| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,1,1,1,1,2,-1,-1,0,2,0,1,1,2,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1878'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+27t^5+125t^3+27t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1878'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 - 1024*K1**2*K2**4 + 1920*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5696*K1**2*K2**2 - 64*K1**2*K2*K4 + 3848*K1**2*K2 - 16*K1**2*K3**2 - 2532*K1**2 + 1408*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4920*K1*K2*K3 + 168*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1432*K2**4 - 624*K2**2*K3**2 - 48*K2**2*K4**2 + 864*K2**2*K4 - 1246*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1256*K3**2 - 126*K4**2 - 28*K5**2 - 2*K6**2 + 1940 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1878'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70497', 'vk6.70508', 'vk6.70556', 'vk6.70571', 'vk6.70703', 'vk6.70724', 'vk6.70808', 'vk6.70823', 'vk6.70974', 'vk6.70995', 'vk6.71052', 'vk6.71079', 'vk6.71193', 'vk6.71214', 'vk6.71272', 'vk6.71285', 'vk6.71759', 'vk6.72180', 'vk6.74064', 'vk6.74149', 'vk6.74625', 'vk6.74717', 'vk6.76209', 'vk6.76226', 'vk6.77554', 'vk6.79068', 'vk6.79168', 'vk6.80648', 'vk6.81260', 'vk6.87021', 'vk6.87940', 'vk6.89132'] |
| 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 | O1O2O3U4U5O6U1O5O4U6U2U3 |
| R3 orbit | {'O1O2O3U4U5O6U1O5O4U6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U2U4O5O6U3O4U6U5 |
| Gauss code of K* | O1O2O3U4U2U3O5O6U1O4U6U5 |
| Gauss code of -K* | O1O2O3U4U5O6U3O5O4U1U2U6 |
| 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 1 -1],[ 0 0 0 1 1 1 -2],[-2 -1 -1 0 -1 -1 -2],[ 0 -2 -1 1 0 0 0],[ 0 -1 -1 1 0 0 -1],[ 1 1 2 2 0 1 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 -1 -1 -1 -1 -2],[ 0 1 0 1 1 0 -2],[ 0 1 -1 0 0 -1 -1],[ 0 1 -1 0 0 -2 0],[ 1 1 0 1 2 0 -1],[ 1 2 2 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,1,1,1,1,2,-1,-1,0,2,0,1,1,2,0,1] |
| Phi over symmetry | [-2,0,0,0,1,1,1,1,1,1,2,-1,-1,0,2,0,1,1,2,0,1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,-1,0,1,1,1,0,-1,2,-1,-1,1,0,1,1] |
| Phi of K* | [-2,0,0,0,1,1,1,1,1,1,2,-1,0,0,0,1,-1,1,1,-1,1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,0,1,2,1,2,1,0,2,1,1,1,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | z^2+6z+9 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+5w^3z^2-16w^3z+22w^2z+9w |
| Inner characteristic polynomial | t^6+21t^4+74t^2+9 |
| Outer characteristic polynomial | t^7+27t^5+125t^3+27t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 - 1024*K1**2*K2**4 + 1920*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5696*K1**2*K2**2 - 64*K1**2*K2*K4 + 3848*K1**2*K2 - 16*K1**2*K3**2 - 2532*K1**2 + 1408*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 64*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4920*K1*K2*K3 + 168*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1432*K2**4 - 624*K2**2*K3**2 - 48*K2**2*K4**2 + 864*K2**2*K4 - 1246*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1256*K3**2 - 126*K4**2 - 28*K5**2 - 2*K6**2 + 1940 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |