| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,1,1,0,1,1,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1830'] |
| 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+31t^5+36t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1535', '6.1830'] |
| 2-strand cable arrow polynomial of the knot is: -48*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 - 4592*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 5016*K1**2*K2 - 80*K1**2*K3**2 - 16*K1**2*K4**2 - 4084*K1**2 + 928*K1*K2**3*K3 - 960*K1*K2**2*K3 - 352*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5296*K1*K2*K3 + 392*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1328*K2**4 - 32*K2**3*K6 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 1272*K2**2*K4 - 2518*K2**2 + 232*K2*K3*K5 + 16*K2*K4*K6 - 1448*K3**2 - 300*K4**2 - 68*K5**2 - 2*K6**2 + 2874 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1830'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71469', 'vk6.71523', 'vk6.71527', 'vk6.72000', 'vk6.72004', 'vk6.72053', 'vk6.72056', 'vk6.73217', 'vk6.73228', 'vk6.73250', 'vk6.73259', 'vk6.73659', 'vk6.73670', 'vk6.75140', 'vk6.75159', 'vk6.77092', 'vk6.77139', 'vk6.77149', 'vk6.77436', 'vk6.77444', 'vk6.78075', 'vk6.78082', 'vk6.78109', 'vk6.78116', 'vk6.81295', 'vk6.81542', 'vk6.81552', 'vk6.85472', 'vk6.85478', 'vk6.86888', 'vk6.87732', 'vk6.89503'] |
| 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 | O1O2O3U1U4O5U2O4O6U5U3U6 |
| R3 orbit | {'O1O2O3U1U4O5U2O4O6U5U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U1U5O4O6U2O5U6U3 |
| Gauss code of K* | O1O2O3U4U5U2O4O6U1O5U6U3 |
| Gauss code of -K* | O1O2O3U1U4O5U3O4O6U2U5U6 |
| 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 0 1 0 -1 2],[ 2 0 1 2 1 1 1],[ 0 -1 0 1 0 0 2],[-1 -2 -1 0 0 -1 2],[ 0 -1 0 0 0 -1 1],[ 1 -1 0 1 1 0 1],[-2 -1 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -2 -1 -1],[-1 2 0 0 -1 -1 -2],[ 0 1 0 0 0 -1 -1],[ 0 2 1 0 0 0 -1],[ 1 1 1 1 0 0 -1],[ 2 1 2 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,2,1,2,1,1,0,1,1,2,0,1,1,0,1,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,1,1,0,1,1,0,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,2,0,1,1,0,0,-1] |
| Phi of K* | [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,1,1,0,1,1,0,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,1,2,1,0,1,1,1,0,1,2,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+23z+23 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+23w^2z+23w |
| Inner characteristic polynomial | t^6+21t^4+16t^2 |
| Outer characteristic polynomial | t^7+31t^5+36t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -48*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 256*K1**2*K2**4 + 608*K1**2*K2**3 - 4592*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 5016*K1**2*K2 - 80*K1**2*K3**2 - 16*K1**2*K4**2 - 4084*K1**2 + 928*K1*K2**3*K3 - 960*K1*K2**2*K3 - 352*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 5296*K1*K2*K3 + 392*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1328*K2**4 - 32*K2**3*K6 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 1272*K2**2*K4 - 2518*K2**2 + 232*K2*K3*K5 + 16*K2*K4*K6 - 1448*K3**2 - 300*K4**2 - 68*K5**2 - 2*K6**2 + 2874 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |