| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,2,2,3,0,0,1,1,0,2,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1671'] |
| 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+42t^5+85t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1671'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1376*K1**4*K2 - 3792*K1**4 + 1056*K1**3*K2*K3 + 32*K1**3*K3*K4 - 960*K1**3*K3 + 128*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4784*K1**2*K2**2 - 608*K1**2*K2*K4 + 9024*K1**2*K2 - 1072*K1**2*K3**2 - 128*K1**2*K4**2 - 5920*K1**2 - 864*K1*K2**2*K3 - 160*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7928*K1*K2*K3 + 1968*K1*K3*K4 + 424*K1*K4*K5 - 384*K2**4 - 528*K2**2*K3**2 - 112*K2**2*K4**2 + 1344*K2**2*K4 - 5316*K2**2 - 96*K2*K3**2*K4 + 656*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2868*K3**2 - 1096*K4**2 - 260*K5**2 - 28*K6**2 + 5454 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1671'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4846', 'vk6.4859', 'vk6.5191', 'vk6.5202', 'vk6.6415', 'vk6.6432', 'vk6.6848', 'vk6.8384', 'vk6.8401', 'vk6.8809', 'vk6.8820', 'vk6.9745', 'vk6.9758', 'vk6.10044', 'vk6.20784', 'vk6.20792', 'vk6.22185', 'vk6.29747', 'vk6.39818', 'vk6.39838', 'vk6.46378', 'vk6.46397', 'vk6.47953', 'vk6.47973', 'vk6.49073', 'vk6.49084', 'vk6.49906', 'vk6.51328', 'vk6.51338', 'vk6.51547', 'vk6.58811', 'vk6.63275'] |
| 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 | O1O2O3U2O4U5U1U6O5O6U4U3 |
| R3 orbit | {'O1O2O3U2O4U5U1U6O5O6U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5O6U5U3U6O4U2 |
| Gauss code of K* | O1O2O3U1U3O4O5U2U6U5O6U4 |
| Gauss code of -K* | O1O2O3U4O5U6U5U2O6O4U1U3 |
| 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 -1 -1 2 1 -2 1],[ 1 0 0 2 0 1 2],[ 1 0 0 1 0 1 1],[-2 -2 -1 0 0 -3 -1],[-1 0 0 0 0 -2 0],[ 2 -1 -1 3 2 0 2],[-1 -2 -1 1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -2 -3],[-1 0 0 0 0 0 -2],[-1 1 0 0 -1 -2 -2],[ 1 1 0 1 0 0 1],[ 1 2 0 2 0 0 1],[ 2 3 2 2 -1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,1,2,3,0,0,0,2,1,2,2,0,-1,-1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,-1,2,2,3,0,0,1,1,0,2,2,0,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,2,2,1,1,1,0,0,2,1,1,2,2,0,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,1,2,1,0,0,1,1,2,2,1,0,2,2] |
| Phi of -K* | [-2,-1,-1,1,1,2,-1,-1,2,2,3,0,0,1,1,0,2,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+30t^4+47t^2 |
| Outer characteristic polynomial | t^7+42t^5+85t^3+3t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -320*K1**4*K2**2 + 1376*K1**4*K2 - 3792*K1**4 + 1056*K1**3*K2*K3 + 32*K1**3*K3*K4 - 960*K1**3*K3 + 128*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 4784*K1**2*K2**2 - 608*K1**2*K2*K4 + 9024*K1**2*K2 - 1072*K1**2*K3**2 - 128*K1**2*K4**2 - 5920*K1**2 - 864*K1*K2**2*K3 - 160*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7928*K1*K2*K3 + 1968*K1*K3*K4 + 424*K1*K4*K5 - 384*K2**4 - 528*K2**2*K3**2 - 112*K2**2*K4**2 + 1344*K2**2*K4 - 5316*K2**2 - 96*K2*K3**2*K4 + 656*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2868*K3**2 - 1096*K4**2 - 260*K5**2 - 28*K6**2 + 5454 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | False |