| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,2,3,1,1,2,1,1,1,0,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1865'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 8*K1*K2 + K1 + 3*K2 + 3*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1080', '6.1837', '6.1841', '6.1865'] |
| Outer characteristic polynomial of the knot is: t^7+32t^5+94t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1865'] |
| 2-strand cable arrow polynomial of the knot is: 2016*K1**4*K2 - 5360*K1**4 + 1376*K1**3*K2*K3 - 1952*K1**3*K3 - 128*K1**2*K2**4 + 800*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6992*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 1152*K1**2*K2*K4 + 10416*K1**2*K2 - 2256*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K4**2 - 4860*K1**2 + 576*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 256*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9656*K1*K2*K3 + 2848*K1*K3*K4 + 280*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1112*K2**4 - 768*K2**2*K3**2 - 64*K2**2*K4**2 + 1760*K2**2*K4 - 4554*K2**2 + 840*K2*K3*K5 + 56*K2*K4*K6 - 64*K3**4 + 48*K3**2*K6 - 2876*K3**2 - 1062*K4**2 - 240*K5**2 - 22*K6**2 + 4964 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1865'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4352', 'vk6.4383', 'vk6.5674', 'vk6.5705', 'vk6.7743', 'vk6.7774', 'vk6.9225', 'vk6.9256', 'vk6.10494', 'vk6.10540', 'vk6.10637', 'vk6.10713', 'vk6.10744', 'vk6.10826', 'vk6.14623', 'vk6.15311', 'vk6.15436', 'vk6.16242', 'vk6.17980', 'vk6.24424', 'vk6.30173', 'vk6.30219', 'vk6.30316', 'vk6.30445', 'vk6.33945', 'vk6.34350', 'vk6.34404', 'vk6.43859', 'vk6.50430', 'vk6.50460', 'vk6.54209', 'vk6.63444'] |
| 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 | O1O2O3U4U5O4U2O6O5U6U1U3 |
| R3 orbit | {'O1O2O3U4U5O4U2O6O5U6U1U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U3U4O5O4U2O6U5U6 |
| Gauss code of K* | O1O2O3U2U4U3O5O6U5O4U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5U6O4O6U1U5U2 |
| 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 0 2 -1 1 -1],[ 1 0 1 2 0 2 -1],[ 0 -1 0 0 0 1 -1],[-2 -2 0 0 -3 0 -1],[ 1 0 0 3 0 1 0],[-1 -2 -1 0 -1 0 -1],[ 1 1 1 1 0 1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 0 -1 -2 -3],[-1 0 0 -1 -1 -2 -1],[ 0 0 1 0 -1 -1 0],[ 1 1 1 1 0 1 0],[ 1 2 2 1 -1 0 0],[ 1 3 1 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,0,1,2,3,1,1,2,1,1,1,0,-1,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,2,3,1,1,2,1,1,1,0,-1,0,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,0,1,1,1,0,0,2,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,2,0,1,2,0,1,0,1,1,0,0,0,0,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,2,2,0,1,1,1,0,1,3,1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+24t^4+65t^2+1 |
| Outer characteristic polynomial | t^7+32t^5+94t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 8*K1*K2 + K1 + 3*K2 + 3*K3 + 4 |
| 2-strand cable arrow polynomial | 2016*K1**4*K2 - 5360*K1**4 + 1376*K1**3*K2*K3 - 1952*K1**3*K3 - 128*K1**2*K2**4 + 800*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6992*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 1152*K1**2*K2*K4 + 10416*K1**2*K2 - 2256*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K4**2 - 4860*K1**2 + 576*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 256*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9656*K1*K2*K3 + 2848*K1*K3*K4 + 280*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1112*K2**4 - 768*K2**2*K3**2 - 64*K2**2*K4**2 + 1760*K2**2*K4 - 4554*K2**2 + 840*K2*K3*K5 + 56*K2*K4*K6 - 64*K3**4 + 48*K3**2*K6 - 2876*K3**2 - 1062*K4**2 - 240*K5**2 - 22*K6**2 + 4964 |
| 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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {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 |