| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,0,2,3,4,0,0,1,1,1,0,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1040'] |
| Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 6*K1*K2 + 7*K2 + 2*K3 + 8 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1040'] |
| Outer characteristic polynomial of the knot is: t^7+50t^5+72t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1040'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 512*K1**4*K2**2 + 2208*K1**4*K2 - 5792*K1**4 + 832*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 256*K1**2*K2**4 + 1696*K1**2*K2**3 - 9824*K1**2*K2**2 - 1024*K1**2*K2*K4 + 13512*K1**2*K2 - 1024*K1**2*K3**2 - 112*K1**2*K4**2 - 6724*K1**2 + 480*K1*K2**3*K3 - 1472*K1*K2**2*K3 - 288*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 10528*K1*K2*K3 + 1904*K1*K3*K4 + 256*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1512*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 24*K2**2*K4**2 + 1992*K2**2*K4 - 5804*K2**2 + 432*K2*K3*K5 + 32*K2*K4*K6 - 2952*K3**2 - 986*K4**2 - 212*K5**2 - 28*K6**2 + 6328 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1040'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4251', 'vk6.4257', 'vk6.4331', 'vk6.4337', 'vk6.5530', 'vk6.5538', 'vk6.5649', 'vk6.5657', 'vk6.7721', 'vk6.7727', 'vk6.9123', 'vk6.9129', 'vk6.9203', 'vk6.9209', 'vk6.19824', 'vk6.19825', 'vk6.26262', 'vk6.26263', 'vk6.26706', 'vk6.26707', 'vk6.38208', 'vk6.38209', 'vk6.44985', 'vk6.44986', 'vk6.48567', 'vk6.48573', 'vk6.49280', 'vk6.49288', 'vk6.50416', 'vk6.50422', 'vk6.66362', 'vk6.66363'] |
| 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 | O1O2O3O4U5U3O5U2O6U1U6U4 |
| R3 orbit | {'O1O2O3O4U5U3O5U2O6U1U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U4O5U3O6U2U6 |
| Gauss code of K* | O1O2O3U1U4U5U3O6O5U6O4U2 |
| Gauss code of -K* | O1O2O3U2O4U5O6O5U1U6U4U3 |
| 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 0 3 -1 1],[ 2 0 0 1 4 1 1],[ 1 0 0 1 2 0 0],[ 0 -1 -1 0 0 0 0],[-3 -4 -2 0 0 -3 0],[ 1 -1 0 0 3 0 1],[-1 -1 0 0 0 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 0 0 -2 -3 -4],[-1 0 0 0 0 -1 -1],[ 0 0 0 0 -1 0 -1],[ 1 2 0 1 0 0 0],[ 1 3 1 0 0 0 -1],[ 2 4 1 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,0,0,2,3,4,0,0,1,1,1,0,1,0,0,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,0,2,3,4,0,0,1,1,1,0,1,0,0,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,1,2,1,0,1,1,1,0,2,2,1,3,2] |
| Phi of K* | [-3,-1,0,1,1,2,2,3,1,2,1,1,1,2,2,1,0,1,0,0,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,1,1,4,0,1,0,2,0,1,3,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+34t^4+39t^2+1 |
| Outer characteristic polynomial | t^7+50t^5+72t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 14*K1**2 - 6*K1*K2 + 7*K2 + 2*K3 + 8 |
| 2-strand cable arrow polynomial | -192*K1**6 - 512*K1**4*K2**2 + 2208*K1**4*K2 - 5792*K1**4 + 832*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 - 256*K1**2*K2**4 + 1696*K1**2*K2**3 - 9824*K1**2*K2**2 - 1024*K1**2*K2*K4 + 13512*K1**2*K2 - 1024*K1**2*K3**2 - 112*K1**2*K4**2 - 6724*K1**2 + 480*K1*K2**3*K3 - 1472*K1*K2**2*K3 - 288*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 10528*K1*K2*K3 + 1904*K1*K3*K4 + 256*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1512*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 24*K2**2*K4**2 + 1992*K2**2*K4 - 5804*K2**2 + 432*K2*K3*K5 + 32*K2*K4*K6 - 2952*K3**2 - 986*K4**2 - 212*K5**2 - 28*K6**2 + 6328 |
| 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}], [{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}, {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}, {4}, {3}, {1, 2}]] |
| If K is slice | False |