| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,0,2,0,1,1,0,0,1,0,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1582'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+17t^5+32t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1582'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 2384*K1**4 + 384*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 + 768*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 5408*K1**2*K2**2 - 448*K1**2*K2*K4 + 8744*K1**2*K2 - 336*K1**2*K3**2 - 5404*K1**2 + 736*K1*K2**3*K3 - 672*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6240*K1*K2*K3 + 624*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 936*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 944*K2**2*K4 - 3822*K2**2 + 232*K2*K3*K5 + 16*K2*K4*K6 - 1740*K3**2 - 342*K4**2 - 40*K5**2 - 2*K6**2 + 4156 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1582'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16510', 'vk6.16601', 'vk6.18088', 'vk6.18426', 'vk6.22937', 'vk6.23032', 'vk6.24539', 'vk6.24958', 'vk6.34916', 'vk6.35025', 'vk6.36670', 'vk6.37094', 'vk6.42483', 'vk6.42594', 'vk6.43950', 'vk6.44267', 'vk6.54753', 'vk6.54848', 'vk6.55904', 'vk6.56190', 'vk6.59213', 'vk6.59276', 'vk6.60433', 'vk6.60789', 'vk6.64767', 'vk6.64830', 'vk6.65538', 'vk6.65850', 'vk6.68065', 'vk6.68128', 'vk6.68620', 'vk6.68835'] |
| 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 | O1O2O3U4O5U3U1O6O4U6U2U5 |
| R3 orbit | {'O1O2O3U4O5U3U1O6O4U6U2U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U5O6O5U3U1O4U6 |
| Gauss code of K* | O1O2O3U4U2U5O6U3O5O4U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5O6U1O4U6U2U5 |
| 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 0 0 2 -1],[ 1 0 0 0 0 2 -1],[ 0 0 0 1 0 1 -1],[ 0 0 -1 0 0 0 -1],[ 0 0 0 0 0 1 -1],[-2 -2 -1 0 -1 0 0],[ 1 1 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -1 0 -2],[ 0 0 0 0 -1 -1 0],[ 0 1 0 0 0 -1 0],[ 0 1 1 0 0 -1 0],[ 1 0 1 1 1 0 1],[ 1 2 0 0 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,1,1,0,2,0,1,1,0,0,1,0,1,0,-1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,0,2,0,1,1,0,0,1,0,1,0,-1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,0,0,3,1,1,1,1,-1,0,1,0,2,1] |
| Phi of K* | [-2,0,0,0,1,1,1,1,2,1,3,0,0,1,0,1,1,0,1,0,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,0,0,0,2,1,1,1,0,-1,0,0,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+22z+33 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+22w^2z+33w |
| Inner characteristic polynomial | t^6+11t^4+15t^2+1 |
| Outer characteristic polynomial | t^7+17t^5+32t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 320*K1**4*K2 - 2384*K1**4 + 384*K1**3*K2*K3 - 512*K1**3*K3 - 320*K1**2*K2**4 + 768*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 5408*K1**2*K2**2 - 448*K1**2*K2*K4 + 8744*K1**2*K2 - 336*K1**2*K3**2 - 5404*K1**2 + 736*K1*K2**3*K3 - 672*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6240*K1*K2*K3 + 624*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 936*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 944*K2**2*K4 - 3822*K2**2 + 232*K2*K3*K5 + 16*K2*K4*K6 - 1740*K3**2 - 342*K4**2 - 40*K5**2 - 2*K6**2 + 4156 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |