| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,0,1,2,1,1,1,1,0,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.955'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+22t^5+46t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.955'] |
| 2-strand cable arrow polynomial of the knot is: -432*K1**4 + 96*K1**3*K3*K4 - 320*K1**3*K3 - 432*K1**2*K2**2 - 128*K1**2*K2*K4 + 2080*K1**2*K2 - 880*K1**2*K3**2 - 96*K1**2*K3*K5 - 208*K1**2*K4**2 - 2596*K1**2 - 352*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2856*K1*K2*K3 + 1800*K1*K3*K4 + 352*K1*K4*K5 - 72*K2**4 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 680*K2**2*K4 - 1996*K2**2 + 168*K2*K3*K5 + 32*K2*K4*K6 - 1496*K3**2 - 894*K4**2 - 124*K5**2 - 12*K6**2 + 2276 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.955'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4708', 'vk6.5017', 'vk6.6220', 'vk6.6681', 'vk6.8203', 'vk6.8630', 'vk6.9579', 'vk6.9914', 'vk6.17407', 'vk6.20930', 'vk6.21085', 'vk6.22341', 'vk6.22515', 'vk6.23581', 'vk6.23918', 'vk6.28407', 'vk6.36188', 'vk6.40077', 'vk6.40327', 'vk6.42127', 'vk6.43405', 'vk6.46603', 'vk6.46793', 'vk6.48050', 'vk6.48746', 'vk6.49759', 'vk6.50756', 'vk6.51442', 'vk6.57737', 'vk6.58937', 'vk6.65301', 'vk6.69775'] |
| 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 | O1O2O3O4U5U1O6O5U6U4U3U2 |
| R3 orbit | {'O1O2O3O4U5U1O6O5U6U4U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U2U1U5O6O5U4U6 |
| Gauss code of K* | O1O2O3O4U5U4U3U2O6O5U1U6 |
| Gauss code of -K* | O1O2O3O4U5U4O6O5U3U2U1U6 |
| 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 0 -1],[ 2 0 2 1 0 2 -1],[-1 -2 0 0 0 0 -1],[-1 -1 0 0 0 0 -1],[-1 0 0 0 0 0 -1],[ 0 -2 0 0 0 0 -1],[ 1 1 1 1 1 1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 0],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[ 0 0 0 0 0 -1 -2],[ 1 1 1 1 1 0 1],[ 2 0 1 2 2 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,1,0,0,0,1,1,0,1,2,1,2,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,0,1,2,1,1,1,1,0,0,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,2,0,1,2,3,0,1,1,1,1,1,1,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,1,1,1,0,1,1,2,1,1,3,0,0,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,2,0,1,2,1,1,1,1,0,0,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+4w^3z^2-12w^3z+17w^2z+11w |
| Inner characteristic polynomial | t^6+14t^4+11t^2 |
| Outer characteristic polynomial | t^7+22t^5+46t^3+6t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -432*K1**4 + 96*K1**3*K3*K4 - 320*K1**3*K3 - 432*K1**2*K2**2 - 128*K1**2*K2*K4 + 2080*K1**2*K2 - 880*K1**2*K3**2 - 96*K1**2*K3*K5 - 208*K1**2*K4**2 - 2596*K1**2 - 352*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2856*K1*K2*K3 + 1800*K1*K3*K4 + 352*K1*K4*K5 - 72*K2**4 - 64*K2**2*K3**2 - 16*K2**2*K4**2 + 680*K2**2*K4 - 1996*K2**2 + 168*K2*K3*K5 + 32*K2*K4*K6 - 1496*K3**2 - 894*K4**2 - 124*K5**2 - 12*K6**2 + 2276 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]] |
| If K is slice | False |