| Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,1,2,3,-1,1,2,3,0,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.773'] |
| 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+51t^5+40t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.773'] |
| 2-strand cable arrow polynomial of the knot is: 352*K1**4*K2 - 3312*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1152*K1**3*K3 + 96*K1**2*K2**2*K4 - 3056*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8664*K1**2*K2 - 1104*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 5872*K1**2 + 64*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7200*K1*K2*K3 + 2200*K1*K3*K4 + 248*K1*K4*K5 - 368*K2**4 - 144*K2**2*K3**2 - 16*K2**2*K4**2 + 1216*K2**2*K4 - 4916*K2**2 + 312*K2*K3*K5 + 32*K2*K4*K6 - 2748*K3**2 - 1000*K4**2 - 148*K5**2 - 12*K6**2 + 5062 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.773'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16968', 'vk6.16970', 'vk6.17211', 'vk6.17213', 'vk6.20880', 'vk6.20884', 'vk6.22289', 'vk6.22293', 'vk6.23368', 'vk6.23669', 'vk6.23671', 'vk6.28353', 'vk6.35424', 'vk6.35857', 'vk6.35859', 'vk6.39991', 'vk6.39995', 'vk6.42059', 'vk6.43168', 'vk6.43170', 'vk6.46531', 'vk6.46535', 'vk6.55129', 'vk6.55131', 'vk6.55388', 'vk6.57691', 'vk6.57695', 'vk6.58884', 'vk6.59847', 'vk6.59849', 'vk6.68404', 'vk6.69745'] |
| 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 | O1O2O3O4U3O5U2U1U4O6U5U6 |
| R3 orbit | {'O1O2O3O4U3O5U2U1U4O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6O5U1U4U3O6U2 |
| Gauss code of K* | O1O2O3U4O5O4U2U1U6U3O6U5 |
| Gauss code of -K* | O1O2O3U4O5U1U5U3U2O6O4U6 |
| 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 -2 -1 2 2 1],[ 2 0 0 0 3 3 1],[ 2 0 0 0 2 2 1],[ 1 0 0 0 1 1 0],[-2 -3 -2 -1 0 1 1],[-2 -3 -2 -1 -1 0 1],[-1 -1 -1 0 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -2 -2],[-2 0 1 1 -1 -2 -3],[-2 -1 0 1 -1 -2 -3],[-1 -1 -1 0 0 -1 -1],[ 1 1 1 0 0 0 0],[ 2 2 2 1 0 0 0],[ 2 3 3 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,2,2,-1,-1,1,2,3,-1,1,2,3,0,1,1,0,0,0] |
| Phi over symmetry | [-2,-2,-1,1,2,2,-1,-1,1,2,3,-1,1,2,3,0,1,1,0,0,0] |
| Phi of -K | [-2,-2,-1,1,2,2,0,1,2,1,1,1,2,2,2,2,2,2,2,2,-1] |
| Phi of K* | [-2,-2,-1,1,2,2,-1,2,2,1,2,2,2,1,2,2,2,2,1,1,0] |
| Phi of -K* | [-2,-2,-1,1,2,2,0,0,1,2,2,0,1,3,3,0,1,1,-1,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+33t^4+8t^2 |
| Outer characteristic polynomial | t^7+51t^5+40t^3+4t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | 352*K1**4*K2 - 3312*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1152*K1**3*K3 + 96*K1**2*K2**2*K4 - 3056*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8664*K1**2*K2 - 1104*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 5872*K1**2 + 64*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7200*K1*K2*K3 + 2200*K1*K3*K4 + 248*K1*K4*K5 - 368*K2**4 - 144*K2**2*K3**2 - 16*K2**2*K4**2 + 1216*K2**2*K4 - 4916*K2**2 + 312*K2*K3*K5 + 32*K2*K4*K6 - 2748*K3**2 - 1000*K4**2 - 148*K5**2 - 12*K6**2 + 5062 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |