| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,0,2,2,0,0,1,1,1,0,0,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.940', '7.16075'] |
| 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+52t^5+131t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.940', '7.16075'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 192*K1**4*K2**2 + 384*K1**4*K2 - 624*K1**4 + 128*K1**3*K2*K3 - 576*K1**2*K2**2 + 800*K1**2*K2 - 208*K1**2*K3**2 - 96*K1**2*K4**2 - 48*K1**2 + 608*K1*K2*K3 + 224*K1*K3*K4 + 64*K1*K4*K5 - 96*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 112*K2**2*K4 - 252*K2**2 + 64*K2*K3*K5 + 32*K2*K4*K6 - 160*K3**2 - 72*K4**2 - 16*K5**2 - 4*K6**2 + 326 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.940'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.310', 'vk6.348', 'vk6.416', 'vk6.702', 'vk6.749', 'vk6.824', 'vk6.865', 'vk6.1125', 'vk6.1498', 'vk6.1570', 'vk6.1667', 'vk6.1947', 'vk6.1985', 'vk6.2047', 'vk6.2170', 'vk6.2279', 'vk6.2652', 'vk6.2729', 'vk6.2791', 'vk6.3120', 'vk6.5252', 'vk6.6509', 'vk6.8881', 'vk6.9798', 'vk6.18309', 'vk6.18647', 'vk6.19400', 'vk6.19695', 'vk6.25201', 'vk6.25869', 'vk6.26184', 'vk6.28493', 'vk6.36928', 'vk6.37393', 'vk6.37980', 'vk6.39867', 'vk6.40277', 'vk6.44857', 'vk6.46937', 'vk6.49124'] |
| 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 | O1O2O3O4U5U6O5O6U2U4U1U3 |
| R3 orbit | {'O1O2O3O4U5U6O5O6U2U4U1U3', 'O1O2O3U4U5O4O5U1U6U2O6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U4U1U3O5O6U5U6 |
| Gauss code of K* | O1O2O3O4U3U1U4U2O5O6U5U6 |
| Gauss code of -K* | O1O2O3O4U5U6O5O6U3U1U4U2 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 -2 2 1 -1 1],[ 1 0 -1 2 1 0 2],[ 2 1 0 2 1 1 3],[-2 -2 -2 0 0 -3 -1],[-1 -1 -1 0 0 -2 0],[ 1 0 -1 3 2 0 1],[-1 -2 -3 1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -2 -3 -2],[-1 0 0 0 -1 -2 -1],[-1 1 0 0 -2 -1 -3],[ 1 2 1 2 0 0 -1],[ 1 3 2 1 0 0 -1],[ 2 2 1 3 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,2,3,2,0,1,2,1,2,1,3,0,1,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,0,0,0,2,2,0,0,1,1,1,0,0,0,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,0,0,0,2,2,0,0,1,1,1,0,0,0,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,0,1,2,0,1,0,0,0,1,2,0,0,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,1,1,3,2,0,1,2,2,2,1,3,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | 7w^2z+15w |
| Inner characteristic polynomial | t^6+40t^4+99t^2 |
| Outer characteristic polynomial | t^7+52t^5+131t^3 |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -256*K1**6 - 192*K1**4*K2**2 + 384*K1**4*K2 - 624*K1**4 + 128*K1**3*K2*K3 - 576*K1**2*K2**2 + 800*K1**2*K2 - 208*K1**2*K3**2 - 96*K1**2*K4**2 - 48*K1**2 + 608*K1*K2*K3 + 224*K1*K3*K4 + 64*K1*K4*K5 - 96*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 112*K2**2*K4 - 252*K2**2 + 64*K2*K3*K5 + 32*K2*K4*K6 - 160*K3**2 - 72*K4**2 - 16*K5**2 - 4*K6**2 + 326 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | True |