| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,3,3,2,3,0,1,0,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1820'] |
| 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+24t^5+100t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1820'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 + 608*K1**2*K2**3 - 3712*K1**2*K2**2 - 96*K1**2*K2*K4 + 4032*K1**2*K2 - 16*K1**2*K3**2 - 3412*K1**2 + 640*K1*K2**3*K3 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 + 4264*K1*K2*K3 + 224*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1208*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 16*K2**2*K4**2 + 1384*K2**2*K4 - 2382*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 1216*K3**2 - 374*K4**2 - 20*K5**2 - 2*K6**2 + 2580 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1820'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11545', 'vk6.11880', 'vk6.12896', 'vk6.13202', 'vk6.20687', 'vk6.22127', 'vk6.28205', 'vk6.29630', 'vk6.31327', 'vk6.31730', 'vk6.32491', 'vk6.32900', 'vk6.39663', 'vk6.41904', 'vk6.46251', 'vk6.47858', 'vk6.52325', 'vk6.52585', 'vk6.53169', 'vk6.53467', 'vk6.57617', 'vk6.58776', 'vk6.62289', 'vk6.63224', 'vk6.63827', 'vk6.63962', 'vk6.64273', 'vk6.64469', 'vk6.67083', 'vk6.67948', 'vk6.69691', 'vk6.70374'] |
| 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 | O1O2O3U1U2O4U3O5O6U4U6U5 |
| R3 orbit | {'O1O2O3U1U2O4U3O5O6U4U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U6O5O4U1O6U2U3 |
| Gauss code of K* | O1O2O3U4U5U6O4O5U1O6U3U2 |
| Gauss code of -K* | O1O2O3U2U1O4U3O5O6U4U5U6 |
| 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 1 2 1 0 0],[ 0 -1 0 1 1 0 0],[-1 -2 -1 0 1 1 1],[ 1 -1 -1 -1 0 2 1],[-1 0 0 -1 -2 0 0],[-1 0 0 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 -1 1 -2],[-1 -1 0 0 0 -1 0],[-1 -1 0 0 0 -2 0],[ 0 1 0 0 0 1 -1],[ 1 -1 1 2 -1 0 -1],[ 2 2 0 0 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,1,-1,2,0,0,1,0,0,2,0,-1,1,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,3,3,2,3,0,1,0,1,1,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,0,1,1,3,3,2,3,0,1,0,1,1,-1,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,0,3,1,0,3,1,1,1,3,2,1,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,1,0,0,2,-1,1,2,-1,0,0,1,0,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w |
| Inner characteristic polynomial | t^6+16t^4+19t^2+1 |
| Outer characteristic polynomial | t^7+24t^5+100t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 + 608*K1**2*K2**3 - 3712*K1**2*K2**2 - 96*K1**2*K2*K4 + 4032*K1**2*K2 - 16*K1**2*K3**2 - 3412*K1**2 + 640*K1*K2**3*K3 - 864*K1*K2**2*K3 - 192*K1*K2**2*K5 + 4264*K1*K2*K3 + 224*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1208*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 16*K2**2*K4**2 + 1384*K2**2*K4 - 2382*K2**2 + 152*K2*K3*K5 + 16*K2*K4*K6 - 1216*K3**2 - 374*K4**2 - 20*K5**2 - 2*K6**2 + 2580 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |