| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,2,1,1,0,0,1,0,2,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1753'] |
| 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+41t^5+152t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1753'] |
| 2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1504*K1**4*K2 - 1424*K1**4 - 384*K1**3*K2**2*K3 + 1216*K1**3*K2*K3 - 608*K1**3*K3 - 512*K1**2*K2**4 + 3296*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8864*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 7624*K1**2*K2 - 496*K1**2*K3**2 - 4604*K1**2 + 1056*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7216*K1*K2*K3 + 456*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 2224*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 16*K2**2*K4**2 + 1464*K2**2*K4 - 2462*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1640*K3**2 - 148*K4**2 - 4*K5**2 - 2*K6**2 + 3370 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1753'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16331', 'vk6.16374', 'vk6.18069', 'vk6.18407', 'vk6.22662', 'vk6.22745', 'vk6.24516', 'vk6.24938', 'vk6.34610', 'vk6.34689', 'vk6.36653', 'vk6.37079', 'vk6.42305', 'vk6.42336', 'vk6.43935', 'vk6.44253', 'vk6.54594', 'vk6.54635', 'vk6.55897', 'vk6.56183', 'vk6.59073', 'vk6.59118', 'vk6.60425', 'vk6.60780', 'vk6.64629', 'vk6.64669', 'vk6.65535', 'vk6.65847', 'vk6.67988', 'vk6.68012', 'vk6.68617', 'vk6.68831'] |
| 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 | O1O2O3U4U5O4O6U1U2O5U6U3 |
| R3 orbit | {'O1O2O3U4U5O4O6U1U2O5U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U2U3O4O6U5U6 |
| Gauss code of K* | O1O2U3O4O5U1U2U5O6O3U6U4 |
| Gauss code of -K* | O1O2U3O4O5U2U6O3O6U1U4U5 |
| 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 2 -1 0 1],[ 2 0 1 2 2 2 1],[ 0 -1 0 1 0 1 0],[-2 -2 -1 0 -3 0 -1],[ 1 -2 0 3 0 0 2],[ 0 -2 -1 0 0 0 1],[-1 -1 0 1 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -3 -2],[-1 1 0 -1 0 -2 -1],[ 0 0 1 0 -1 0 -2],[ 0 1 0 1 0 0 -1],[ 1 3 2 0 0 0 -2],[ 2 2 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,0,1,3,2,1,0,2,1,1,0,2,0,1,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,0,1,2,2,1,1,0,0,1,0,2,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,-1,0,1,2,2,1,1,0,0,1,0,2,1,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,1,2,0,2,1,0,0,2,1,1,1,1,0,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,1,2,1,2,0,0,2,3,1,0,1,1,0,1] |
| 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+31t^4+108t^2+4 |
| Outer characteristic polynomial | t^7+41t^5+152t^3+10t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1504*K1**4*K2 - 1424*K1**4 - 384*K1**3*K2**2*K3 + 1216*K1**3*K2*K3 - 608*K1**3*K3 - 512*K1**2*K2**4 + 3296*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8864*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 7624*K1**2*K2 - 496*K1**2*K3**2 - 4604*K1**2 + 1056*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7216*K1*K2*K3 + 456*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 2224*K2**4 - 32*K2**3*K6 - 432*K2**2*K3**2 - 16*K2**2*K4**2 + 1464*K2**2*K4 - 2462*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1640*K3**2 - 148*K4**2 - 4*K5**2 - 2*K6**2 + 3370 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {1, 5}, {4}, {2, 3}]] |
| If K is slice | False |