| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,3,1,2,1,2,1,1,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1907'] |
| 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+31t^5+86t^3+19t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1907'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 656*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 + 1344*K1**2*K2**3 - 5040*K1**2*K2**2 - 544*K1**2*K2*K4 + 4760*K1**2*K2 - 48*K1**2*K3**2 - 3364*K1**2 + 704*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4920*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 672*K2**4*K4 - 3136*K2**4 - 1408*K2**2*K3**2 - 688*K2**2*K4**2 + 2464*K2**2*K4 - 1478*K2**2 + 712*K2*K3*K5 + 200*K2*K4*K6 - 1188*K3**2 - 556*K4**2 - 72*K5**2 - 10*K6**2 + 2818 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1907'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10137', 'vk6.10194', 'vk6.10337', 'vk6.10424', 'vk6.17665', 'vk6.17712', 'vk6.24232', 'vk6.24279', 'vk6.29920', 'vk6.29967', 'vk6.30030', 'vk6.30079', 'vk6.36496', 'vk6.36590', 'vk6.43593', 'vk6.43703', 'vk6.51619', 'vk6.51654', 'vk6.51699', 'vk6.51722', 'vk6.55695', 'vk6.55752', 'vk6.60265', 'vk6.60327', 'vk6.63334', 'vk6.63361', 'vk6.63382', 'vk6.63401', 'vk6.65405', 'vk6.65444', 'vk6.68545', 'vk6.68576'] |
| 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 | O1O2O3U1U2U4O5O4O6U3U5U6 |
| R3 orbit | {'O1O2O3U1U2U4O5O4O6U3U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U4U5O6O4O5U1U3U6 |
| Gauss code of K* | O1O2O3U4U5U1O4O5O6U2U6U3 |
| Gauss code of -K* | O1O2O3U1U4U2O4O5O6U3U5U6 |
| 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 0 1 -1 2],[ 2 0 1 2 2 1 1],[ 0 -1 0 1 0 1 1],[ 0 -2 -1 0 0 0 2],[-1 -2 0 0 0 -1 1],[ 1 -1 -1 0 1 0 1],[-2 -1 -1 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -1 -1],[-1 1 0 0 0 -1 -2],[ 0 1 0 0 1 1 -1],[ 0 2 0 -1 0 0 -2],[ 1 1 1 -1 0 0 -1],[ 2 1 2 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,1,2,1,1,0,0,1,2,-1,-1,1,0,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,1,3,1,2,1,2,1,1,0,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,1,3,1,2,1,2,1,1,0,1,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,0,1,2,3,1,1,1,1,-1,1,0,2,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,2,1,-1,0,1,1,1,0,1,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-8w^3z+24w^2z+21w |
| Inner characteristic polynomial | t^6+21t^4+36t^2+4 |
| Outer characteristic polynomial | t^7+31t^5+86t^3+19t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 352*K1**4*K2 - 656*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 + 1344*K1**2*K2**3 - 5040*K1**2*K2**2 - 544*K1**2*K2*K4 + 4760*K1**2*K2 - 48*K1**2*K3**2 - 3364*K1**2 + 704*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4920*K1*K2*K3 + 392*K1*K3*K4 + 24*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 672*K2**4*K4 - 3136*K2**4 - 1408*K2**2*K3**2 - 688*K2**2*K4**2 + 2464*K2**2*K4 - 1478*K2**2 + 712*K2*K3*K5 + 200*K2*K4*K6 - 1188*K3**2 - 556*K4**2 - 72*K5**2 - 10*K6**2 + 2818 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}]] |
| If K is slice | False |