| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,-1,1,1,3,0,1,2,3,-1,1,0,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1000'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+53t^5+78t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1000'] |
| 2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 272*K1**4 + 32*K1**3*K2*K3 - 128*K1**3*K3 + 192*K1**2*K2**3 - 3264*K1**2*K2**2 - 64*K1**2*K2*K4 + 4528*K1**2*K2 - 16*K1**2*K3**2 - 3724*K1**2 + 384*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 + 4544*K1*K2*K3 + 384*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 760*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 992*K2**2*K4 - 2678*K2**2 + 208*K2*K3*K5 + 16*K2*K4*K6 - 1440*K3**2 - 294*K4**2 - 68*K5**2 - 2*K6**2 + 2740 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1000'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71627', 'vk6.71794', 'vk6.72216', 'vk6.72356', 'vk6.73407', 'vk6.73601', 'vk6.73870', 'vk6.74286', 'vk6.74912', 'vk6.75378', 'vk6.75673', 'vk6.75877', 'vk6.76466', 'vk6.77247', 'vk6.77333', 'vk6.77583', 'vk6.77687', 'vk6.78342', 'vk6.78869', 'vk6.79325', 'vk6.80122', 'vk6.80294', 'vk6.80423', 'vk6.80790', 'vk6.82025', 'vk6.82763', 'vk6.85379', 'vk6.86700', 'vk6.86934', 'vk6.87033', 'vk6.87605', 'vk6.89477'] |
| 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 | O1O2O3O4U2U1O5U4O6U3U5U6 |
| R3 orbit | {'O1O2O3O4U2U1O5U4O6U3U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6U2O5U1O6U4U3 |
| Gauss code of K* | O1O2O3U4U5U1U6O5O4U2O6U3 |
| Gauss code of -K* | O1O2O3U1O4U2O5O6U4U3U6U5 |
| 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 0 1 1 2],[ 2 0 0 3 2 2 1],[ 2 0 0 2 1 2 1],[ 0 -3 -2 0 0 2 2],[-1 -2 -1 0 0 1 1],[-1 -2 -2 -2 -1 0 1],[-2 -1 -1 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 -1 -1 -2 -1 -1],[-1 1 0 1 0 -1 -2],[-1 1 -1 0 -2 -2 -2],[ 0 2 0 2 0 -2 -3],[ 2 1 1 2 2 0 0],[ 2 1 2 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,1,1,2,1,1,-1,0,1,2,2,2,2,2,3,0] |
| Phi over symmetry | [-2,-2,0,1,1,2,0,-1,1,1,3,0,1,2,3,-1,1,0,1,0,0] |
| Phi of -K | [-2,-2,0,1,1,2,0,-1,1,1,3,0,1,2,3,-1,1,0,1,0,0] |
| Phi of K* | [-2,-1,-1,0,2,2,0,0,0,3,3,-1,-1,1,1,1,1,2,-1,0,0] |
| Phi of -K* | [-2,-2,0,1,1,2,0,2,1,2,1,3,2,2,1,0,2,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+23w^2z+23w |
| Inner characteristic polynomial | t^6+39t^4+15t^2 |
| Outer characteristic polynomial | t^7+53t^5+78t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 96*K1**4*K2 - 272*K1**4 + 32*K1**3*K2*K3 - 128*K1**3*K3 + 192*K1**2*K2**3 - 3264*K1**2*K2**2 - 64*K1**2*K2*K4 + 4528*K1**2*K2 - 16*K1**2*K3**2 - 3724*K1**2 + 384*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 + 4544*K1*K2*K3 + 384*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 760*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 992*K2**2*K4 - 2678*K2**2 + 208*K2*K3*K5 + 16*K2*K4*K6 - 1440*K3**2 - 294*K4**2 - 68*K5**2 - 2*K6**2 + 2740 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |