| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,3,2,1,0,1,1,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.681'] |
| 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+48t^5+37t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.681'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 352*K1**4*K2 - 1216*K1**4 + 352*K1**3*K2*K3 - 224*K1**3*K3 + 128*K1**2*K2**3 - 2064*K1**2*K2**2 - 96*K1**2*K2*K4 + 3120*K1**2*K2 - 352*K1**2*K3**2 - 1724*K1**2 - 128*K1*K2**2*K3 + 2344*K1*K2*K3 + 344*K1*K3*K4 - 184*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 248*K2**2*K4 - 1510*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 780*K3**2 - 162*K4**2 - 80*K5**2 - 18*K6**2 + 1616 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.681'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13932', 'vk6.13933', 'vk6.14027', 'vk6.14029', 'vk6.15000', 'vk6.15003', 'vk6.15122', 'vk6.15123', 'vk6.16544', 'vk6.16637', 'vk6.17459', 'vk6.17485', 'vk6.17491', 'vk6.23972', 'vk6.23982', 'vk6.24005', 'vk6.24015', 'vk6.24128', 'vk6.25995', 'vk6.26381', 'vk6.33743', 'vk6.33819', 'vk6.34944', 'vk6.35064', 'vk6.36269', 'vk6.36379', 'vk6.37588', 'vk6.37677', 'vk6.43424', 'vk6.44584', 'vk6.53884', 'vk6.54431', 'vk6.54774', 'vk6.54866', 'vk6.55604', 'vk6.56426', 'vk6.56552', 'vk6.60113', 'vk6.60119', 'vk6.60200'] |
| 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 | O1O2O3O4U3O5U1U5O6U4U2U6 |
| R3 orbit | {'O1O2O3U2O4O5U1U5O6U3U4U6', 'O1O2O3O4U3O5U1U5O6U4U2U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U1O5U6U4O6U2 |
| Gauss code of K* | O1O2O3U4U2U5U1O5U6O4O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U4O6U3U6U2U5 |
| 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 3 1 2],[ 0 -3 0 -1 1 0 2],[ 1 0 1 0 1 0 1],[-1 -3 -1 -1 0 0 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -2 -1 -2],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -1 -3],[ 0 2 0 1 0 -1 -3],[ 1 1 0 1 1 0 0],[ 3 2 1 3 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,1,2,1,2,0,0,0,1,1,1,3,1,3,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,3,1,3,2,1,0,1,1,0,1,2,0,0,1] |
| Phi of -K | [-3,-1,0,1,1,2,2,0,1,3,3,0,1,2,2,0,1,0,0,0,1] |
| Phi of K* | [-2,-1,-1,0,1,3,0,1,0,2,3,0,0,1,1,1,2,3,0,0,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,3,1,3,2,1,0,1,1,0,1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+32t^4+12t^2 |
| Outer characteristic polynomial | t^7+48t^5+37t^3+3t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 352*K1**4*K2 - 1216*K1**4 + 352*K1**3*K2*K3 - 224*K1**3*K3 + 128*K1**2*K2**3 - 2064*K1**2*K2**2 - 96*K1**2*K2*K4 + 3120*K1**2*K2 - 352*K1**2*K3**2 - 1724*K1**2 - 128*K1*K2**2*K3 + 2344*K1*K2*K3 + 344*K1*K3*K4 - 184*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 248*K2**2*K4 - 1510*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 780*K3**2 - 162*K4**2 - 80*K5**2 - 18*K6**2 + 1616 |
| 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 |