| Min(phi) over symmetries of the knot is: [-3,-1,0,2,2,0,1,2,3,0,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.662', '7.10890'] |
| Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384'] |
| Outer characteristic polynomial of the knot is: t^6+37t^4+11t^2 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.662', '7.10890'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 288*K1**4*K2 - 1344*K1**4 + 96*K1**3*K2*K3 - 1056*K1**2*K2**2 + 2472*K1**2*K2 - 160*K1**2*K3**2 - 1204*K1**2 + 1248*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 160*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 128*K2**2*K4 - 1070*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 400*K3**2 - 72*K4**2 - 12*K5**2 - 2*K6**2 + 1174 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.662'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16515', 'vk6.16608', 'vk6.18078', 'vk6.18416', 'vk6.22946', 'vk6.23043', 'vk6.23504', 'vk6.23843', 'vk6.24525', 'vk6.24944', 'vk6.35032', 'vk6.35653', 'vk6.36668', 'vk6.37092', 'vk6.39445', 'vk6.41646', 'vk6.42492', 'vk6.42605', 'vk6.43944', 'vk6.44261', 'vk6.46029', 'vk6.47697', 'vk6.54742', 'vk6.54839', 'vk6.56196', 'vk6.57451', 'vk6.59206', 'vk6.59271', 'vk6.59654', 'vk6.60002', 'vk6.60791', 'vk6.62122', 'vk6.64821', 'vk6.65054', 'vk6.65552', 'vk6.65864', 'vk6.68058', 'vk6.68123', 'vk6.68630', 'vk6.68845'] |
| 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 | O1O2O3O4U2O5U4U1O6U3U6U5 |
| R3 orbit | {'O1O2O3O4U2O5U4U1O6U3U6U5', 'O1O2O3U1O4O5U2U4O6U3U6U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U6U2O6U4U1O5U3 |
| Gauss code of K* | O1O2O3U4U5U1U6O5U3O6O4U2 |
| Gauss code of -K* | O1O2O3U2O4O5U1O6U5U3U6U4 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -2 0 0 3 1],[ 2 0 -1 2 1 3 1],[ 2 1 0 2 1 2 1],[ 0 -2 -2 0 0 3 1],[ 0 -1 -1 0 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 3 1 0 -2 -2],[-3 0 0 -1 -2 -3],[-1 0 0 0 -1 -1],[ 0 1 0 0 -1 -1],[ 2 2 1 1 0 1],[ 2 3 1 1 -1 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-3,-1,0,2,2,0,1,2,3,0,1,1,1,1,-1] |
| Phi over symmetry | [-3,-1,0,2,2,0,1,2,3,0,1,1,1,1,-1] |
| Phi of -K | [-2,-2,0,1,3,-1,1,2,3,1,2,2,1,2,2] |
| Phi of K* | [-3,-1,0,2,2,2,2,2,3,1,2,2,1,1,-1] |
| Phi of -K* | [-2,-2,0,1,3,-1,1,1,3,1,1,2,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 11z+23 |
| Enhanced Jones-Krushkal polynomial | 11w^2z+23w |
| Inner characteristic polynomial | t^5+19t^3+3t |
| Outer characteristic polynomial | t^6+37t^4+11t^2 |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 288*K1**4*K2 - 1344*K1**4 + 96*K1**3*K2*K3 - 1056*K1**2*K2**2 + 2472*K1**2*K2 - 160*K1**2*K3**2 - 1204*K1**2 + 1248*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 160*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 128*K2**2*K4 - 1070*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 400*K3**2 - 72*K4**2 - 12*K5**2 - 2*K6**2 + 1174 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}]] |
| If K is slice | False |