| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,4,1,0,1,1,0,1,2,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.661'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+44t^5+55t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.661'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 1056*K1**4*K2 - 1264*K1**4 - 256*K1**3*K2**2*K3 + 352*K1**3*K2*K3 - 672*K1**3*K3 - 256*K1**2*K2**4 + 3392*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 9024*K1**2*K2**2 - 992*K1**2*K2*K4 + 8520*K1**2*K2 - 16*K1**2*K3**2 - 80*K1**2*K4**2 - 5560*K1**2 + 1152*K1*K2**3*K3 - 2144*K1*K2**2*K3 - 288*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 7600*K1*K2*K3 + 752*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3136*K2**4 - 32*K2**3*K6 - 448*K2**2*K3**2 - 16*K2**2*K4**2 + 2712*K2**2*K4 - 3014*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 1748*K3**2 - 636*K4**2 - 36*K5**2 - 2*K6**2 + 4074 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.661'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4198', 'vk6.4277', 'vk6.5452', 'vk6.5564', 'vk6.7559', 'vk6.7644', 'vk6.9063', 'vk6.9142', 'vk6.18242', 'vk6.18577', 'vk6.24714', 'vk6.25127', 'vk6.36837', 'vk6.37300', 'vk6.44073', 'vk6.44412', 'vk6.48510', 'vk6.48589', 'vk6.49202', 'vk6.49308', 'vk6.50295', 'vk6.50369', 'vk6.51060', 'vk6.51091', 'vk6.56037', 'vk6.56311', 'vk6.60586', 'vk6.60925', 'vk6.65699', 'vk6.65993', 'vk6.68744', 'vk6.68952'] |
| 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 | O1O2O3O4U2O5U3U4O6U5U1U6 |
| R3 orbit | {'O1O2O3O4U2O5U3U4O6U5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4U6O5U1U2O6U3 |
| Gauss code of K* | O1O2O3U2U4U5U6O4U1O5O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U4U5U6U2 |
| 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 -1 -2 -1 1 1 2],[ 1 0 -2 -1 1 2 2],[ 2 2 0 1 2 2 0],[ 1 1 -1 0 1 2 1],[-1 -1 -2 -1 0 1 1],[-1 -2 -2 -2 -1 0 1],[-2 -2 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -1 -2 0],[-1 1 0 1 -1 -1 -2],[-1 1 -1 0 -2 -2 -2],[ 1 1 1 2 0 1 -1],[ 1 2 1 2 -1 0 -2],[ 2 0 2 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,1,1,1,2,0,-1,1,1,2,2,2,2,-1,1,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,1,4,1,0,1,1,0,1,2,1,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,0,1,1,4,1,0,1,1,0,1,2,1,0,0] |
| Phi of K* | [-2,-1,-1,1,1,2,0,0,1,2,4,-1,0,0,1,1,1,1,-1,-1,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,2,2,2,0,1,1,2,1,1,2,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+32t^4+15t^2+1 |
| Outer characteristic polynomial | t^7+44t^5+55t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -448*K1**4*K2**2 + 1056*K1**4*K2 - 1264*K1**4 - 256*K1**3*K2**2*K3 + 352*K1**3*K2*K3 - 672*K1**3*K3 - 256*K1**2*K2**4 + 3392*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 9024*K1**2*K2**2 - 992*K1**2*K2*K4 + 8520*K1**2*K2 - 16*K1**2*K3**2 - 80*K1**2*K4**2 - 5560*K1**2 + 1152*K1*K2**3*K3 - 2144*K1*K2**2*K3 - 288*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 7600*K1*K2*K3 + 752*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 3136*K2**4 - 32*K2**3*K6 - 448*K2**2*K3**2 - 16*K2**2*K4**2 + 2712*K2**2*K4 - 3014*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 1748*K3**2 - 636*K4**2 - 36*K5**2 - 2*K6**2 + 4074 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |