| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,3,3,2,1,2,2,1,-1,-1,0,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.539'] |
| Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369'] |
| Outer characteristic polynomial of the knot is: t^7+56t^5+65t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.539'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 224*K1**4*K2 - 1648*K1**4 + 96*K1**3*K2*K3 - 32*K1**3*K3 - 1616*K1**2*K2**2 + 3168*K1**2*K2 - 400*K1**2*K3**2 - 1504*K1**2 - 256*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 2168*K1*K2*K3 + 680*K1*K3*K4 + 184*K1*K4*K5 + 24*K1*K5*K6 - 208*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 536*K2**2*K4 - 1630*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 820*K3**2 - 448*K4**2 - 156*K5**2 - 18*K6**2 + 1758 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.539'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13942', 'vk6.13951', 'vk6.14036', 'vk6.14047', 'vk6.15009', 'vk6.15022', 'vk6.15129', 'vk6.15144', 'vk6.16548', 'vk6.16640', 'vk6.17443', 'vk6.17462', 'vk6.17492', 'vk6.23955', 'vk6.23981', 'vk6.23986', 'vk6.24012', 'vk6.24109', 'vk6.26000', 'vk6.26384', 'vk6.33752', 'vk6.33839', 'vk6.34941', 'vk6.35059', 'vk6.36270', 'vk6.36364', 'vk6.37590', 'vk6.37679', 'vk6.43416', 'vk6.44581', 'vk6.53878', 'vk6.54424', 'vk6.54779', 'vk6.54869', 'vk6.55605', 'vk6.56446', 'vk6.56555', 'vk6.60100', 'vk6.60110', 'vk6.60185'] |
| 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 | O1O2O3O4O5U3U1U5O6U4U6U2 |
| R3 orbit | {'O1O2O3O4O5U3U1U5O6U4U6U2', 'O1O2O3O4U5U1U4O6U3U6O5U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U6U2O6U1U5U3 |
| Gauss code of K* | O1O2O3U4O5O4O6U2U6U1U5U3 |
| Gauss code of -K* | O1O2O3U2O4O5O6U4U3U6U1U5 |
| 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 1 -2 1 2 1],[ 3 0 3 0 3 2 1],[-1 -3 0 -2 0 1 1],[ 2 0 2 0 2 1 1],[-1 -3 0 -2 0 0 1],[-2 -2 -1 -1 0 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 -1 -1 -2],[-1 0 0 1 0 -2 -3],[-1 0 -1 0 -1 -1 -1],[-1 1 0 1 0 -2 -3],[ 2 1 2 1 2 0 0],[ 3 2 3 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,-1,2,3,0,0,1,1,2,-1,0,2,3,1,1,1,2,3,0] |
| Phi over symmetry | [-3,-2,1,1,1,2,0,1,3,3,2,1,2,2,1,-1,-1,0,0,0,1] |
| Phi of -K | [-3,-2,1,1,1,2,1,1,1,3,3,1,1,2,3,0,-1,0,-1,1,1] |
| Phi of K* | [-2,-1,-1,-1,2,3,0,1,1,3,3,0,1,1,1,1,1,1,2,3,1] |
| Phi of -K* | [-3,-2,1,1,1,2,0,1,3,3,2,1,2,2,1,-1,-1,0,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+36t^4+25t^2+1 |
| Outer characteristic polynomial | t^7+56t^5+65t^3+4t |
| Flat arrow polynomial | -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 224*K1**4*K2 - 1648*K1**4 + 96*K1**3*K2*K3 - 32*K1**3*K3 - 1616*K1**2*K2**2 + 3168*K1**2*K2 - 400*K1**2*K3**2 - 1504*K1**2 - 256*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 2168*K1*K2*K3 + 680*K1*K3*K4 + 184*K1*K4*K5 + 24*K1*K5*K6 - 208*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 536*K2**2*K4 - 1630*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 820*K3**2 - 448*K4**2 - 156*K5**2 - 18*K6**2 + 1758 |
| 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}], [{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}], [{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}, {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 |