K4**2 - 52*K5**2 - 2*K6**2 + 2044 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {3, 5}, {1, 4}, {2}]] |
| If K is slice | False |