| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,2,0,2,1,0,1,1,1,0,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1543'] |
| 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+39t^5+120t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1543'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 + 384*K1**2*K2**5 - 1344*K1**2*K2**4 + 3104*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 6064*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1024*K1**2*K2*K4 + 5352*K1**2*K2 - 160*K1**2*K3**2 - 3628*K1**2 - 640*K1*K2**4*K3 + 1472*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4384*K1*K2*K3 - 96*K1*K2*K4*K5 + 864*K1*K3*K4 - 288*K2**6 + 448*K2**4*K4 - 1776*K2**4 - 736*K2**2*K3**2 - 336*K2**2*K4**2 + 1288*K2**2*K4 - 1502*K2**2 - 96*K2*K3**2*K4 + 200*K2*K3*K5 + 120*K2*K4*K6 - 968*K3**2 - 488*K4**2 - 12*K5**2 - 2*K6**2 + 2534 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1543'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11453', 'vk6.11752', 'vk6.12768', 'vk6.13110', 'vk6.20673', 'vk6.22111', 'vk6.28176', 'vk6.29599', 'vk6.31211', 'vk6.31556', 'vk6.32385', 'vk6.32792', 'vk6.39624', 'vk6.41863', 'vk6.46232', 'vk6.47837', 'vk6.52209', 'vk6.52474', 'vk6.53040', 'vk6.53362', 'vk6.57610', 'vk6.58769', 'vk6.62274', 'vk6.63213', 'vk6.63778', 'vk6.63891', 'vk6.64206', 'vk6.64392', 'vk6.67072', 'vk6.67937', 'vk6.69686', 'vk6.70367'] |
| 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 | O1O2O3U4O5U1U6O4O6U5U2U3 |
| R3 orbit | {'O1O2O3U4O5U1U6O4O6U5U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2U4O5O6U5U3O4U6 |
| Gauss code of K* | O1O2O3U4U2U3O5U1O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U3O5U1U2U6 |
| 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 0 3],[ 0 -1 0 1 0 -1 1],[-2 -2 -1 0 -2 -1 -1],[ 1 -2 0 2 0 1 1],[ 0 0 1 1 -1 0 0],[-1 -3 -1 1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -1 -3],[ 0 1 0 0 1 -1 0],[ 0 1 1 -1 0 0 -1],[ 1 2 1 1 0 0 -2],[ 2 2 3 0 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,1,1,2,2,0,1,1,3,-1,1,0,0,1,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,1,2,0,2,1,0,1,1,1,0,1,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,-1,1,2,0,2,1,0,1,1,1,0,1,1,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,1,1,1,2,0,1,1,0,-1,1,1,0,2,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,0,1,3,2,1,0,1,2,1,0,1,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w |
| Inner characteristic polynomial | t^6+29t^4+76t^2+1 |
| Outer characteristic polynomial | t^7+39t^5+120t^3+9t |
| 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**2 + 448*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 + 384*K1**2*K2**5 - 1344*K1**2*K2**4 + 3104*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 6064*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1024*K1**2*K2*K4 + 5352*K1**2*K2 - 160*K1**2*K3**2 - 3628*K1**2 - 640*K1*K2**4*K3 + 1472*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4384*K1*K2*K3 - 96*K1*K2*K4*K5 + 864*K1*K3*K4 - 288*K2**6 + 448*K2**4*K4 - 1776*K2**4 - 736*K2**2*K3**2 - 336*K2**2*K4**2 + 1288*K2**2*K4 - 1502*K2**2 - 96*K2*K3**2*K4 + 200*K2*K3*K5 + 120*K2*K4*K6 - 968*K3**2 - 488*K4**2 - 12*K5**2 - 2*K6**2 + 2534 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |