| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,0,2,3,-1,0,1,1,0,1,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1577'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+29t^5+62t^3+14t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1577'] |
| 2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 512*K1**4*K2**2 + 800*K1**4*K2 - 816*K1**4 - 256*K1**3*K2**2*K3 + 480*K1**3*K2*K3 - 544*K1**3*K3 - 512*K1**2*K2**4 + 2464*K1**2*K2**3 - 7568*K1**2*K2**2 - 256*K1**2*K2*K4 + 6504*K1**2*K2 - 304*K1**2*K3**2 - 4544*K1**2 + 2208*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7280*K1*K2*K3 + 408*K1*K3*K4 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 3072*K2**4 - 32*K2**3*K6 - 1520*K2**2*K3**2 - 112*K2**2*K4**2 + 1864*K2**2*K4 - 1950*K2**2 + 560*K2*K3*K5 + 88*K2*K4*K6 - 1852*K3**2 - 216*K4**2 - 60*K5**2 - 18*K6**2 + 3382 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1577'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72568', 'vk6.72573', 'vk6.72671', 'vk6.72678', 'vk6.72986', 'vk6.72992', 'vk6.73145', 'vk6.73148', 'vk6.74211', 'vk6.74213', 'vk6.74843', 'vk6.74844', 'vk6.76402', 'vk6.76406', 'vk6.76892', 'vk6.77855', 'vk6.77886', 'vk6.77896', 'vk6.78001', 'vk6.79249', 'vk6.79257', 'vk6.79735', 'vk6.80744', 'vk6.80749', 'vk6.81153', 'vk6.81156', 'vk6.82313', 'vk6.83991', 'vk6.86354', 'vk6.87275', 'vk6.88253', 'vk6.88254'] |
| 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 | O1O2O3U4O5U2U3O6O4U1U6U5 |
| R3 orbit | {'O1O2O3U4O5U2U3O6O4U1U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U3O6O5U1U2O4U6 |
| Gauss code of K* | O1O2O3U1U4U5O6U3O4O5U2U6 |
| Gauss code of -K* | O1O2O3U4U2O5O6U1O4U5U6U3 |
| 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 0 2 0],[ 2 0 0 2 0 3 0],[ 1 0 0 1 0 1 -1],[-1 -2 -1 0 -1 0 -1],[ 0 0 0 1 0 1 0],[-2 -3 -1 0 -1 0 0],[ 0 0 1 1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -1 -3],[-1 0 0 -1 -1 -1 -2],[ 0 0 1 0 0 1 0],[ 0 1 1 0 0 0 0],[ 1 1 1 -1 0 0 0],[ 2 3 2 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,1,3,1,1,1,2,0,-1,0,0,0,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,0,2,3,-1,0,1,1,0,1,0,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,2,2,1,1,1,2,1,2,0,0,1,0,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,2,1,0,0,1,1,0,1,2,2,2,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,0,0,2,3,-1,0,1,1,0,1,0,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+19t^4+28t^2+1 |
| Outer characteristic polynomial | t^7+29t^5+62t^3+14t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | 128*K1**4*K2**3 - 512*K1**4*K2**2 + 800*K1**4*K2 - 816*K1**4 - 256*K1**3*K2**2*K3 + 480*K1**3*K2*K3 - 544*K1**3*K3 - 512*K1**2*K2**4 + 2464*K1**2*K2**3 - 7568*K1**2*K2**2 - 256*K1**2*K2*K4 + 6504*K1**2*K2 - 304*K1**2*K3**2 - 4544*K1**2 + 2208*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7280*K1*K2*K3 + 408*K1*K3*K4 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 3072*K2**4 - 32*K2**3*K6 - 1520*K2**2*K3**2 - 112*K2**2*K4**2 + 1864*K2**2*K4 - 1950*K2**2 + 560*K2*K3*K5 + 88*K2*K4*K6 - 1852*K3**2 - 216*K4**2 - 60*K5**2 - 18*K6**2 + 3382 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |