| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,0,2,2,0,1,0,0,0,2,1,-1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1548'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+54t^5+290t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1548'] |
| 2-strand cable arrow polynomial of the knot is: -832*K1**4 + 960*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 - 2448*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 5616*K1**2*K2 - 1664*K1**2*K3**2 - 160*K1**2*K3*K5 - 48*K1**2*K4**2 - 32*K1**2*K5**2 - 5720*K1**2 + 96*K1*K2**3*K3 - 736*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6872*K1*K2*K3 + 1952*K1*K3*K4 + 320*K1*K4*K5 + 80*K1*K5*K6 - 112*K2**4 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 624*K2**2*K4 - 4132*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2776*K3**2 - 748*K4**2 - 192*K5**2 - 44*K6**2 + 4394 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1548'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11008', 'vk6.11089', 'vk6.12176', 'vk6.12285', 'vk6.18200', 'vk6.18535', 'vk6.24660', 'vk6.25084', 'vk6.30579', 'vk6.30676', 'vk6.31851', 'vk6.31900', 'vk6.36794', 'vk6.37246', 'vk6.44037', 'vk6.44377', 'vk6.51807', 'vk6.51876', 'vk6.52673', 'vk6.52769', 'vk6.56006', 'vk6.56279', 'vk6.60547', 'vk6.60887', 'vk6.63493', 'vk6.63539', 'vk6.63973', 'vk6.64019', 'vk6.65671', 'vk6.65955', 'vk6.68719', 'vk6.68927'] |
| 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 | O1O2O3U4O5U2U6O4O6U1U3U5 |
| R3 orbit | {'O1O2O3U4O5U2U6O4O6U1U3U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1U3O5O6U5U2O4U6 |
| Gauss code of K* | O1O2O3U1U4U2O5U3O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U1O5U2U6U3 |
| 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 -1 2 1],[ 2 0 1 2 0 2 3],[ 1 -1 0 0 1 1 2],[-1 -2 0 0 -2 0 0],[ 1 0 -1 2 0 3 1],[-2 -2 -1 0 -3 0 -2],[-1 -3 -2 0 -1 2 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -3 -2],[-1 0 0 0 0 -2 -2],[-1 2 0 0 -2 -1 -3],[ 1 1 0 2 0 1 -1],[ 1 3 2 1 -1 0 0],[ 2 2 2 3 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,2,1,3,2,0,0,2,2,2,1,3,-1,1,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,1,0,2,2,0,1,0,0,0,2,1,-1,1,0] |
| Phi of -K | [-2,-1,-1,1,1,2,0,1,0,1,2,-1,0,2,2,1,0,0,0,-1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,-1,1,0,2,2,0,1,0,0,0,2,1,-1,1,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,1,2,3,2,-1,2,1,3,0,2,1,0,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+23w^2z+31w |
| Inner characteristic polynomial | t^6+42t^4+216t^2 |
| Outer characteristic polynomial | t^7+54t^5+290t^3+5t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -832*K1**4 + 960*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 - 2448*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 5616*K1**2*K2 - 1664*K1**2*K3**2 - 160*K1**2*K3*K5 - 48*K1**2*K4**2 - 32*K1**2*K5**2 - 5720*K1**2 + 96*K1*K2**3*K3 - 736*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 6872*K1*K2*K3 + 1952*K1*K3*K4 + 320*K1*K4*K5 + 80*K1*K5*K6 - 112*K2**4 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 624*K2**2*K4 - 4132*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2776*K3**2 - 748*K4**2 - 192*K5**2 - 44*K6**2 + 4394 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |