| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,1,2,4,0,1,0,1,1,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1183'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+47t^5+77t^3+19t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1183'] |
| 2-strand cable arrow polynomial of the knot is: -688*K1**2*K2**2 + 480*K1**2*K2 - 80*K1**2*K3**2 - 1500*K1**2 + 448*K1*K2**3*K3 - 160*K1*K2**2*K3 - 416*K1*K2**2*K5 - 64*K1*K2*K3*K6 + 3392*K1*K2*K3 + 168*K1*K3*K4 + 56*K1*K4*K5 + 48*K1*K5*K6 - 744*K2**4 - 960*K2**2*K3**2 - 8*K2**2*K4**2 + 648*K2**2*K4 - 1634*K2**2 + 1400*K2*K3*K5 + 48*K2*K4*K6 - 16*K3**4 + 88*K3**2*K6 - 1748*K3**2 - 146*K4**2 - 448*K5**2 - 70*K6**2 + 1904 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1183'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72630', 'vk6.72635', 'vk6.72785', 'vk6.72794', 'vk6.73100', 'vk6.73107', 'vk6.73175', 'vk6.73178', 'vk6.73808', 'vk6.73811', 'vk6.73947', 'vk6.73949', 'vk6.73962', 'vk6.73963', 'vk6.75757', 'vk6.75760', 'vk6.75777', 'vk6.75784', 'vk6.77876', 'vk6.77932', 'vk6.77992', 'vk6.78019', 'vk6.78765', 'vk6.78776', 'vk6.80364', 'vk6.80366', 'vk6.80372', 'vk6.80377', 'vk6.81780', 'vk6.87793', 'vk6.89154', 'vk6.89330'] |
| 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 | O1O2O3O4U1U5U4O6O5U3U2U6 |
| R3 orbit | {'O1O2O3O4U1U5U4O6O5U3U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U2O6O5U1U6U4 |
| Gauss code of K* | O1O2O3U4U2O5O6O4U1U6U5U3 |
| Gauss code of -K* | O1O2O3U4U1O5O4O6U5U3U2U6 |
| 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 0 2 0 1],[ 3 0 3 2 1 2 2],[ 0 -3 0 0 1 -1 1],[ 0 -2 0 0 1 -1 0],[-2 -1 -1 -1 0 -2 -1],[ 0 -2 1 1 2 0 1],[-1 -2 -1 0 1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 0 -3],[-2 0 -1 -1 -1 -2 -1],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 0 -1 -2],[ 0 1 1 0 0 -1 -3],[ 0 2 1 1 1 0 -2],[ 3 1 2 2 3 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,0,3,1,1,1,2,1,0,1,1,2,0,1,2,1,3,2] |
| Phi over symmetry | [-3,0,0,0,1,2,0,1,1,2,4,0,1,0,1,1,1,1,0,0,0] |
| Phi of -K | [-3,0,0,0,1,2,0,1,1,2,4,0,1,0,1,1,1,1,0,0,0] |
| Phi of K* | [-2,-1,0,0,0,3,0,0,1,1,4,0,0,1,2,1,1,1,0,0,1] |
| Phi of -K* | [-3,0,0,0,1,2,2,2,3,2,1,-1,0,0,1,1,1,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+4w^3z^2-16w^3z+21w^2z+11w |
| Inner characteristic polynomial | t^6+33t^4+28t^2+1 |
| Outer characteristic polynomial | t^7+47t^5+77t^3+19t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -688*K1**2*K2**2 + 480*K1**2*K2 - 80*K1**2*K3**2 - 1500*K1**2 + 448*K1*K2**3*K3 - 160*K1*K2**2*K3 - 416*K1*K2**2*K5 - 64*K1*K2*K3*K6 + 3392*K1*K2*K3 + 168*K1*K3*K4 + 56*K1*K4*K5 + 48*K1*K5*K6 - 744*K2**4 - 960*K2**2*K3**2 - 8*K2**2*K4**2 + 648*K2**2*K4 - 1634*K2**2 + 1400*K2*K3*K5 + 48*K2*K4*K6 - 16*K3**4 + 88*K3**2*K6 - 1748*K3**2 - 146*K4**2 - 448*K5**2 - 70*K6**2 + 1904 |
| 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}], [{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}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |