| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,1,3,0,1,1,1,1,1,0,-1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1706'] |
| 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+28t^5+55t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1706'] |
| 2-strand cable arrow polynomial of the knot is: -2368*K1**2*K2**4 + 3552*K1**2*K2**3 - 7408*K1**2*K2**2 - 224*K1**2*K2*K4 + 4840*K1**2*K2 - 2896*K1**2 - 384*K1*K2**4*K3 + 3040*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 480*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5384*K1*K2*K3 + 264*K1*K3*K4 + 64*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2904*K2**4 - 32*K2**3*K6 - 1360*K2**2*K3**2 - 112*K2**2*K4**2 + 1832*K2**2*K4 - 910*K2**2 + 800*K2*K3*K5 + 24*K2*K4*K6 - 1144*K3**2 - 282*K4**2 - 176*K5**2 - 2*K6**2 + 2224 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1706'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16786', 'vk6.16795', 'vk6.16819', 'vk6.16826', 'vk6.18160', 'vk6.18164', 'vk6.18495', 'vk6.18499', 'vk6.23198', 'vk6.23207', 'vk6.24619', 'vk6.25031', 'vk6.25038', 'vk6.35217', 'vk6.35248', 'vk6.36759', 'vk6.37179', 'vk6.37185', 'vk6.42694', 'vk6.42707', 'vk6.44336', 'vk6.44340', 'vk6.54979', 'vk6.55014', 'vk6.55960', 'vk6.55970', 'vk6.59367', 'vk6.59380', 'vk6.60496', 'vk6.65626', 'vk6.68168', 'vk6.68177'] |
| 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 | O1O2O3U1U4O5O4U2U3O6U5U6 |
| R3 orbit | {'O1O2O3U1U4O5O4U2U3O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U1U2O6O5U6U3 |
| Gauss code of K* | O1O2U3O4O3U5U1U2O5O6U4U6 |
| Gauss code of -K* | O1O2U1O3O4U5U2O5O6U3U4U6 |
| 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 1 2 2 2 0],[ 1 -1 0 1 1 1 1],[-1 -2 -1 0 -1 0 1],[-1 -2 -1 1 0 0 0],[ 0 -2 -1 0 0 0 1],[-1 0 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 1 0 -1 -2],[-1 0 -1 0 -1 -1 0],[ 0 0 0 1 0 -1 -2],[ 1 1 1 1 1 0 -1],[ 2 2 2 0 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,2,-1,0,1,2,1,1,0,1,2,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,1,3,0,1,1,1,1,1,0,-1,0,-1] |
| Phi of -K | [-2,-1,0,1,1,1,0,0,1,1,3,0,1,1,1,1,1,0,-1,0,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,3,-1,1,1,1,1,1,1,0,0,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,2,0,2,2,1,1,1,1,1,0,0,-1,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w |
| Inner characteristic polynomial | t^6+20t^4+26t^2+1 |
| Outer characteristic polynomial | t^7+28t^5+55t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -2368*K1**2*K2**4 + 3552*K1**2*K2**3 - 7408*K1**2*K2**2 - 224*K1**2*K2*K4 + 4840*K1**2*K2 - 2896*K1**2 - 384*K1*K2**4*K3 + 3040*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 480*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5384*K1*K2*K3 + 264*K1*K3*K4 + 64*K1*K4*K5 - 288*K2**6 + 448*K2**4*K4 - 2904*K2**4 - 32*K2**3*K6 - 1360*K2**2*K3**2 - 112*K2**2*K4**2 + 1832*K2**2*K4 - 910*K2**2 + 800*K2*K3*K5 + 24*K2*K4*K6 - 1144*K3**2 - 282*K4**2 - 176*K5**2 - 2*K6**2 + 2224 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |