| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,1,1,0,1,1,-1,2,2,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1593'] |
| 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+37t^5+172t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1593'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 512*K1**4*K2 - 2160*K1**4 + 160*K1**3*K2*K3 - 416*K1**3*K3 - 2112*K1**2*K2**2 - 256*K1**2*K2*K4 + 6560*K1**2*K2 - 336*K1**2*K3**2 - 5364*K1**2 + 96*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 - 32*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 5080*K1*K2*K3 - 96*K1*K2*K4*K5 + 1640*K1*K3*K4 + 416*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**4 - 304*K2**2*K3**2 - 112*K2**2*K4**2 + 1296*K2**2*K4 - 4748*K2**2 + 800*K2*K3*K5 + 112*K2*K4*K6 - 2456*K3**2 - 1180*K4**2 - 380*K5**2 - 28*K6**2 + 4770 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1593'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17107', 'vk6.17350', 'vk6.20575', 'vk6.21984', 'vk6.23500', 'vk6.23839', 'vk6.28041', 'vk6.29500', 'vk6.35641', 'vk6.36082', 'vk6.39451', 'vk6.41652', 'vk6.43007', 'vk6.43319', 'vk6.46039', 'vk6.47707', 'vk6.55258', 'vk6.55510', 'vk6.57441', 'vk6.58612', 'vk6.59666', 'vk6.60014', 'vk6.62116', 'vk6.63086', 'vk6.65058', 'vk6.65253', 'vk6.66977', 'vk6.67842', 'vk6.68321', 'vk6.68471', 'vk6.69596', 'vk6.70289'] |
| 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 | O1O2O3U4O5U6U2O6O4U1U5U3 |
| R3 orbit | {'O1O2O3U4O5U6U2O6O4U1U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U3O5O6U2U6O4U5 |
| Gauss code of K* | O1O2O3U1U4U3O5U2O6O4U6U5 |
| Gauss code of -K* | O1O2O3U4U5O6O5U2O4U1U6U3 |
| 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 0 1 -1],[ 2 0 2 3 1 1 1],[ 0 -2 0 0 1 -1 0],[-2 -3 0 0 -1 -1 -2],[ 0 -1 -1 1 0 1 -1],[-1 -1 1 1 -1 0 -1],[ 1 -1 0 2 1 1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -2 -3],[-1 1 0 1 -1 -1 -1],[ 0 0 -1 0 1 0 -2],[ 0 1 1 -1 0 -1 -1],[ 1 2 1 0 1 0 -1],[ 2 3 1 2 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,0,1,2,3,-1,1,1,1,-1,0,2,1,1,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,2,1,1,0,1,1,-1,2,2,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,2,1,1,0,1,1,-1,2,2,0,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,1,2,1,1,0,2,1,2,-1,0,1,1,0,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,1,3,1,0,1,2,-1,1,1,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | z^2+20z+37 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+20w^2z+37w |
| Inner characteristic polynomial | t^6+27t^4+106t^2 |
| Outer characteristic polynomial | t^7+37t^5+172t^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**4*K2**2 + 512*K1**4*K2 - 2160*K1**4 + 160*K1**3*K2*K3 - 416*K1**3*K3 - 2112*K1**2*K2**2 - 256*K1**2*K2*K4 + 6560*K1**2*K2 - 336*K1**2*K3**2 - 5364*K1**2 + 96*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 - 32*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 5080*K1*K2*K3 - 96*K1*K2*K4*K5 + 1640*K1*K3*K4 + 416*K1*K4*K5 + 24*K1*K5*K6 - 112*K2**4 - 304*K2**2*K3**2 - 112*K2**2*K4**2 + 1296*K2**2*K4 - 4748*K2**2 + 800*K2*K3*K5 + 112*K2*K4*K6 - 2456*K3**2 - 1180*K4**2 - 380*K5**2 - 28*K6**2 + 4770 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |