| Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,1,2,2,1,2,1,2,0,0,-1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1387'] |
| 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+34t^5+52t^3+17t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1387'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 256*K1**4*K2 - 2336*K1**4 + 704*K1**3*K2*K3 - 768*K1**3*K3 + 128*K1**2*K2**3 - 2784*K1**2*K2**2 - 640*K1**2*K2*K4 + 6416*K1**2*K2 - 1184*K1**2*K3**2 - 4624*K1**2 - 192*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 5920*K1*K2*K3 + 1280*K1*K3*K4 + 80*K1*K4*K5 - 80*K2**4 - 32*K2**2*K3**2 - 48*K2**2*K4**2 + 544*K2**2*K4 - 4068*K2**2 + 400*K2*K3*K5 + 208*K2*K4*K6 + 96*K3**2*K6 - 2312*K3**2 - 588*K4**2 - 216*K5**2 - 140*K6**2 + 4170 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1387'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4847', 'vk6.5190', 'vk6.6416', 'vk6.6847', 'vk6.8385', 'vk6.8808', 'vk6.9746', 'vk6.10043', 'vk6.20796', 'vk6.22192', 'vk6.29755', 'vk6.39834', 'vk6.46393', 'vk6.47969', 'vk6.49072', 'vk6.49905', 'vk6.51329', 'vk6.51546', 'vk6.58813', 'vk6.63277'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is +. |
| The reverse -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 | O1O2O3U4O5O6U5U2U6O4U1U3 |
| R3 orbit | {'O1O2O3U4O5O6U5U2U6O4U1U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U3O4U5U2U6O5O6U4 |
| Gauss code of K* | Same |
| Gauss code of -K* | O1O2O3U1U3O4U5U2U6O5O6U4 |
| Diagrammatic symmetry type | + |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 -1 2 -1 -1 2],[ 1 0 0 2 0 -1 2],[ 1 0 0 1 0 0 2],[-2 -2 -1 0 -2 -1 1],[ 1 0 0 2 0 0 1],[ 1 1 0 1 0 0 1],[-2 -2 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 -1 -1 -1 -1],[-2 0 1 -1 -1 -2 -2],[-2 -1 0 -1 -2 -1 -2],[ 1 1 1 0 0 0 1],[ 1 1 2 0 0 0 0],[ 1 2 1 0 0 0 0],[ 1 2 2 -1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,1,1,1,1,-1,1,1,2,2,1,2,1,2,0,0,-1,0,0,0] |
| Phi over symmetry | [-2,-2,1,1,1,1,-1,1,1,2,2,1,2,1,2,0,0,-1,0,0,0] |
| Phi of -K | [-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,1,1,0,1,2,2,1,-1] |
| Phi of K* | [-2,-2,1,1,1,1,-1,1,1,2,2,1,2,1,2,0,0,-1,0,0,0] |
| Phi of -K* | [-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,1,1,0,1,2,2,1,-1] |
| Symmetry type of based matrix | + |
| u-polynomial | -2t^2+4t |
| Normalized Jones-Krushkal polynomial | 17z+35 |
| Enhanced Jones-Krushkal polynomial | -4w^3z+21w^2z+35w |
| Inner characteristic polynomial | t^6+22t^4+30t^2+9 |
| Outer characteristic polynomial | t^7+34t^5+52t^3+17t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 256*K1**4*K2 - 2336*K1**4 + 704*K1**3*K2*K3 - 768*K1**3*K3 + 128*K1**2*K2**3 - 2784*K1**2*K2**2 - 640*K1**2*K2*K4 + 6416*K1**2*K2 - 1184*K1**2*K3**2 - 4624*K1**2 - 192*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 5920*K1*K2*K3 + 1280*K1*K3*K4 + 80*K1*K4*K5 - 80*K2**4 - 32*K2**2*K3**2 - 48*K2**2*K4**2 + 544*K2**2*K4 - 4068*K2**2 + 400*K2*K3*K5 + 208*K2*K4*K6 + 96*K3**2*K6 - 2312*K3**2 - 588*K4**2 - 216*K5**2 - 140*K6**2 + 4170 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |