| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,1,1,1,1,0,0,1,0,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1122', '7.40805'] |
| 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+20t^5+23t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1122', '7.40805'] |
| 2-strand cable arrow polynomial of the knot is: -1280*K1**6 - 1344*K1**4*K2**2 + 3648*K1**4*K2 - 4912*K1**4 + 1152*K1**3*K2*K3 - 992*K1**3*K3 - 448*K1**2*K2**4 + 1888*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 7120*K1**2*K2**2 - 928*K1**2*K2*K4 + 7168*K1**2*K2 - 752*K1**2*K3**2 - 96*K1**2*K3*K5 - 112*K1**2*K4**2 - 900*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 352*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5088*K1*K2*K3 + 1072*K1*K3*K4 + 296*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 920*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 112*K2**2*K4**2 + 880*K2**2*K4 - 1638*K2**2 + 344*K2*K3*K5 + 88*K2*K4*K6 - 916*K3**2 - 402*K4**2 - 112*K5**2 - 18*K6**2 + 2088 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1122'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.61', 'vk6.116', 'vk6.215', 'vk6.262', 'vk6.301', 'vk6.687', 'vk6.1213', 'vk6.1260', 'vk6.1353', 'vk6.1400', 'vk6.1445', 'vk6.1931', 'vk6.2385', 'vk6.2449', 'vk6.2933', 'vk6.2991', 'vk6.5745', 'vk6.5778', 'vk6.7814', 'vk6.7847', 'vk6.13291', 'vk6.13324', 'vk6.14776', 'vk6.14789', 'vk6.15936', 'vk6.15947', 'vk6.18057', 'vk6.24499', 'vk6.33040', 'vk6.33373', 'vk6.43919', 'vk6.50509'] |
| 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 | O1O2O3O4U3U4O5U1U5O6U2U6 |
| R3 orbit | {'O1O2O3O4U3U4O5U1U5O6U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3O5U6U4O6U1U2 |
| Gauss code of K* | O1O2U3O4O3U5O6O5U4U6U1U2 |
| Gauss code of -K* | O1O2U1O3O4U3O5O6U5U6U2U4 |
| 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 0 -1 1 1 1],[ 2 0 2 -1 1 1 1],[ 0 -2 0 -1 1 0 1],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 0 1 1 0 -1 -2],[ 1 0 0 1 1 0 1],[ 2 1 1 1 2 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,0,1,0,1,0,1,1,1,1,1,2,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,1,1,1,1,0,0,1,0,1,1,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,2,0,2,2,2,0,1,2,2,0,0,1,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,1,2,0,0,2,2,1,2,2,0,0,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,2,1,1,1,1,0,0,1,0,1,1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+12t^4+10t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+23t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -1280*K1**6 - 1344*K1**4*K2**2 + 3648*K1**4*K2 - 4912*K1**4 + 1152*K1**3*K2*K3 - 992*K1**3*K3 - 448*K1**2*K2**4 + 1888*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 7120*K1**2*K2**2 - 928*K1**2*K2*K4 + 7168*K1**2*K2 - 752*K1**2*K3**2 - 96*K1**2*K3*K5 - 112*K1**2*K4**2 - 900*K1**2 + 768*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 352*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5088*K1*K2*K3 + 1072*K1*K3*K4 + 296*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 920*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 112*K2**2*K4**2 + 880*K2**2*K4 - 1638*K2**2 + 344*K2*K3*K5 + 88*K2*K4*K6 - 916*K3**2 - 402*K4**2 - 112*K5**2 - 18*K6**2 + 2088 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {4, 5}, {2, 3}, {1}]] |
| If K is slice | False |