| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,1,3,2,0,0,1,1,1,1,0,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1365'] |
| 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+32t^5+43t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1365'] |
| 2-strand cable arrow polynomial of the knot is: 1504*K1**4*K2 - 3360*K1**4 + 832*K1**3*K2*K3 - 288*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5744*K1**2*K2**2 - 384*K1**2*K2*K4 + 7352*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K4**2 - 3152*K1**2 + 320*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 256*K1*K2**2*K5 + 5720*K1*K2*K3 + 1152*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 592*K2**4 - 32*K2**3*K6 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 1160*K2**2*K4 - 3310*K2**2 + 336*K2*K3*K5 + 16*K2*K4*K6 - 1460*K3**2 - 424*K4**2 - 76*K5**2 - 2*K6**2 + 3166 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1365'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4353', 'vk6.4386', 'vk6.5671', 'vk6.5704', 'vk6.7744', 'vk6.7777', 'vk6.9222', 'vk6.9255', 'vk6.10497', 'vk6.10552', 'vk6.10649', 'vk6.10718', 'vk6.10751', 'vk6.10838', 'vk6.14619', 'vk6.15296', 'vk6.15423', 'vk6.16238', 'vk6.17965', 'vk6.24407', 'vk6.30184', 'vk6.30239', 'vk6.30336', 'vk6.30465', 'vk6.33942', 'vk6.34343', 'vk6.34400', 'vk6.43842', 'vk6.50431', 'vk6.50463', 'vk6.54220', 'vk6.63440'] |
| 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 | O1O2O3U2O4O5U4U1U5O6U3U6 |
| R3 orbit | {'O1O2O3U2O4O5U4U1U5O6U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1O4U5U3U6O5O6U2 |
| Gauss code of K* | O1O2O3U4O5O4U2U6U5O6U1U3 |
| Gauss code of -K* | O1O2O3U1U3O4U5U4U2O6O5U6 |
| 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 2 1],[ 2 0 0 3 0 2 1],[ 1 0 0 1 0 0 1],[-1 -3 -1 0 -1 1 1],[ 1 0 0 1 0 1 0],[-2 -2 0 -1 -1 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 0 -1 -2],[-1 0 0 -1 -1 0 -1],[-1 1 1 0 -1 -1 -3],[ 1 0 1 1 0 0 0],[ 1 1 0 1 0 0 0],[ 2 2 1 3 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,0,1,2,1,1,0,1,1,1,3,0,0,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,0,0,1,3,2,0,0,1,1,1,1,0,-1,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,1,1,0,2,2,0,1,1,3,1,2,2,-1,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,2,3,2,1,1,1,0,2,1,2,0,1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,0,1,3,2,0,0,1,1,1,1,0,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+25z+27 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+25w^2z+27w |
| Inner characteristic polynomial | t^6+20t^4+21t^2 |
| Outer characteristic polynomial | t^7+32t^5+43t^3+5t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | 1504*K1**4*K2 - 3360*K1**4 + 832*K1**3*K2*K3 - 288*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5744*K1**2*K2**2 - 384*K1**2*K2*K4 + 7352*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K4**2 - 3152*K1**2 + 320*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 256*K1*K2**2*K5 + 5720*K1*K2*K3 + 1152*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 592*K2**4 - 32*K2**3*K6 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 1160*K2**2*K4 - 3310*K2**2 + 336*K2*K3*K5 + 16*K2*K4*K6 - 1460*K3**2 - 424*K4**2 - 76*K5**2 - 2*K6**2 + 3166 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |