| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,1,3,1,0,1,1,0,1,2,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1289'] |
| 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+44t^5+43t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1289'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 3040*K1**4*K2 - 6464*K1**4 + 928*K1**3*K2*K3 - 1248*K1**3*K3 - 128*K1**2*K2**4 + 896*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7312*K1**2*K2**2 - 576*K1**2*K2*K4 + 9552*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 - 2396*K1**2 + 224*K1*K2**3*K3 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5608*K1*K2*K3 + 504*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 632*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 512*K2**2*K4 - 2808*K2**2 + 40*K2*K3*K5 - 1052*K3**2 - 94*K4**2 + 3028 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1289'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13917', 'vk6.14014', 'vk6.14166', 'vk6.14407', 'vk6.14986', 'vk6.15109', 'vk6.15634', 'vk6.16090', 'vk6.16718', 'vk6.16749', 'vk6.16851', 'vk6.18798', 'vk6.19271', 'vk6.19565', 'vk6.23160', 'vk6.23234', 'vk6.25392', 'vk6.26458', 'vk6.33728', 'vk6.33805', 'vk6.34284', 'vk6.35152', 'vk6.37525', 'vk6.42745', 'vk6.44680', 'vk6.54117', 'vk6.54923', 'vk6.54952', 'vk6.56392', 'vk6.56608', 'vk6.59351', 'vk6.64596'] |
| 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 | O1O2O3U2O4O5U1U5O6U3U6U4 |
| R3 orbit | {'O1O2O3U2O4O5U1U5O6U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U1O5U6U3O6O4U2 |
| Gauss code of K* | O1O2O3U4U5U1O5U3U6O4O6U2 |
| Gauss code of -K* | O1O2O3U2O4O5U4U1O6U3U6U5 |
| 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 0 2 1 1],[ 3 0 0 3 3 1 1],[ 1 0 0 1 1 0 1],[ 0 -3 -1 0 2 0 1],[-2 -3 -1 -2 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -2 -1 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 2 0 1 0 -1 -3],[ 1 1 0 1 1 0 0],[ 3 3 1 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,0,2,1,3,0,0,0,1,1,1,1,1,3,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,3,1,1,3,1,0,1,1,0,1,2,0,0,0] |
| Phi of -K | [-3,-1,0,1,1,2,2,0,3,3,2,0,1,2,2,0,1,0,0,1,1] |
| Phi of K* | [-2,-1,-1,0,1,3,1,1,0,2,2,0,0,1,3,1,2,3,0,0,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,3,1,1,3,1,0,1,1,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 7z^2+28z+29 |
| Enhanced Jones-Krushkal polynomial | 7w^3z^2+28w^2z+29w |
| Inner characteristic polynomial | t^6+28t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+44t^5+43t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 3040*K1**4*K2 - 6464*K1**4 + 928*K1**3*K2*K3 - 1248*K1**3*K3 - 128*K1**2*K2**4 + 896*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7312*K1**2*K2**2 - 576*K1**2*K2*K4 + 9552*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 - 2396*K1**2 + 224*K1*K2**3*K3 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5608*K1*K2*K3 + 504*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 632*K2**4 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 512*K2**2*K4 - 2808*K2**2 + 40*K2*K3*K5 - 1052*K3**2 - 94*K4**2 + 3028 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |