| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,0,2,2,4,0,0,0,1,1,1,1,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.533'] |
| Arrow polynomial of the knot is: -8*K1**2 - 6*K1*K2 + 3*K1 + 4*K2 + 3*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.248', '6.533', '6.1091'] |
| Outer characteristic polynomial of the knot is: t^7+70t^5+49t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.533'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 480*K1**4*K2 - 1552*K1**4 + 192*K1**3*K2*K3 - 448*K1**3*K3 - 1728*K1**2*K2**2 - 224*K1**2*K2*K4 + 3672*K1**2*K2 - 400*K1**2*K3**2 - 48*K1**2*K4**2 - 2364*K1**2 - 192*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3080*K1*K2*K3 - 32*K1*K3**2*K5 + 848*K1*K3*K4 + 192*K1*K4*K5 + 32*K1*K5*K6 - 128*K2**4 - 96*K2**2*K3**2 - 56*K2**2*K4**2 + 600*K2**2*K4 - 2178*K2**2 + 336*K2*K3*K5 + 40*K2*K4*K6 + 32*K3**2*K6 - 1200*K3**2 - 500*K4**2 - 164*K5**2 - 22*K6**2 + 2242 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.533'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4079', 'vk6.4112', 'vk6.4250', 'vk6.4330', 'vk6.5321', 'vk6.5354', 'vk6.5523', 'vk6.5537', 'vk6.5644', 'vk6.5658', 'vk6.7480', 'vk6.7714', 'vk6.8952', 'vk6.8985', 'vk6.9112', 'vk6.9192', 'vk6.14532', 'vk6.15284', 'vk6.15413', 'vk6.15756', 'vk6.16171', 'vk6.26277', 'vk6.26720', 'vk6.29843', 'vk6.29876', 'vk6.33922', 'vk6.34207', 'vk6.38213', 'vk6.38239', 'vk6.44990', 'vk6.45012', 'vk6.48560', 'vk6.49168', 'vk6.49265', 'vk6.49279', 'vk6.50246', 'vk6.51595', 'vk6.53981', 'vk6.54482', 'vk6.63307'] |
| 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 | O1O2O3O4O5U2U6U5O6U4U1U3 |
| R3 orbit | {'O1O2O3O4O5U2U6U5O6U4U1U3', 'O1O2O3O4U1U5U4O5U6U2O6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U5U2O6U1U6U4 |
| Gauss code of K* | O1O2O3U2O4O5O6U5U1U6U4U3 |
| Gauss code of -K* | O1O2O3U4O5O4O6U5U3U1U6U2 |
| 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 -3 2 1 2 -1],[ 1 0 -2 2 1 2 0],[ 3 2 0 3 2 1 2],[-2 -2 -3 0 0 1 -3],[-1 -1 -2 0 0 1 -2],[-2 -2 -1 -1 -1 0 -2],[ 1 0 -2 3 2 2 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -1 -3],[-2 0 1 0 -2 -3 -3],[-2 -1 0 -1 -2 -2 -1],[-1 0 1 0 -1 -2 -2],[ 1 2 2 1 0 0 -2],[ 1 3 2 2 0 0 -2],[ 3 3 1 2 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,1,3,-1,0,2,3,3,1,2,2,1,1,2,2,0,2,2] |
| Phi over symmetry | [-3,-1,-1,1,2,2,0,0,2,2,4,0,0,0,1,1,1,1,1,0,-1] |
| Phi of -K | [-3,-1,-1,1,2,2,0,0,2,2,4,0,0,0,1,1,1,1,1,0,-1] |
| Phi of K* | [-2,-2,-1,1,1,3,-1,0,1,1,4,1,0,1,2,0,1,2,0,0,0] |
| Phi of -K* | [-3,-1,-1,1,2,2,2,2,2,1,3,0,1,2,2,2,2,3,1,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+50t^4+27t^2+1 |
| Outer characteristic polynomial | t^7+70t^5+49t^3+4t |
| Flat arrow polynomial | -8*K1**2 - 6*K1*K2 + 3*K1 + 4*K2 + 3*K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 - 64*K1**4*K2**2 + 480*K1**4*K2 - 1552*K1**4 + 192*K1**3*K2*K3 - 448*K1**3*K3 - 1728*K1**2*K2**2 - 224*K1**2*K2*K4 + 3672*K1**2*K2 - 400*K1**2*K3**2 - 48*K1**2*K4**2 - 2364*K1**2 - 192*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3080*K1*K2*K3 - 32*K1*K3**2*K5 + 848*K1*K3*K4 + 192*K1*K4*K5 + 32*K1*K5*K6 - 128*K2**4 - 96*K2**2*K3**2 - 56*K2**2*K4**2 + 600*K2**2*K4 - 2178*K2**2 + 336*K2*K3*K5 + 40*K2*K4*K6 + 32*K3**2*K6 - 1200*K3**2 - 500*K4**2 - 164*K5**2 - 22*K6**2 + 2242 |
| 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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {4}, {1, 3}, {2}]] |
| If K is slice | False |