| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,1,1,-1,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1025', '7.35745'] |
| 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+22t^5+34t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1025', '7.35745'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 928*K1**4*K2 - 2432*K1**4 + 480*K1**3*K2*K3 - 160*K1**3*K3 - 384*K1**2*K2**4 + 1568*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 5344*K1**2*K2**2 - 288*K1**2*K2*K4 + 4520*K1**2*K2 - 256*K1**2*K3**2 - 668*K1**2 + 640*K1*K2**3*K3 - 928*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3192*K1*K2*K3 + 216*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1448*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 976*K2**2*K4 - 574*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 384*K3**2 - 94*K4**2 - 12*K5**2 - 2*K6**2 + 1140 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1025'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.332', 'vk6.371', 'vk6.452', 'vk6.729', 'vk6.780', 'vk6.901', 'vk6.1462', 'vk6.1521', 'vk6.1600', 'vk6.1635', 'vk6.1744', 'vk6.1814', 'vk6.1962', 'vk6.2001', 'vk6.2081', 'vk6.2132', 'vk6.2231', 'vk6.2266', 'vk6.3011', 'vk6.3134', 'vk6.3797', 'vk6.3988', 'vk6.7181', 'vk6.7356', 'vk6.18786', 'vk6.19863', 'vk6.24916', 'vk6.25377', 'vk6.25920', 'vk6.26304', 'vk6.26749', 'vk6.37995', 'vk6.38050', 'vk6.39196', 'vk6.39721', 'vk6.41967', 'vk6.45041', 'vk6.46284', 'vk6.57642', 'vk6.58524'] |
| 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 | O1O2O3O4U3U4O5U2O6U5U1U6 |
| R3 orbit | {'O1O2O3O4U3U4U1O5O6U2U5U6', 'O1O2O3O4U3U4O5U2O6U5U1U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U4U6O5U3O6U1U2 |
| Gauss code of K* | O1O2O3U2U4U5U6O5O6U1O4U3 |
| Gauss code of -K* | O1O2O3U1O4U3O5O6U5U6U4U2 |
| 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 -1 -1 1 0 2],[ 1 0 -1 -1 1 1 2],[ 1 1 0 -1 1 1 1],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 0 -1 -1 0 0 0 1],[-2 -2 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 -1 -1],[ 1 0 1 0 0 1 1],[ 1 1 1 1 -1 0 1],[ 1 2 1 1 -1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,1,1,-1,-1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,1,1,-1,-1,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,-1,1,1,3,-1,0,1,2,0,1,1,1,1,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,1,1,2,3,1,1,1,1,0,0,1,-1,-1,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,-1,1,1,2,-1,1,1,1,0,1,0,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+14t^4+15t^2 |
| Outer characteristic polynomial | t^7+22t^5+34t^3+3t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -384*K1**4*K2**2 + 928*K1**4*K2 - 2432*K1**4 + 480*K1**3*K2*K3 - 160*K1**3*K3 - 384*K1**2*K2**4 + 1568*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 5344*K1**2*K2**2 - 288*K1**2*K2*K4 + 4520*K1**2*K2 - 256*K1**2*K3**2 - 668*K1**2 + 640*K1*K2**3*K3 - 928*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3192*K1*K2*K3 + 216*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1448*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 976*K2**2*K4 - 574*K2**2 + 144*K2*K3*K5 + 16*K2*K4*K6 - 384*K3**2 - 94*K4**2 - 12*K5**2 - 2*K6**2 + 1140 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |