| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,2,0,1,0,1,0,1,0,0,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1772'] |
| Arrow polynomial of the knot is: 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.543', '6.1656', '6.1696', '6.1770', '6.1772', '6.1794'] |
| Outer characteristic polynomial of the knot is: t^7+21t^5+64t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1772'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 768*K1**4*K2 - 1760*K1**4 + 224*K1**3*K2*K3 - 256*K1**3*K3 + 1536*K1**2*K2**3 - 7920*K1**2*K2**2 - 448*K1**2*K2*K4 + 9288*K1**2*K2 - 128*K1**2*K3**2 - 5684*K1**2 + 448*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6840*K1*K2*K3 + 496*K1*K3*K4 + 48*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2312*K2**4 - 64*K2**3*K6 - 416*K2**2*K3**2 - 128*K2**2*K4**2 + 2096*K2**2*K4 - 3540*K2**2 + 288*K2*K3*K5 + 48*K2*K4*K6 - 1532*K3**2 - 478*K4**2 - 64*K5**2 - 4*K6**2 + 4252 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1772'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11275', 'vk6.11355', 'vk6.12538', 'vk6.12651', 'vk6.18357', 'vk6.18697', 'vk6.24801', 'vk6.25260', 'vk6.30958', 'vk6.31084', 'vk6.32135', 'vk6.32256', 'vk6.36991', 'vk6.37443', 'vk6.44171', 'vk6.44492', 'vk6.52043', 'vk6.52128', 'vk6.52884', 'vk6.52949', 'vk6.56141', 'vk6.56369', 'vk6.60662', 'vk6.61009', 'vk6.63662', 'vk6.63709', 'vk6.64092', 'vk6.64139', 'vk6.65805', 'vk6.66059', 'vk6.68801', 'vk6.69011'] |
| 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 | O1O2O3U4U2O5O4U6U1O6U5U3 |
| R3 orbit | {'O1O2O3U4U2O5O4U6U1O6U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U3U5O6O4U2U6 |
| Gauss code of K* | O1O2U1O3O4U2U5U4O6O5U3U6 |
| Gauss code of -K* | O1O2U3O4O3U5U2O6O5U1U6U4 |
| 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 0 0 -1],[ 1 0 1 2 0 -1 1],[ 0 -1 0 0 0 -1 0],[-2 -2 0 0 -1 -1 -2],[ 0 0 0 1 0 0 -1],[ 0 1 1 1 0 0 0],[ 1 -1 0 2 1 0 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -1 -2 -2],[ 0 0 0 0 -1 0 -1],[ 0 1 0 0 0 -1 0],[ 0 1 1 0 0 0 1],[ 1 2 0 1 0 0 -1],[ 1 2 1 0 -1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,1,1,2,2,0,1,0,1,0,1,0,0,-1,1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,2,2,0,1,0,1,0,1,0,0,-1,1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,1,2,1,1,0,1,1,0,1,2,0,1,1] |
| Phi of K* | [-2,0,0,0,1,1,1,1,2,1,1,0,0,0,1,1,1,2,1,0,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,0,0,1,2,-1,1,0,2,1,0,1,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-2w^3z+25w^2z+31w |
| Inner characteristic polynomial | t^6+15t^4+35t^2+4 |
| Outer characteristic polynomial | t^7+21t^5+64t^3+13t |
| Flat arrow polynomial | 8*K1**3 - 10*K1**2 - 8*K1*K2 - 2*K1 + 5*K2 + 2*K3 + 6 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 768*K1**4*K2 - 1760*K1**4 + 224*K1**3*K2*K3 - 256*K1**3*K3 + 1536*K1**2*K2**3 - 7920*K1**2*K2**2 - 448*K1**2*K2*K4 + 9288*K1**2*K2 - 128*K1**2*K3**2 - 5684*K1**2 + 448*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6840*K1*K2*K3 + 496*K1*K3*K4 + 48*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2312*K2**4 - 64*K2**3*K6 - 416*K2**2*K3**2 - 128*K2**2*K4**2 + 2096*K2**2*K4 - 3540*K2**2 + 288*K2*K3*K5 + 48*K2*K4*K6 - 1532*K3**2 - 478*K4**2 - 64*K5**2 - 4*K6**2 + 4252 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |