| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,0,0,1,0,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1558'] |
| 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+40t^5+202t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1558'] |
| 2-strand cable arrow polynomial of the knot is: 1152*K1**4*K2 - 1840*K1**4 - 384*K1**3*K2**2*K3 + 960*K1**3*K2*K3 - 544*K1**3*K3 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3872*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 5544*K1**2*K2 - 752*K1**2*K3**2 - 3772*K1**2 + 576*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5016*K1*K2*K3 + 552*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 664*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 920*K2**2*K4 - 3142*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 - 1528*K3**2 - 230*K4**2 - 4*K5**2 - 2*K6**2 + 3116 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1558'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10140', 'vk6.10201', 'vk6.10346', 'vk6.10429', 'vk6.16683', 'vk6.19077', 'vk6.19124', 'vk6.19262', 'vk6.19555', 'vk6.22998', 'vk6.23117', 'vk6.25706', 'vk6.25752', 'vk6.26077', 'vk6.26451', 'vk6.29927', 'vk6.29982', 'vk6.30086', 'vk6.34985', 'vk6.35108', 'vk6.37806', 'vk6.37866', 'vk6.42557', 'vk6.44675', 'vk6.51624', 'vk6.51731', 'vk6.54904', 'vk6.56593', 'vk6.59330', 'vk6.64865', 'vk6.66194', 'vk6.66223'] |
| 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 | O1O2O3U4O5U3U6O4O6U2U1U5 |
| R3 orbit | {'O1O2O3U4O5U3U6O4O6U2U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U3U2O5O6U5U1O4U6 |
| Gauss code of K* | O1O2O3U2U1U4O5U3O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U1O5U6U3U2 |
| 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 0 -1 2 1],[ 1 0 0 1 -1 2 2],[ 1 0 0 1 -1 1 2],[ 0 -1 -1 0 0 0 1],[ 1 1 1 0 0 3 1],[-2 -2 -1 0 -3 0 -2],[-1 -2 -2 -1 -1 2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -2 0 -1 -2 -3],[-1 2 0 -1 -2 -2 -1],[ 0 0 1 0 -1 -1 0],[ 1 1 2 1 0 0 -1],[ 1 2 2 1 0 0 -1],[ 1 3 1 0 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,2,0,1,2,3,1,2,2,1,1,1,0,0,1,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,0,0,1,0,0,1,1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,-1,1,1,0,0,0,0,1,0,0,2,0,2,-1] |
| Phi of K* | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,0,0,1,0,0,1,1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,2,1,1,0,1,3,1,2,2,1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 7z^2+28z+29 |
| Enhanced Jones-Krushkal polynomial | 7w^3z^2+28w^2z+29w |
| Inner characteristic polynomial | t^6+32t^4+151t^2+4 |
| Outer characteristic polynomial | t^7+40t^5+202t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 1152*K1**4*K2 - 1840*K1**4 - 384*K1**3*K2**2*K3 + 960*K1**3*K2*K3 - 544*K1**3*K3 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3872*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 96*K1**2*K2*K4 + 5544*K1**2*K2 - 752*K1**2*K3**2 - 3772*K1**2 + 576*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5016*K1*K2*K3 + 552*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 664*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 16*K2**2*K4**2 + 920*K2**2*K4 - 3142*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 - 1528*K3**2 - 230*K4**2 - 4*K5**2 - 2*K6**2 + 3116 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |