| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,2,3,1,1,1,1,0,0,0,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1074'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+22t^5+33t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1074'] |
| 2-strand cable arrow polynomial of the knot is: -784*K1**4 + 576*K1**3*K2*K3 - 288*K1**3*K3 - 704*K1**2*K2**2 - 992*K1**2*K2*K4 + 1864*K1**2*K2 - 1264*K1**2*K3**2 - 2548*K1**2 + 96*K1*K2**2*K3*K4 - 192*K1*K2**2*K3 - 64*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4040*K1*K2*K3 + 2232*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 8*K2**4 - 80*K2**2*K3**2 - 112*K2**2*K4**2 + 664*K2**2*K4 - 2188*K2**2 + 104*K2*K3*K5 + 112*K2*K4*K6 - 1912*K3**2 - 894*K4**2 - 20*K5**2 - 28*K6**2 + 2452 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1074'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10147', 'vk6.10214', 'vk6.10357', 'vk6.10434', 'vk6.16682', 'vk6.19074', 'vk6.19123', 'vk6.19266', 'vk6.19558', 'vk6.22999', 'vk6.23116', 'vk6.25699', 'vk6.25748', 'vk6.26081', 'vk6.26455', 'vk6.29930', 'vk6.29987', 'vk6.30087', 'vk6.34986', 'vk6.35107', 'vk6.37797', 'vk6.37859', 'vk6.42558', 'vk6.44671', 'vk6.51639', 'vk6.51742', 'vk6.54903', 'vk6.56595', 'vk6.59329', 'vk6.64864', 'vk6.66183', 'vk6.66216'] |
| 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 | O1O2O3O4U5U4O6U1O5U6U3U2 |
| R3 orbit | {'O1O2O3O4U5U4O6U1O5U6U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U2U5O6U4O5U1U6 |
| Gauss code of K* | O1O2O3U4U3U2U5O6O5U1O4U6 |
| Gauss code of -K* | O1O2O3U4O5U3O6O4U6U2U1U5 |
| 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 -1 0],[ 2 0 2 1 0 1 0],[-1 -2 0 0 1 -1 -1],[-1 -1 0 0 1 -1 -1],[-1 0 -1 -1 0 -1 -1],[ 1 -1 1 1 1 0 0],[ 0 0 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 -1 -1 -2],[ 0 1 1 1 0 0 0],[ 1 1 1 1 0 0 -1],[ 2 1 0 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,1,1,1,1,0,1,1,2,0,0,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,2,3,1,1,1,1,0,0,0,0,-1,-1] |
| Phi of -K | [-2,-1,0,1,1,1,0,2,1,2,3,1,1,1,1,0,0,0,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,-1,0,1,3,0,0,1,1,0,1,2,1,2,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,0,0,1,2,0,1,1,1,1,1,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2-8w^3z+24w^2z+21w |
| Inner characteristic polynomial | t^6+14t^4+12t^2 |
| Outer characteristic polynomial | t^7+22t^5+33t^3+7t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -784*K1**4 + 576*K1**3*K2*K3 - 288*K1**3*K3 - 704*K1**2*K2**2 - 992*K1**2*K2*K4 + 1864*K1**2*K2 - 1264*K1**2*K3**2 - 2548*K1**2 + 96*K1*K2**2*K3*K4 - 192*K1*K2**2*K3 - 64*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 4040*K1*K2*K3 + 2232*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 8*K2**4 - 80*K2**2*K3**2 - 112*K2**2*K4**2 + 664*K2**2*K4 - 2188*K2**2 + 104*K2*K3*K5 + 112*K2*K4*K6 - 1912*K3**2 - 894*K4**2 - 20*K5**2 - 28*K6**2 + 2452 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |