| Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,1,0,-1,0,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.695'] |
| 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^7+45t^5+49t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.695'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 + 2976*K1**4*K2 - 7936*K1**4 + 672*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2272*K1**3*K3 - 4176*K1**2*K2**2 - 672*K1**2*K2*K4 + 11720*K1**2*K2 - 800*K1**2*K3**2 - 80*K1**2*K4**2 - 4328*K1**2 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5824*K1*K2*K3 + 1128*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 336*K2**2*K4 - 4398*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 1688*K3**2 - 396*K4**2 - 32*K5**2 - 2*K6**2 + 4490 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.695'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18876', 'vk6.18882', 'vk6.18889', 'vk6.18895', 'vk6.18902', 'vk6.18904', 'vk6.18952', 'vk6.18958', 'vk6.18967', 'vk6.18973', 'vk6.18978', 'vk6.18980', 'vk6.25579', 'vk6.25581', 'vk6.25590', 'vk6.25600', 'vk6.25601', 'vk6.25607', 'vk6.37609', 'vk6.37611', 'vk6.37616', 'vk6.37622', 'vk6.37631', 'vk6.37637', 'vk6.56418', 'vk6.56423', 'vk6.56456', 'vk6.56457', 'vk6.56463', 'vk6.56467', 'vk6.56469', 'vk6.56478'] |
| 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 | O1O2O3O4U3O5U6U2O6U4U1U5 |
| R3 orbit | {'O1O2O3O4U3O5U6U2O6U4U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4U1O6U3U6O5U2 |
| Gauss code of K* | O1O2O3U2U4U5U1O5U3O6O4U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1O6U3U6U5U2 |
| 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 -1 -1 1 3 -1],[ 1 0 0 -1 2 3 0],[ 1 0 0 0 1 1 1],[ 1 1 0 0 1 1 1],[-1 -2 -1 -1 0 1 -1],[-3 -3 -1 -1 -1 0 -3],[ 1 0 -1 -1 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 -1 -1 -1 -1],[-3 0 -1 -1 -1 -3 -3],[-1 1 0 -1 -1 -1 -2],[ 1 1 1 0 0 1 1],[ 1 1 1 0 0 1 0],[ 1 3 1 -1 -1 0 0],[ 1 3 2 -1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,1,1,1,1,1,1,1,3,3,1,1,1,2,0,-1,-1,-1,0,0] |
| Phi over symmetry | [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,1,0,-1,0,-1,-1,0] |
| Phi of -K | [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,0,0,1,1,1,1,1,3,1] |
| Phi of K* | [-3,-1,1,1,1,1,1,1,1,3,3,0,1,1,1,0,-1,0,-1,-1,0] |
| Phi of -K* | [-1,-1,-1,-1,1,3,-1,-1,0,1,3,0,0,1,1,1,1,1,2,3,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+3t |
| Normalized Jones-Krushkal polynomial | z^2+22z+41 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+22w^2z+41w |
| Inner characteristic polynomial | t^6+31t^4+29t^2+1 |
| Outer characteristic polynomial | t^7+45t^5+49t^3+5t |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -256*K1**6 + 2976*K1**4*K2 - 7936*K1**4 + 672*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2272*K1**3*K3 - 4176*K1**2*K2**2 - 672*K1**2*K2*K4 + 11720*K1**2*K2 - 800*K1**2*K3**2 - 80*K1**2*K4**2 - 4328*K1**2 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5824*K1*K2*K3 + 1128*K1*K3*K4 + 72*K1*K4*K5 - 32*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 336*K2**2*K4 - 4398*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 1688*K3**2 - 396*K4**2 - 32*K5**2 - 2*K6**2 + 4490 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |