| Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,1,3,1,0,1,1,1,1,1,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.646'] |
| 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+36t^5+45t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.646'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 896*K1**4*K2 - 2080*K1**4 + 128*K1**3*K2*K3 - 384*K1**3*K3 - 1984*K1**2*K2**2 - 32*K1**2*K2*K4 + 4160*K1**2*K2 - 160*K1**2*K3**2 - 1980*K1**2 - 352*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2544*K1*K2*K3 + 384*K1*K3*K4 + 16*K1*K4*K5 - 96*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 392*K2**2*K4 - 2006*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 828*K3**2 - 236*K4**2 - 40*K5**2 - 2*K6**2 + 1946 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.646'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4125', 'vk6.4158', 'vk6.5363', 'vk6.5396', 'vk6.5459', 'vk6.5572', 'vk6.7485', 'vk6.7653', 'vk6.8986', 'vk6.9019', 'vk6.11178', 'vk6.12264', 'vk6.12373', 'vk6.12447', 'vk6.12478', 'vk6.13359', 'vk6.13582', 'vk6.13615', 'vk6.14259', 'vk6.14706', 'vk6.14747', 'vk6.15183', 'vk6.15862', 'vk6.15901', 'vk6.26196', 'vk6.26641', 'vk6.30852', 'vk6.30883', 'vk6.32036', 'vk6.32067', 'vk6.33083', 'vk6.33116', 'vk6.38174', 'vk6.38180', 'vk6.44831', 'vk6.44925', 'vk6.49221', 'vk6.49330', 'vk6.52762', 'vk6.53531'] |
| 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 | O1O2O3O4U1O5U4U5O6U3U6U2 |
| R3 orbit | {'O1O2O3O4U1O5U4U5U2O6U3U6', 'O1O2O3O4U1O5U4U5O6U3U6U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U3U5U2O5U6U1O6U4 |
| Gauss code of K* | O1O2O3U4U3U1U5O4U6O5O6U2 |
| Gauss code of -K* | O1O2O3U2O4O5U4O6U5U3U1U6 |
| 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 0 1 1],[ 3 0 3 2 1 1 1],[-1 -3 0 -1 -1 1 1],[ 0 -2 1 0 -1 1 1],[ 0 -1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 0 -3],[-1 0 1 1 -1 -1 -3],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 -1 -1 -1],[ 0 1 0 1 0 1 -1],[ 0 1 1 1 -1 0 -2],[ 3 3 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,0,3,-1,-1,1,1,3,0,0,1,1,1,1,1,-1,1,2] |
| Phi over symmetry | [-3,0,0,1,1,1,1,2,1,1,3,1,0,1,1,1,1,1,0,-1,-1] |
| Phi of -K | [-3,0,0,1,1,1,1,2,1,3,3,1,0,0,0,0,0,1,-1,-1,0] |
| Phi of K* | [-1,-1,-1,0,0,3,-1,0,0,0,3,1,0,0,1,0,1,3,-1,1,2] |
| Phi of -K* | [-3,0,0,1,1,1,1,2,1,1,3,1,0,1,1,1,1,1,0,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+24t^4+17t^2 |
| Outer characteristic polynomial | t^7+36t^5+45t^3+3t |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -192*K1**6 - 128*K1**4*K2**2 + 896*K1**4*K2 - 2080*K1**4 + 128*K1**3*K2*K3 - 384*K1**3*K3 - 1984*K1**2*K2**2 - 32*K1**2*K2*K4 + 4160*K1**2*K2 - 160*K1**2*K3**2 - 1980*K1**2 - 352*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2544*K1*K2*K3 + 384*K1*K3*K4 + 16*K1*K4*K5 - 96*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 392*K2**2*K4 - 2006*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 828*K3**2 - 236*K4**2 - 40*K5**2 - 2*K6**2 + 1946 |
| 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}], [{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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |