| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,0,2,2,1,1,1,1,1,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1123', '7.40806'] |
| 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+24t^5+52t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1123', '7.40806'] |
| 2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 2176*K1**4*K2**2 + 3712*K1**4*K2 - 4384*K1**4 + 1856*K1**3*K2*K3 - 576*K1**3*K3 - 1216*K1**2*K2**4 + 3904*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10240*K1**2*K2**2 - 928*K1**2*K2*K4 + 8288*K1**2*K2 - 704*K1**2*K3**2 - 1660*K1**2 + 2144*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 - 448*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7016*K1*K2*K3 + 888*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2632*K2**4 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 112*K2**2*K4**2 + 2096*K2**2*K4 - 1734*K2**2 - 96*K2*K3**2*K4 + 688*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1260*K3**2 - 382*K4**2 - 96*K5**2 - 18*K6**2 + 2660 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1123'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.401', 'vk6.432', 'vk6.438', 'vk6.839', 'vk6.849', 'vk6.878', 'vk6.888', 'vk6.1583', 'vk6.2029', 'vk6.2036', 'vk6.2056', 'vk6.2065', 'vk6.2704', 'vk6.2737', 'vk6.2740', 'vk6.3141', 'vk6.13528', 'vk6.13548', 'vk6.13717', 'vk6.13737', 'vk6.19464', 'vk6.19469', 'vk6.19757', 'vk6.19763', 'vk6.25799', 'vk6.25807', 'vk6.26632', 'vk6.37909', 'vk6.37915', 'vk6.44910', 'vk6.53670', 'vk6.66242'] |
| 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 | O1O2O3O4U3U4O5U6U2O6U1U5 |
| R3 orbit | {'O1O2O3O4U3U4O5U6U2O6U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4O6U3U6O5U1U2 |
| Gauss code of K* | O1O2U3O4O5U4O6O3U6U5U1U2 |
| Gauss code of -K* | O1O2U3O4O3U1O5O6U5U6U4U2 |
| 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 -1 1 2 -1],[ 1 0 1 -1 1 2 0],[ 0 -1 0 -1 1 0 0],[ 1 1 1 0 1 0 1],[-1 -1 -1 -1 0 0 -1],[-2 -2 0 0 0 0 -2],[ 1 0 0 -1 1 2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 0 0 -2 -2],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 -1 0 -1],[ 1 0 1 1 0 1 1],[ 1 2 1 0 -1 0 0],[ 1 2 1 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,0,0,2,2,1,1,1,1,1,0,1,-1,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,0,2,2,1,1,1,1,1,0,1,-1,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,-1,0,1,3,0,0,1,1,1,1,1,0,2,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,2,1,1,3,0,1,1,1,0,1,0,0,-1,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,1,1,1,0,1,1,2,1,0,0] |
| 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+16t^4+31t^2+4 |
| Outer characteristic polynomial | t^7+24t^5+52t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -1024*K1**6 - 2176*K1**4*K2**2 + 3712*K1**4*K2 - 4384*K1**4 + 1856*K1**3*K2*K3 - 576*K1**3*K3 - 1216*K1**2*K2**4 + 3904*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10240*K1**2*K2**2 - 928*K1**2*K2*K4 + 8288*K1**2*K2 - 704*K1**2*K3**2 - 1660*K1**2 + 2144*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 - 448*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7016*K1*K2*K3 + 888*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 2632*K2**4 - 32*K2**3*K6 - 1200*K2**2*K3**2 - 112*K2**2*K4**2 + 2096*K2**2*K4 - 1734*K2**2 - 96*K2*K3**2*K4 + 688*K2*K3*K5 + 88*K2*K4*K6 + 24*K3**2*K6 - 1260*K3**2 - 382*K4**2 - 96*K5**2 - 18*K6**2 + 2660 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |