| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,2,0,0,-1,0,-1,-1,0,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1784'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874'] |
| Outer characteristic polynomial of the knot is: t^7+27t^5+68t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1784'] |
| 2-strand cable arrow polynomial of the knot is: -1088*K1**4*K2**2 + 2176*K1**4*K2 - 3680*K1**4 + 96*K1**3*K2*K3 - 320*K1**3*K3 - 256*K1**2*K2**4 + 2752*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9376*K1**2*K2**2 - 512*K1**2*K2*K4 + 10856*K1**2*K2 - 160*K1**2*K3**2 - 64*K1**2*K4**2 - 5828*K1**2 + 384*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7368*K1*K2*K3 + 832*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1480*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 1424*K2**2*K4 - 4174*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 1720*K3**2 - 546*K4**2 - 36*K5**2 - 2*K6**2 + 4776 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1784'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4754', 'vk6.5081', 'vk6.6300', 'vk6.6739', 'vk6.8261', 'vk6.8710', 'vk6.9643', 'vk6.9958', 'vk6.20397', 'vk6.21744', 'vk6.27731', 'vk6.29271', 'vk6.39175', 'vk6.41405', 'vk6.45903', 'vk6.47542', 'vk6.48794', 'vk6.49005', 'vk6.49618', 'vk6.49821', 'vk6.50818', 'vk6.51033', 'vk6.51293', 'vk6.51488', 'vk6.57258', 'vk6.58473', 'vk6.61898', 'vk6.63009', 'vk6.66875', 'vk6.67747', 'vk6.69499', 'vk6.70219'] |
| 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 | O1O2O3U4U5O6O4U6U1O5U2U3 |
| R3 orbit | {'O1O2O3U4U5O6O4U6U1O5U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2O4U3U5O6O5U4U6 |
| Gauss code of K* | O1O2U3O4O5U2U4U5O6O3U1U6 |
| Gauss code of -K* | O1O2U3O4O5U6U5O3O6U1U2U4 |
| 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 0 2 0 0 -1],[ 1 0 0 1 2 2 -1],[ 0 0 0 1 0 1 -1],[-2 -1 -1 0 -2 -1 -1],[ 0 -2 0 2 0 0 -1],[ 0 -2 -1 1 0 0 -1],[ 1 1 1 1 1 1 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 -1 -1 -2 -1 -1],[ 0 1 0 1 0 0 -1],[ 0 1 -1 0 0 -2 -1],[ 0 2 0 0 0 -2 -1],[ 1 1 0 2 2 0 -1],[ 1 1 1 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,1,1,2,1,1,-1,0,0,1,0,2,1,2,1,1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,2,2,0,0,-1,0,-1,-1,0,1,0,-1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,0,0,2,-1,-1,1,2,0,0,0,1,1,1] |
| Phi of K* | [-2,0,0,0,1,1,0,1,1,2,2,0,0,-1,0,-1,-1,0,1,0,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,0,2,2,1,1,1,1,1,0,1,1,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+21t^4+39t^2+1 |
| Outer characteristic polynomial | t^7+27t^5+68t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -1088*K1**4*K2**2 + 2176*K1**4*K2 - 3680*K1**4 + 96*K1**3*K2*K3 - 320*K1**3*K3 - 256*K1**2*K2**4 + 2752*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9376*K1**2*K2**2 - 512*K1**2*K2*K4 + 10856*K1**2*K2 - 160*K1**2*K3**2 - 64*K1**2*K4**2 - 5828*K1**2 + 384*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 128*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7368*K1*K2*K3 + 832*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1480*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 1424*K2**2*K4 - 4174*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 1720*K3**2 - 546*K4**2 - 36*K5**2 - 2*K6**2 + 4776 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |