| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,3,-1,0,1,1,0,0,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1580'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+27t^5+55t^3+21t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1580'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 224*K1**4*K2 - 1008*K1**4 + 96*K1**3*K2*K3 - 32*K1**3*K3 + 544*K1**2*K2**3 - 3392*K1**2*K2**2 - 320*K1**2*K2*K4 + 5472*K1**2*K2 - 144*K1**2*K3**2 - 4156*K1**2 + 96*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4344*K1*K2*K3 + 864*K1*K3*K4 + 160*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1424*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 112*K2**2*K4**2 + 2088*K2**2*K4 - 3382*K2**2 + 576*K2*K3*K5 + 112*K2*K4*K6 - 1528*K3**2 - 904*K4**2 - 236*K5**2 - 42*K6**2 + 3630 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1580'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11279', 'vk6.11359', 'vk6.12540', 'vk6.12653', 'vk6.18358', 'vk6.18696', 'vk6.24802', 'vk6.25259', 'vk6.30955', 'vk6.31080', 'vk6.32131', 'vk6.32252', 'vk6.36992', 'vk6.37442', 'vk6.44172', 'vk6.44491', 'vk6.52047', 'vk6.52132', 'vk6.52886', 'vk6.52951', 'vk6.56142', 'vk6.56368', 'vk6.60663', 'vk6.61008', 'vk6.63660', 'vk6.63707', 'vk6.64088', 'vk6.64135', 'vk6.65806', 'vk6.66058', 'vk6.68802', 'vk6.69010'] |
| 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 | O1O2O3U4O5U2U5O6O4U1U6U3 |
| R3 orbit | {'O1O2O3U4O5U2U5O6O4U1U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U3O5O4U6U2O6U5 |
| Gauss code of K* | O1O2O3U1U4U3O5U6O4O6U2U5 |
| Gauss code of -K* | O1O2O3U4U2O5O6U5O4U1U6U3 |
| 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 2 0 1 0],[ 2 0 0 3 1 1 0],[ 1 0 0 1 0 1 -1],[-2 -3 -1 0 -1 0 -1],[ 0 -1 0 1 0 1 0],[-1 -1 -1 0 -1 0 0],[ 0 0 1 1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -1 -3],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 1 0],[ 0 1 1 0 0 0 -1],[ 1 1 1 -1 0 0 0],[ 2 3 1 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,0,-1,0,0,1,0] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,1,3,-1,0,1,1,0,0,1,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,1,1,2,2,1,1,2,1,2,0,0,1,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,1,2,1,0,1,1,2,0,1,1,2,2,1] |
| Phi of -K* | [-2,-1,0,0,1,2,0,0,1,1,3,-1,0,1,1,0,0,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+20z+29 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2-4w^3z+24w^2z+29w |
| Inner characteristic polynomial | t^6+17t^4+27t^2+9 |
| Outer characteristic polynomial | t^7+27t^5+55t^3+21t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 224*K1**4*K2 - 1008*K1**4 + 96*K1**3*K2*K3 - 32*K1**3*K3 + 544*K1**2*K2**3 - 3392*K1**2*K2**2 - 320*K1**2*K2*K4 + 5472*K1**2*K2 - 144*K1**2*K3**2 - 4156*K1**2 + 96*K1*K2**3*K3 - 896*K1*K2**2*K3 - 64*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4344*K1*K2*K3 + 864*K1*K3*K4 + 160*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1424*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 112*K2**2*K4**2 + 2088*K2**2*K4 - 3382*K2**2 + 576*K2*K3*K5 + 112*K2*K4*K6 - 1528*K3**2 - 904*K4**2 - 236*K5**2 - 42*K6**2 + 3630 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |