| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,0,-1,0,-1,0,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.692'] |
| 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+60t^5+101t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.692'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 256*K1**4*K2**2 + 800*K1**4*K2 - 1472*K1**4 + 256*K1**3*K2*K3 - 64*K1**3*K3 - 2000*K1**2*K2**2 - 64*K1**2*K2*K4 + 3552*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K4**2 - 1936*K1**2 - 544*K1*K2**2*K3 + 2536*K1*K2*K3 + 472*K1*K3*K4 + 56*K1*K4*K5 - 208*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 488*K2**2*K4 - 1854*K2**2 + 104*K2*K3*K5 + 8*K2*K4*K6 - 844*K3**2 - 272*K4**2 - 36*K5**2 - 2*K6**2 + 1846 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.692'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11055', 'vk6.11133', 'vk6.11555', 'vk6.11895', 'vk6.12221', 'vk6.12328', 'vk6.13210', 'vk6.19231', 'vk6.19323', 'vk6.19526', 'vk6.19616', 'vk6.22392', 'vk6.22711', 'vk6.22812', 'vk6.26043', 'vk6.26087', 'vk6.26509', 'vk6.28424', 'vk6.30624', 'vk6.30719', 'vk6.31344', 'vk6.31362', 'vk6.31755', 'vk6.31930', 'vk6.32528', 'vk6.32927', 'vk6.34762', 'vk6.38106', 'vk6.40134', 'vk6.40160', 'vk6.42380', 'vk6.44630', 'vk6.44741', 'vk6.46666', 'vk6.52343', 'vk6.52603', 'vk6.52812', 'vk6.56633', 'vk6.64719', 'vk6.66284'] |
| 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 | O1O2O3O4U3O5U6U1O6U2U5U4 |
| R3 orbit | {'O1O2O3O4U3O5U6U1O6U2U5U4', 'O1O2O3O4U3O5U2U6U1O6U5U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U1U5U3O6U4U6O5U2 |
| Gauss code of K* | O1O2O3U4U1U5U3O5U2O6O4U6 |
| Gauss code of -K* | O1O2O3U4O5O4U2O6U1U6U3U5 |
| 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 3 2 -1],[ 2 0 0 0 3 1 2],[ 1 0 0 0 3 1 1],[ 1 0 0 0 1 0 1],[-3 -3 -3 -1 0 0 -3],[-2 -1 -1 0 0 0 -2],[ 1 -2 -1 -1 3 2 0]] |
| Primitive based matrix | [[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -1 -3 -3 -3],[-2 0 0 0 -1 -2 -1],[ 1 1 0 0 0 1 0],[ 1 3 1 0 0 1 0],[ 1 3 2 -1 -1 0 -2],[ 2 3 1 0 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,0,-1,0,-1,0,2] |
| Phi over symmetry | [-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,0,-1,0,-1,0,2] |
| Phi of -K | [-2,-1,-1,-1,2,3,-1,1,1,3,2,1,1,1,1,0,2,1,3,3,1] |
| Phi of K* | [-3,-2,1,1,1,2,1,1,1,3,2,1,2,3,3,-1,-1,-1,0,1,1] |
| Phi of -K* | [-2,-1,-1,-1,2,3,0,0,2,1,3,0,1,0,1,1,1,3,2,3,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+3t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+40t^4+41t^2+1 |
| Outer characteristic polynomial | t^7+60t^5+101t^3+4t |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -256*K1**6 - 256*K1**4*K2**2 + 800*K1**4*K2 - 1472*K1**4 + 256*K1**3*K2*K3 - 64*K1**3*K3 - 2000*K1**2*K2**2 - 64*K1**2*K2*K4 + 3552*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K4**2 - 1936*K1**2 - 544*K1*K2**2*K3 + 2536*K1*K2*K3 + 472*K1*K3*K4 + 56*K1*K4*K5 - 208*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 488*K2**2*K4 - 1854*K2**2 + 104*K2*K3*K5 + 8*K2*K4*K6 - 844*K3**2 - 272*K4**2 - 36*K5**2 - 2*K6**2 + 1846 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |