| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,2,1,1,1,1,1,1,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.391'] |
| 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+34t^5+40t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.391'] |
| 2-strand cable arrow polynomial of the knot is: 1952*K1**4*K2 - 5024*K1**4 + 832*K1**3*K2*K3 - 1472*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5648*K1**2*K2**2 - 608*K1**2*K2*K4 + 9744*K1**2*K2 - 1344*K1**2*K3**2 - 128*K1**2*K4**2 - 4648*K1**2 + 320*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7320*K1*K2*K3 + 1736*K1*K3*K4 + 216*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 496*K2**4 - 304*K2**2*K3**2 - 48*K2**2*K4**2 + 856*K2**2*K4 - 4110*K2**2 + 248*K2*K3*K5 + 16*K2*K4*K6 - 2148*K3**2 - 600*K4**2 - 84*K5**2 - 2*K6**2 + 4350 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.391'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4358', 'vk6.4389', 'vk6.5676', 'vk6.5707', 'vk6.7741', 'vk6.7772', 'vk6.9219', 'vk6.9250', 'vk6.10474', 'vk6.10547', 'vk6.10642', 'vk6.10693', 'vk6.10724', 'vk6.10829', 'vk6.14626', 'vk6.15326', 'vk6.15453', 'vk6.16245', 'vk6.17995', 'vk6.24433', 'vk6.30161', 'vk6.30234', 'vk6.30329', 'vk6.30456', 'vk6.33960', 'vk6.34365', 'vk6.34421', 'vk6.43860', 'vk6.50428', 'vk6.50458', 'vk6.54214', 'vk6.63434'] |
| 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 | O1O2O3O4O5U4U2O6U5U6U1U3 |
| R3 orbit | {'O1O2O3O4O5U4U2O6U5U6U1U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U3U5U6U1O6U4U2 |
| Gauss code of K* | O1O2O3O4U3U5U4U6U1O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U4U6U1U5U2 |
| 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 -2 2 -1 1 1],[ 1 0 -1 2 -1 1 1],[ 2 1 0 2 0 2 1],[-2 -2 -2 0 -1 0 1],[ 1 1 0 1 0 1 1],[-1 -1 -2 0 -1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 -1 -1 -2],[ 1 1 1 1 0 1 0],[ 1 2 1 1 -1 0 -1],[ 2 2 1 2 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,-1,0,1,2,2,1,1,1,1,1,1,2,-1,0,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,2,2,1,1,1,1,1,1,2,-1,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,0,1,1,2,2,1,1,1,1,1,1,2,-1,1,2] |
| Phi of K* | [-2,-1,-1,1,1,2,1,2,1,2,2,1,1,1,1,1,1,2,-1,0,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,1,1,2,2,1,1,1,1,1,1,2,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+22t^4+16t^2+1 |
| Outer characteristic polynomial | t^7+34t^5+40t^3+5t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | 1952*K1**4*K2 - 5024*K1**4 + 832*K1**3*K2*K3 - 1472*K1**3*K3 - 128*K1**2*K2**4 + 416*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5648*K1**2*K2**2 - 608*K1**2*K2*K4 + 9744*K1**2*K2 - 1344*K1**2*K3**2 - 128*K1**2*K4**2 - 4648*K1**2 + 320*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7320*K1*K2*K3 + 1736*K1*K3*K4 + 216*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 496*K2**4 - 304*K2**2*K3**2 - 48*K2**2*K4**2 + 856*K2**2*K4 - 4110*K2**2 + 248*K2*K3*K5 + 16*K2*K4*K6 - 2148*K3**2 - 600*K4**2 - 84*K5**2 - 2*K6**2 + 4350 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |