| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,3,3,0,0,1,1,1,1,1,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.609'] |
| Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354'] |
| Outer characteristic polynomial of the knot is: t^7+44t^5+32t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.609'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 + 576*K1**4*K2 - 2320*K1**4 + 96*K1**3*K2*K3 - 704*K1**3*K3 - 1008*K1**2*K2**2 - 320*K1**2*K2*K4 + 3920*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 1824*K1**2 - 64*K1*K2**2*K3 - 32*K1*K2**2*K5 + 2328*K1*K2*K3 + 584*K1*K3*K4 + 56*K1*K4*K5 - 56*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 216*K2**2*K4 - 1670*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 760*K3**2 - 214*K4**2 - 24*K5**2 - 2*K6**2 + 1724 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.609'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4442', 'vk6.4537', 'vk6.5824', 'vk6.5951', 'vk6.6382', 'vk6.6813', 'vk6.8000', 'vk6.8339', 'vk6.9311', 'vk6.9430', 'vk6.11634', 'vk6.11987', 'vk6.12980', 'vk6.13426', 'vk6.13523', 'vk6.13714', 'vk6.14084', 'vk6.15057', 'vk6.15177', 'vk6.17777', 'vk6.17808', 'vk6.18833', 'vk6.19440', 'vk6.19735', 'vk6.24324', 'vk6.25428', 'vk6.25459', 'vk6.26618', 'vk6.33272', 'vk6.33333', 'vk6.37552', 'vk6.39288', 'vk6.39757', 'vk6.41468', 'vk6.44893', 'vk6.46317', 'vk6.47894', 'vk6.48645', 'vk6.49887', 'vk6.53229'] |
| 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 | O1O2O3O4U3O5U1O6U5U6U4U2 |
| R3 orbit | {'O1O2O3U2O4O5U1O6U5U6U3U4', 'O1O2O3O4U3O5U1O6U5U6U4U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U3U1U5U6O5U4O6U2 |
| Gauss code of K* | O1O2O3O4U5U4U6U3O6U1O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5U4O6U2U6U1U5 |
| 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 -1 2 0 1],[ 3 0 3 0 3 1 1],[-1 -3 0 -1 1 -1 1],[ 1 0 1 0 1 0 0],[-2 -3 -1 -1 0 -1 1],[ 0 -1 1 0 1 0 1],[-1 -1 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 -1 -1 -3],[-1 -1 0 -1 -1 0 -1],[-1 1 1 0 -1 -1 -3],[ 0 1 1 1 0 0 -1],[ 1 1 0 1 0 0 0],[ 3 3 1 3 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,-1,1,1,1,3,1,1,0,1,1,1,3,0,1,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,1,1,3,3,0,0,1,1,1,1,1,-1,-1,1] |
| Phi of -K | [-3,-1,0,1,1,2,2,2,1,3,2,1,1,2,2,0,0,1,-1,0,2] |
| Phi of K* | [-2,-1,-1,0,1,3,0,2,1,2,2,1,0,1,1,0,2,3,1,2,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,1,1,3,3,0,0,1,1,1,1,1,-1,-1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+28t^4+11t^2 |
| Outer characteristic polynomial | t^7+44t^5+32t^3+3t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -128*K1**6 + 576*K1**4*K2 - 2320*K1**4 + 96*K1**3*K2*K3 - 704*K1**3*K3 - 1008*K1**2*K2**2 - 320*K1**2*K2*K4 + 3920*K1**2*K2 - 336*K1**2*K3**2 - 48*K1**2*K4**2 - 1824*K1**2 - 64*K1*K2**2*K3 - 32*K1*K2**2*K5 + 2328*K1*K2*K3 + 584*K1*K3*K4 + 56*K1*K4*K5 - 56*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 216*K2**2*K4 - 1670*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 760*K3**2 - 214*K4**2 - 24*K5**2 - 2*K6**2 + 1724 |
| 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}], [{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}, {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 |