| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,1,0,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1854'] |
| Arrow polynomial of the knot is: -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.425', '6.655', '6.755', '6.769', '6.792', '6.1240', '6.1494', '6.1522', '6.1534', '6.1587', '6.1707', '6.1746', '6.1747', '6.1786', '6.1814', '6.1828', '6.1835', '6.1854', '6.1870'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+44t^3+15t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1854'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 1376*K1**4*K2 - 3872*K1**4 + 640*K1**3*K2*K3 + 192*K1**3*K3*K4 - 544*K1**3*K3 + 640*K1**2*K2**3 - 5248*K1**2*K2**2 - 640*K1**2*K2*K4 + 8544*K1**2*K2 - 1120*K1**2*K3**2 - 96*K1**2*K3*K5 - 272*K1**2*K4**2 - 4668*K1**2 - 1056*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 6648*K1*K2*K3 + 1880*K1*K3*K4 + 296*K1*K4*K5 - 392*K2**4 - 96*K2**2*K3**2 - 16*K2**2*K4**2 + 776*K2**2*K4 - 4036*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2184*K3**2 - 694*K4**2 - 124*K5**2 - 12*K6**2 + 4340 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1854'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4073', 'vk6.4106', 'vk6.5311', 'vk6.5344', 'vk6.7443', 'vk6.7474', 'vk6.8942', 'vk6.8975', 'vk6.10115', 'vk6.10280', 'vk6.10305', 'vk6.14537', 'vk6.15270', 'vk6.15399', 'vk6.15759', 'vk6.16176', 'vk6.29863', 'vk6.29896', 'vk6.33904', 'vk6.33989', 'vk6.34204', 'vk6.34374', 'vk6.48471', 'vk6.49174', 'vk6.50223', 'vk6.50256', 'vk6.51607', 'vk6.53963', 'vk6.54028', 'vk6.54175', 'vk6.54468', 'vk6.63326'] |
| 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 | O1O2O3U4U1O4U5O6O5U2U6U3 |
| R3 orbit | {'O1O2O3U4U1O4U5O6O5U2U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U2O5O4U5O6U3U6 |
| Gauss code of K* | O1O2O3U4U1U3O5O4U5O6U2U6 |
| Gauss code of -K* | O1O2O3U4U2O4U5O6O5U1U3U6 |
| 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 -1 2 -1 1 0],[ 1 0 0 1 1 1 1],[ 1 0 0 2 1 1 0],[-2 -1 -2 0 -2 -1 -1],[ 1 -1 -1 2 0 2 0],[-1 -1 -1 1 -2 0 0],[ 0 -1 0 1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 -1 0 0],[ 1 1 1 1 0 0 1],[ 1 2 1 0 0 0 1],[ 1 2 2 0 -1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,1,1,2,2,0,1,1,2,1,0,0,0,-1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,1,0,-1,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,2,1,1,0,1,1,1,1,1,1,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,1,1,0,-1,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,-1,0,2,2,0,0,1,2,1,1,1,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+20t^4+29t^2+9 |
| Outer characteristic polynomial | t^7+28t^5+44t^3+15t |
| Flat arrow polynomial | -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| 2-strand cable arrow polynomial | -192*K1**6 - 192*K1**4*K2**2 + 1376*K1**4*K2 - 3872*K1**4 + 640*K1**3*K2*K3 + 192*K1**3*K3*K4 - 544*K1**3*K3 + 640*K1**2*K2**3 - 5248*K1**2*K2**2 - 640*K1**2*K2*K4 + 8544*K1**2*K2 - 1120*K1**2*K3**2 - 96*K1**2*K3*K5 - 272*K1**2*K4**2 - 4668*K1**2 - 1056*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 6648*K1*K2*K3 + 1880*K1*K3*K4 + 296*K1*K4*K5 - 392*K2**4 - 96*K2**2*K3**2 - 16*K2**2*K4**2 + 776*K2**2*K4 - 4036*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2184*K3**2 - 694*K4**2 - 124*K5**2 - 12*K6**2 + 4340 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]] |
| If K is slice | False |