| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,0,1,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1803'] |
| 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+20t^5+43t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1803'] |
| 2-strand cable arrow polynomial of the knot is: -768*K1**4*K2**2 + 352*K1**4*K2 - 976*K1**4 + 1280*K1**3*K2*K3 - 512*K1**3*K3 - 768*K1**2*K2**4 + 1504*K1**2*K2**3 - 5376*K1**2*K2**2 - 608*K1**2*K2*K4 + 5304*K1**2*K2 - 1264*K1**2*K3**2 - 3580*K1**2 + 1344*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 6112*K1*K2*K3 - 96*K1*K2*K4*K5 + 1520*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 744*K2**4 - 656*K2**2*K3**2 - 112*K2**2*K4**2 + 736*K2**2*K4 - 2364*K2**2 + 352*K2*K3*K5 + 112*K2*K4*K6 - 1872*K3**2 - 526*K4**2 - 68*K5**2 - 28*K6**2 + 2924 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1803'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19914', 'vk6.19961', 'vk6.21137', 'vk6.21220', 'vk6.26831', 'vk6.26962', 'vk6.28609', 'vk6.28702', 'vk6.38267', 'vk6.38370', 'vk6.40395', 'vk6.40531', 'vk6.45138', 'vk6.45239', 'vk6.46990', 'vk6.47050', 'vk6.56697', 'vk6.56755', 'vk6.57783', 'vk6.57868', 'vk6.61099', 'vk6.61222', 'vk6.62353', 'vk6.62442', 'vk6.66385', 'vk6.66465', 'vk6.67147', 'vk6.67248', 'vk6.69044', 'vk6.69113', 'vk6.69834', 'vk6.69886'] |
| 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 | O1O2O3U4U3O5O6U2U1O4U5U6 |
| R3 orbit | {'O1O2O3U4U3O5O6U2U1O4U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O6U3U2O4O5U1U6 |
| Gauss code of K* | O1O2U3O4O5U2U1U6O3O6U4U5 |
| Gauss code of -K* | O1O2U3O4O5U1U2O6O3U6U5U4 |
| 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 1 -1 0 2],[ 1 0 0 0 -1 1 2],[ 1 0 0 0 0 0 1],[-1 0 0 0 -1 -1 -1],[ 1 1 0 1 0 0 1],[ 0 -1 0 1 0 0 1],[-2 -2 -1 1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -1 -1 -1 -2],[-1 -1 0 -1 0 -1 0],[ 0 1 1 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 1 1 0 0 0 1],[ 1 2 0 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,0,1,0,0,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,0,1,0,0,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,2,1,1,2,2,0,1,2] |
| Phi of K* | [-2,-1,0,1,1,1,2,1,1,2,2,0,2,1,2,0,1,1,-1,0,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,0,2,0,0,1,1,0,0,1,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 7z^2+26z+25 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+9w^3z^2+26w^2z+25w |
| Inner characteristic polynomial | t^6+12t^4+22t^2+4 |
| Outer characteristic polynomial | t^7+20t^5+43t^3+13t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -768*K1**4*K2**2 + 352*K1**4*K2 - 976*K1**4 + 1280*K1**3*K2*K3 - 512*K1**3*K3 - 768*K1**2*K2**4 + 1504*K1**2*K2**3 - 5376*K1**2*K2**2 - 608*K1**2*K2*K4 + 5304*K1**2*K2 - 1264*K1**2*K3**2 - 3580*K1**2 + 1344*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 - 64*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 6112*K1*K2*K3 - 96*K1*K2*K4*K5 + 1520*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 744*K2**4 - 656*K2**2*K3**2 - 112*K2**2*K4**2 + 736*K2**2*K4 - 2364*K2**2 + 352*K2*K3*K5 + 112*K2*K4*K6 - 1872*K3**2 - 526*K4**2 - 68*K5**2 - 28*K6**2 + 2924 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {3, 5}, {2, 4}, {1}]] |
| If K is slice | False |