| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,0,1,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1642'] |
| Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717'] |
| Outer characteristic polynomial of the knot is: t^7+40t^5+34t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1642'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 1216*K1**4*K2 - 4432*K1**4 + 320*K1**3*K2*K3 - 512*K1**3*K3 - 3408*K1**2*K2**2 - 128*K1**2*K2*K4 + 6760*K1**2*K2 - 688*K1**2*K3**2 - 2260*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3800*K1*K2*K3 + 824*K1*K3*K4 + 32*K1*K4*K5 - 160*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 312*K2**2*K4 - 2612*K2**2 + 128*K2*K3*K5 + 32*K2*K4*K6 - 1136*K3**2 - 320*K4**2 - 52*K5**2 - 12*K6**2 + 2774 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1642'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4084', 'vk6.4115', 'vk6.5326', 'vk6.5357', 'vk6.7446', 'vk6.7475', 'vk6.8949', 'vk6.8980', 'vk6.10112', 'vk6.10283', 'vk6.10306', 'vk6.14542', 'vk6.15274', 'vk6.15403', 'vk6.15762', 'vk6.16177', 'vk6.29862', 'vk6.29893', 'vk6.33916', 'vk6.34001', 'vk6.34217', 'vk6.34386', 'vk6.48466', 'vk6.49171', 'vk6.50214', 'vk6.50241', 'vk6.51604', 'vk6.53951', 'vk6.54016', 'vk6.54170', 'vk6.54456', 'vk6.63323'] |
| 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 | O1O2O3U1O4U2U5U4O5O6U3U6 |
| R3 orbit | {'O1O2O3U1O4U2U5U4O5O6U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1O4O5U6U5U2O6U3 |
| Gauss code of K* | O1O2O3U2U4O5O4U6U1U5O6U3 |
| Gauss code of -K* | O1O2O3U1O4U5U3U4O6O5U6U2 |
| 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 2 -1 1],[ 2 0 1 2 1 2 1],[ 1 -1 0 2 1 0 1],[-1 -2 -2 0 1 -2 1],[-2 -1 -1 -1 0 -2 0],[ 1 -2 0 2 2 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -2 -1],[-1 0 0 -1 -1 -1 -1],[-1 1 1 0 -2 -2 -2],[ 1 1 1 2 0 0 -1],[ 1 2 1 2 0 0 -2],[ 2 1 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,1,2,1,1,1,1,1,2,2,2,0,1,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,0,1,2,-1,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,0,1,2,-1,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,1,2,3,1,0,0,1,1,1,2,0,-1,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,2,1,2,1,0,1,2,1,1,2,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | z^2+18z+33 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+18w^2z+33w |
| Inner characteristic polynomial | t^6+28t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+40t^5+34t^3+5t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 - 64*K1**4*K2**2 + 1216*K1**4*K2 - 4432*K1**4 + 320*K1**3*K2*K3 - 512*K1**3*K3 - 3408*K1**2*K2**2 - 128*K1**2*K2*K4 + 6760*K1**2*K2 - 688*K1**2*K3**2 - 2260*K1**2 - 224*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3800*K1*K2*K3 + 824*K1*K3*K4 + 32*K1*K4*K5 - 160*K2**4 - 80*K2**2*K3**2 - 16*K2**2*K4**2 + 312*K2**2*K4 - 2612*K2**2 + 128*K2*K3*K5 + 32*K2*K4*K6 - 1136*K3**2 - 320*K4**2 - 52*K5**2 - 12*K6**2 + 2774 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |