| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,1,2,1,0,1,1,0,1,0,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.568', '7.24819'] |
| 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+40t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.568', '7.24819'] |
| 2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 3424*K1**4 + 768*K1**3*K2*K3 - 704*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3760*K1**2*K2**2 - 576*K1**2*K2*K4 + 3704*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 + 208*K1**2 + 320*K1*K2**3*K3 - 544*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2720*K1*K2*K3 + 520*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 600*K2**4 - 272*K2**2*K3**2 - 48*K2**2*K4**2 + 600*K2**2*K4 - 694*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 412*K3**2 - 146*K4**2 - 36*K5**2 - 2*K6**2 + 840 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.568'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.412', 'vk6.461', 'vk6.466', 'vk6.857', 'vk6.910', 'vk6.915', 'vk6.1616', 'vk6.2093', 'vk6.2466', 'vk6.2500', 'vk6.2504', 'vk6.2709', 'vk6.2754', 'vk6.2757', 'vk6.3020', 'vk6.3149', 'vk6.3227', 'vk6.3238', 'vk6.3326', 'vk6.3339', 'vk6.3354', 'vk6.3441', 'vk6.15216', 'vk6.15233', 'vk6.15242', 'vk6.15243', 'vk6.15247', 'vk6.15261', 'vk6.26355', 'vk6.26358', 'vk6.26798', 'vk6.26803', 'vk6.33870', 'vk6.33884', 'vk6.33901', 'vk6.37932', 'vk6.37949', 'vk6.45095', 'vk6.45100', 'vk6.54449'] |
| 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 | O1O2O3O4O5U4U5U3O6U1U6U2 |
| R3 orbit | {'O1O2O3O4O5U4U5U3O6U1U6U2', 'O1O2O3O4U3O5U4U5O6U1U6U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U6U5O6U3U1U2 |
| Gauss code of K* | O1O2O3U4O5O4O6U5U6U3U1U2 |
| Gauss code of -K* | O1O2O3U2O4O5O6U5U6U4U1U3 |
| 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 0 -1 1 1],[ 2 0 2 0 -1 1 1],[-1 -2 0 0 -1 1 0],[ 0 0 0 0 -1 1 0],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 0 -1],[ 0 0 1 0 0 -1 0],[ 1 1 1 0 1 0 1],[ 2 2 1 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,1,0,0,1,1,0,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,0,1,1,2,1,0,1,1,0,1,0,0,0,-1] |
| Phi of -K | [-2,-1,0,1,1,1,2,2,1,2,2,0,1,1,2,1,0,1,-1,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,1,1,2,2,0,2,2] |
| Phi of -K* | [-2,-1,0,1,1,1,-1,0,1,1,2,1,0,1,1,0,1,0,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+19w^2z+15w |
| Inner characteristic polynomial | t^6+12t^4+17t^2 |
| Outer characteristic polynomial | t^7+20t^5+40t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 1536*K1**4*K2 - 3424*K1**4 + 768*K1**3*K2*K3 - 704*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3760*K1**2*K2**2 - 576*K1**2*K2*K4 + 3704*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 + 208*K1**2 + 320*K1*K2**3*K3 - 544*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2720*K1*K2*K3 + 520*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 600*K2**4 - 272*K2**2*K3**2 - 48*K2**2*K4**2 + 600*K2**2*K4 - 694*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 412*K3**2 - 146*K4**2 - 36*K5**2 - 2*K6**2 + 840 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]] |
| If K is slice | False |