| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,3,-1,0,1,2,-1,1,0,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1202'] |
| 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+58t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1202'] |
| 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 + 640*K1**2*K2**3 - 2688*K1**2*K2**2 - 64*K1**2*K2*K4 + 2848*K1**2*K2 - 48*K1**2*K3**2 - 1548*K1**2 + 224*K1*K2**3*K3 - 736*K1*K2**2*K3 - 96*K1*K2**2*K5 + 2352*K1*K2*K3 + 200*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 704*K2**2*K4 - 1158*K2**2 + 272*K2*K3*K5 + 16*K2*K4*K6 - 568*K3**2 - 160*K4**2 - 60*K5**2 - 2*K6**2 + 1302 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1202'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4677', 'vk6.4972', 'vk6.6149', 'vk6.6630', 'vk6.8150', 'vk6.8558', 'vk6.9532', 'vk6.9883', 'vk6.17670', 'vk6.17719', 'vk6.22141', 'vk6.24233', 'vk6.28236', 'vk6.29659', 'vk6.29905', 'vk6.29938', 'vk6.30003', 'vk6.30066', 'vk6.36503', 'vk6.39696', 'vk6.41935', 'vk6.43602', 'vk6.46268', 'vk6.47873', 'vk6.48709', 'vk6.48916', 'vk6.49483', 'vk6.49700', 'vk6.51616', 'vk6.51649', 'vk6.51696', 'vk6.51721', 'vk6.55712', 'vk6.58786', 'vk6.60282', 'vk6.63245', 'vk6.63348', 'vk6.63396', 'vk6.65414', 'vk6.68554'] |
| 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 | O1O2O3O4U2U4U1O5O6U3U5U6 |
| R3 orbit | {'O1O2O3O4U2U4U1O5O6U3U5U6', 'O1O2O3U1O4U2U4O5O6U3U5U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U6U2O5O6U4U1U3 |
| Gauss code of K* | O1O2O3U4U5O6O4O5U3U1U6U2 |
| Gauss code of -K* | O1O2O3U1U2O4O5O6U5U3U6U4 |
| 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 -2 0 1 0 2],[ 1 0 -1 2 1 1 1],[ 2 1 0 2 1 1 1],[ 0 -2 -2 0 0 1 2],[-1 -1 -1 0 0 0 0],[ 0 -1 -1 -1 0 0 1],[-2 -1 -1 -2 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -1 -1],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 -1 -1 -1],[ 0 2 0 1 0 -2 -2],[ 1 1 1 1 2 0 -1],[ 2 1 1 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,2,1,1,0,0,1,1,1,1,1,2,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,2,3,-1,0,1,2,-1,1,0,1,1,1] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,2,3,-1,0,1,2,-1,1,0,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,0,1,2,3,1,1,1,2,1,-1,0,0,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,1,1,1,2,1,1,-1,0,1,0,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+21t^4+20t^2 |
| Outer characteristic polynomial | t^7+31t^5+58t^3+3t |
| 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 + 640*K1**2*K2**3 - 2688*K1**2*K2**2 - 64*K1**2*K2*K4 + 2848*K1**2*K2 - 48*K1**2*K3**2 - 1548*K1**2 + 224*K1*K2**3*K3 - 736*K1*K2**2*K3 - 96*K1*K2**2*K5 + 2352*K1*K2*K3 + 200*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 688*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 704*K2**2*K4 - 1158*K2**2 + 272*K2*K3*K5 + 16*K2*K4*K6 - 568*K3**2 - 160*K4**2 - 60*K5**2 - 2*K6**2 + 1302 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | False |