| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,2,1,2,2,0,2,1,2,0,1,-1,-1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1866'] |
| Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866'] |
| Outer characteristic polynomial of the knot is: t^7+34t^5+194t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1866'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 128*K1**4*K2 - 304*K1**4 + 288*K1**3*K2*K3 - 256*K1**3*K3 - 1024*K1**2*K2**4 + 1856*K1**2*K2**3 - 6032*K1**2*K2**2 - 352*K1**2*K2*K4 + 5432*K1**2*K2 - 272*K1**2*K3**2 - 4352*K1**2 + 2336*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 288*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 6744*K1*K2*K3 - 96*K1*K2*K4*K5 + 760*K1*K3*K4 + 80*K1*K4*K5 + 24*K1*K5*K6 - 2120*K2**4 - 1616*K2**2*K3**2 - 112*K2**2*K4**2 + 1664*K2**2*K4 - 2460*K2**2 + 888*K2*K3*K5 + 112*K2*K4*K6 - 2060*K3**2 - 506*K4**2 - 156*K5**2 - 28*K6**2 + 3448 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1866'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71565', 'vk6.71674', 'vk6.72090', 'vk6.72305', 'vk6.74037', 'vk6.74598', 'vk6.76085', 'vk6.76795', 'vk6.77181', 'vk6.77278', 'vk6.77477', 'vk6.77644', 'vk6.79029', 'vk6.79605', 'vk6.80564', 'vk6.81014', 'vk6.81102', 'vk6.81142', 'vk6.81164', 'vk6.81212', 'vk6.81306', 'vk6.81453', 'vk6.82259', 'vk6.83505', 'vk6.83836', 'vk6.83979', 'vk6.85405', 'vk6.86328', 'vk6.87102', 'vk6.88023', 'vk6.88339', 'vk6.88970'] |
| 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 | O1O2O3U4U5O6U1O4O5U2U6U3 |
| R3 orbit | {'O1O2O3U4U5O6U1O4O5U2U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U2O5O6U3O4U5U6 |
| Gauss code of K* | O1O2O3U4U1U3O5O6U2O4U5U6 |
| Gauss code of -K* | O1O2O3U4U5O6U2O4O5U1U3U6 |
| 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 2 -1 1 0],[ 1 0 -1 1 1 2 0],[ 1 1 0 2 0 2 -1],[-2 -1 -2 0 -2 0 -2],[ 1 -1 0 2 0 1 1],[-1 -2 -2 0 -1 0 0],[ 0 0 1 2 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -2 -1 -2 -2],[-1 0 0 0 -2 -1 -2],[ 0 2 0 0 0 -1 1],[ 1 1 2 0 0 1 -1],[ 1 2 1 1 -1 0 0],[ 1 2 2 -1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,2,1,2,2,0,2,1,2,0,1,-1,-1,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,2,1,2,2,0,2,1,2,0,1,-1,-1,1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,2,0,1,-1,1,0,2,0,1,1,1,0,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,0,1,1,2,1,0,1,0,2,0,1,0,1,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,1,2,-1,0,2,1,-1,2,2,0,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+8w^3z^2-2w^3z+25w^2z+23w |
| Inner characteristic polynomial | t^6+26t^4+129t^2+1 |
| Outer characteristic polynomial | t^7+34t^5+194t^3+9t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 128*K1**4*K2 - 304*K1**4 + 288*K1**3*K2*K3 - 256*K1**3*K3 - 1024*K1**2*K2**4 + 1856*K1**2*K2**3 - 6032*K1**2*K2**2 - 352*K1**2*K2*K4 + 5432*K1**2*K2 - 272*K1**2*K3**2 - 4352*K1**2 + 2336*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 - 288*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 6744*K1*K2*K3 - 96*K1*K2*K4*K5 + 760*K1*K3*K4 + 80*K1*K4*K5 + 24*K1*K5*K6 - 2120*K2**4 - 1616*K2**2*K3**2 - 112*K2**2*K4**2 + 1664*K2**2*K4 - 2460*K2**2 + 888*K2*K3*K5 + 112*K2*K4*K6 - 2060*K3**2 - 506*K4**2 - 156*K5**2 - 28*K6**2 + 3448 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |