| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,1,4,2,1,1,1,0,1,0,1,-1,-1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.728'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+58t^5+66t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.728'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 1888*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 - 1088*K1**2*K2**4 + 2432*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6944*K1**2*K2**2 - 160*K1**2*K2*K4 + 7680*K1**2*K2 - 64*K1**2*K3**2 - 3948*K1**2 + 1120*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 96*K1*K2**2*K5 + 4792*K1*K2*K3 + 176*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1864*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 1064*K2**2*K4 - 2008*K2**2 + 24*K2*K3*K5 - 932*K3**2 - 134*K4**2 + 2948 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.728'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17080', 'vk6.17323', 'vk6.20250', 'vk6.21556', 'vk6.23464', 'vk6.23805', 'vk6.27479', 'vk6.29075', 'vk6.35597', 'vk6.36053', 'vk6.38897', 'vk6.41099', 'vk6.42974', 'vk6.43289', 'vk6.45654', 'vk6.47388', 'vk6.55219', 'vk6.55473', 'vk6.57080', 'vk6.58235', 'vk6.59619', 'vk6.59964', 'vk6.61627', 'vk6.62809', 'vk6.65022', 'vk6.65227', 'vk6.66716', 'vk6.67574', 'vk6.68294', 'vk6.68444', 'vk6.69363', 'vk6.70106'] |
| 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 | O1O2O3O4U1O5U4U2U3O6U5U6 |
| R3 orbit | {'O1O2O3O4U1O5U4U2U3O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6O5U2U3U1O6U4 |
| Gauss code of K* | O1O2O3U4O5O4U6U2U3U1O6U5 |
| Gauss code of -K* | O1O2O3U4O5U3U1U2U5O6O4U6 |
| 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 -3 -1 1 0 2 1],[ 3 0 2 3 1 3 0],[ 1 -2 0 1 0 3 1],[-1 -3 -1 0 0 2 1],[ 0 -1 0 0 0 1 1],[-2 -3 -3 -2 -1 0 1],[-1 0 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 1 -2 -1 -3 -3],[-1 -1 0 -1 -1 -1 0],[-1 2 1 0 0 -1 -3],[ 0 1 1 0 0 0 -1],[ 1 3 1 1 0 0 -2],[ 3 3 0 3 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,-1,2,1,3,3,1,1,1,0,0,1,3,0,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,2,1,4,2,1,1,1,0,1,0,1,-1,-1,2] |
| Phi of -K | [-3,-1,0,1,1,2,0,2,1,4,2,1,1,1,0,1,0,1,-1,-1,2] |
| Phi of K* | [-2,-1,-1,0,1,3,-1,2,1,0,2,1,1,1,1,0,1,4,1,2,0] |
| Phi of -K* | [-3,-1,0,1,1,2,2,1,0,3,3,0,1,1,3,1,0,1,-1,-1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-4w^3z+25w^2z+27w |
| Inner characteristic polynomial | t^6+42t^4+25t^2+1 |
| Outer characteristic polynomial | t^7+58t^5+66t^3+13t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 320*K1**4*K2 - 1888*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 - 1088*K1**2*K2**4 + 2432*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6944*K1**2*K2**2 - 160*K1**2*K2*K4 + 7680*K1**2*K2 - 64*K1**2*K3**2 - 3948*K1**2 + 1120*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 96*K1*K2**2*K5 + 4792*K1*K2*K3 + 176*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1864*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 1064*K2**2*K4 - 2008*K2**2 + 24*K2*K3*K5 - 932*K3**2 - 134*K4**2 + 2948 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |