| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,2,2,2,1,1,1,0,0,1,2,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.677'] |
| Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 6*K1*K2 - 3*K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.677', '6.696', '6.738', '6.758'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+63t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.677'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 576*K1**2*K2**4 + 704*K1**2*K2**3 - 5168*K1**2*K2**2 - 352*K1**2*K2*K4 + 4272*K1**2*K2 - 112*K1**2*K3**2 - 3580*K1**2 + 1024*K1*K2**3*K3 - 704*K1*K2**2*K3 - 320*K1*K2**2*K5 + 6056*K1*K2*K3 - 96*K1*K2*K4*K5 + 872*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**6 + 384*K2**4*K4 - 1944*K2**4 - 128*K2**3*K6 - 480*K2**2*K3**2 - 248*K2**2*K4**2 + 2272*K2**2*K4 - 2678*K2**2 + 360*K2*K3*K5 + 216*K2*K4*K6 + 8*K3**2*K6 - 1920*K3**2 - 878*K4**2 - 92*K5**2 - 42*K6**2 + 3236 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.677'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11061', 'vk6.11139', 'vk6.12227', 'vk6.12334', 'vk6.18342', 'vk6.18680', 'vk6.24783', 'vk6.25242', 'vk6.30632', 'vk6.30727', 'vk6.31868', 'vk6.31938', 'vk6.36961', 'vk6.37420', 'vk6.44152', 'vk6.44473', 'vk6.51862', 'vk6.51907', 'vk6.52729', 'vk6.52836', 'vk6.56118', 'vk6.56340', 'vk6.60637', 'vk6.60975', 'vk6.63519', 'vk6.63563', 'vk6.64001', 'vk6.64045', 'vk6.65763', 'vk6.66026', 'vk6.68772', 'vk6.68981'] |
| 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 | O1O2O3O4U3O5U1U4O6U5U2U6 |
| R3 orbit | {'O1O2O3O4U3O5U1U4O6U5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U6O5U1U4O6U2 |
| Gauss code of K* | O1O2O3U4U2U5U6O5U1O4O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U4U6U2U5 |
| 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 0 -1 1 1 2],[ 3 0 3 0 2 2 2],[ 0 -3 0 -1 0 1 2],[ 1 0 1 0 1 1 0],[-1 -2 0 -1 0 1 1],[-1 -2 -1 -1 -1 0 1],[-2 -2 -2 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -2 0 -2],[-1 1 0 1 0 -1 -2],[-1 1 -1 0 -1 -1 -2],[ 0 2 0 1 0 -1 -3],[ 1 0 1 1 1 0 0],[ 3 2 2 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,1,1,2,0,2,-1,0,1,2,1,1,2,1,3,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,3,2,2,2,1,1,1,0,0,1,2,1,1,1] |
| Phi of -K | [-3,-1,0,1,1,2,2,0,2,2,3,0,1,1,3,0,1,0,1,0,0] |
| Phi of K* | [-2,-1,-1,0,1,3,0,0,0,3,3,-1,0,1,2,1,1,2,0,0,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,3,2,2,2,1,1,1,0,0,1,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+32t^4+24t^2 |
| Outer characteristic polynomial | t^7+48t^5+63t^3+11t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 6*K1*K2 - 3*K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 + 32*K1**3*K2*K3 - 576*K1**2*K2**4 + 704*K1**2*K2**3 - 5168*K1**2*K2**2 - 352*K1**2*K2*K4 + 4272*K1**2*K2 - 112*K1**2*K3**2 - 3580*K1**2 + 1024*K1*K2**3*K3 - 704*K1*K2**2*K3 - 320*K1*K2**2*K5 + 6056*K1*K2*K3 - 96*K1*K2*K4*K5 + 872*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**6 + 384*K2**4*K4 - 1944*K2**4 - 128*K2**3*K6 - 480*K2**2*K3**2 - 248*K2**2*K4**2 + 2272*K2**2*K4 - 2678*K2**2 + 360*K2*K3*K5 + 216*K2*K4*K6 + 8*K3**2*K6 - 1920*K3**2 - 878*K4**2 - 92*K5**2 - 42*K6**2 + 3236 |
| 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}], [{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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |