| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,3,2,2,0,1,1,1,1,1,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1300'] |
| 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+52t^5+81t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1300'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 320*K1**4*K2 - 2688*K1**4 + 96*K1**3*K2*K3 - 128*K1**3*K3 + 96*K1**2*K2**3 - 3568*K1**2*K2**2 + 6416*K1**2*K2 - 928*K1**2*K3**2 - 3496*K1**2 - 224*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 4696*K1*K2*K3 + 992*K1*K3*K4 + 104*K1*K4*K5 + 24*K1*K5*K6 - 216*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 480*K2**2*K4 - 3142*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 1516*K3**2 - 430*K4**2 - 116*K5**2 - 18*K6**2 + 3316 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1300'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4427', 'vk6.4524', 'vk6.5813', 'vk6.5942', 'vk6.7870', 'vk6.7979', 'vk6.9292', 'vk6.9413', 'vk6.10170', 'vk6.10243', 'vk6.10384', 'vk6.17871', 'vk6.17934', 'vk6.18268', 'vk6.18603', 'vk6.24374', 'vk6.25158', 'vk6.30061', 'vk6.30124', 'vk6.36886', 'vk6.37344', 'vk6.43809', 'vk6.44107', 'vk6.44430', 'vk6.48632', 'vk6.50529', 'vk6.50616', 'vk6.51138', 'vk6.51671', 'vk6.55838', 'vk6.56072', 'vk6.65506'] |
| 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 | O1O2O3U2O4O5U6U4O6U1U3U5 |
| R3 orbit | {'O1O2O3U2O4O5U6U4O6U1U3U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1U3O5U6U5O4O6U2 |
| Gauss code of K* | O1O2O3U1U4U2O4U5U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1U5O6U2U6U3 |
| 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 -2 -1 1 0 3 -1],[ 2 0 0 2 1 3 1],[ 1 0 0 1 0 1 1],[-1 -2 -1 0 1 2 -2],[ 0 -1 0 -1 0 0 0],[-3 -3 -1 -2 0 0 -3],[ 1 -1 -1 2 0 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -2 0 -1 -3 -3],[-1 2 0 1 -1 -2 -2],[ 0 0 -1 0 0 0 -1],[ 1 1 1 0 0 1 0],[ 1 3 2 0 -1 0 -1],[ 2 3 2 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,2,0,1,3,3,-1,1,2,2,0,0,1,-1,0,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,3,1,3,2,2,0,1,1,1,1,1,-1,0,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,1,1,2,1,1,0,1,1,1,3,2,3,0] |
| Phi of K* | [-3,-1,0,1,1,2,0,3,1,3,2,2,0,1,1,1,1,1,-1,0,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,1,2,3,1,0,1,1,0,2,3,-1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+22z+33 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+22w^2z+33w |
| Inner characteristic polynomial | t^6+36t^4+44t^2+1 |
| Outer characteristic polynomial | t^7+52t^5+81t^3+6t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 320*K1**4*K2 - 2688*K1**4 + 96*K1**3*K2*K3 - 128*K1**3*K3 + 96*K1**2*K2**3 - 3568*K1**2*K2**2 + 6416*K1**2*K2 - 928*K1**2*K3**2 - 3496*K1**2 - 224*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 4696*K1*K2*K3 + 992*K1*K3*K4 + 104*K1*K4*K5 + 24*K1*K5*K6 - 216*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 480*K2**2*K4 - 3142*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 1516*K3**2 - 430*K4**2 - 116*K5**2 - 18*K6**2 + 3316 |
| 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}, {4}, {2, 3}, {1}]] |
| If K is slice | False |