| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,1,0,1,2,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1292'] |
| 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+32t^5+37t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1292'] |
| 2-strand cable arrow polynomial of the knot is: 3200*K1**4*K2 - 7712*K1**4 + 960*K1**3*K2*K3 - 2144*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6672*K1**2*K2**2 - 800*K1**2*K2*K4 + 11120*K1**2*K2 - 1088*K1**2*K3**2 - 32*K1**2*K4**2 - 3280*K1**2 + 288*K1*K2**3*K3 - 640*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7016*K1*K2*K3 + 1208*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 496*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 744*K2**2*K4 - 3790*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 1700*K3**2 - 408*K4**2 - 20*K5**2 - 2*K6**2 + 3950 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1292'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4367', 'vk6.4400', 'vk6.5689', 'vk6.5722', 'vk6.7758', 'vk6.7791', 'vk6.9240', 'vk6.9273', 'vk6.10483', 'vk6.10537', 'vk6.10632', 'vk6.10700', 'vk6.10733', 'vk6.10823', 'vk6.14599', 'vk6.15316', 'vk6.15443', 'vk6.16218', 'vk6.17989', 'vk6.24431', 'vk6.30170', 'vk6.30224', 'vk6.30319', 'vk6.30450', 'vk6.33954', 'vk6.34355', 'vk6.34411', 'vk6.43866', 'vk6.50444', 'vk6.50477', 'vk6.54201', 'vk6.63430'] |
| 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 | O1O2O3U2O4O5U3U5O6U1U6U4 |
| R3 orbit | {'O1O2O3U2O4O5U3U5O6U1U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U3O5U6U1O6O4U2 |
| Gauss code of K* | O1O2O3U1U4U5O4U3U6O5O6U2 |
| Gauss code of -K* | O1O2O3U2O4O5U4U1O6U5U6U3 |
| 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 -1 -1 2 1 1],[ 2 0 -1 0 3 1 1],[ 1 1 0 1 1 1 0],[ 1 0 -1 0 2 1 0],[-2 -3 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 -1 -2 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 1 1 0 1 0 1 1],[ 1 2 0 1 -1 0 0],[ 2 3 1 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,0,1,2,3,0,0,0,1,1,1,1,-1,-1,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,1,0,1,2,0,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,1,2,2,2,1,1,1,2,1,1,2,2,0,1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,1,1,1,2,1,0,1,1,2,2,2,2,-1,1,2] |
| Phi of -K* | [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,1,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+20t^4+15t^2+1 |
| Outer characteristic polynomial | t^7+32t^5+37t^3+5t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | 3200*K1**4*K2 - 7712*K1**4 + 960*K1**3*K2*K3 - 2144*K1**3*K3 - 128*K1**2*K2**4 + 544*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6672*K1**2*K2**2 - 800*K1**2*K2*K4 + 11120*K1**2*K2 - 1088*K1**2*K3**2 - 32*K1**2*K4**2 - 3280*K1**2 + 288*K1*K2**3*K3 - 640*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7016*K1*K2*K3 + 1208*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 496*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 744*K2**2*K4 - 3790*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 1700*K3**2 - 408*K4**2 - 20*K5**2 - 2*K6**2 + 3950 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]] |
| If K is slice | False |