| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,1,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1648', '7.37424'] |
| 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+34t^5+48t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1648', '7.37424'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 928*K1**4*K2 - 2048*K1**4 + 480*K1**3*K2*K3 - 192*K1**3*K3 - 384*K1**2*K2**4 + 1568*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4704*K1**2*K2**2 - 288*K1**2*K2*K4 + 4040*K1**2*K2 - 224*K1**2*K3**2 - 860*K1**2 + 640*K1*K2**3*K3 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2808*K1*K2*K3 + 184*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1224*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 848*K2**2*K4 - 670*K2**2 + 112*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 :['6.1648'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.335', 'vk6.376', 'vk6.441', 'vk6.730', 'vk6.783', 'vk6.890', 'vk6.1475', 'vk6.1532', 'vk6.1597', 'vk6.1634', 'vk6.1741', 'vk6.1805', 'vk6.1969', 'vk6.2010', 'vk6.2074', 'vk6.2127', 'vk6.2228', 'vk6.2261', 'vk6.3010', 'vk6.3131', 'vk6.3796', 'vk6.3989', 'vk6.7180', 'vk6.7357', 'vk6.18791', 'vk6.19852', 'vk6.24917', 'vk6.25380', 'vk6.25918', 'vk6.26295', 'vk6.26738', 'vk6.37997', 'vk6.38052', 'vk6.39195', 'vk6.39712', 'vk6.41958', 'vk6.45036', 'vk6.46277', 'vk6.57647', 'vk6.58529'] |
| 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 | O1O2O3U1O4U3U5U6O5O6U4U2 |
| R3 orbit | {'O1O2O3U1U2O4U5U6O5O6U3U4', 'O1O2O3U1O4U3U5U6O5O6U4U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U2U4O5O6U5U6U1O4U3 |
| Gauss code of K* | O1O2O3U2U3O4O5U6U5U1O6U4 |
| Gauss code of -K* | O1O2O3U4O5U3U6U5O6O4U1U2 |
| 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 -2 1 0 1 -1 1],[ 2 0 2 1 1 2 2],[-1 -2 0 -1 1 -2 0],[ 0 -1 1 0 1 0 0],[-1 -1 -1 -1 0 -2 0],[ 1 -2 2 0 2 0 1],[-1 -2 0 0 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -2 -2],[-1 -1 0 0 -1 -2 -1],[-1 0 0 0 0 -1 -2],[ 0 1 1 0 0 0 -1],[ 1 2 2 1 0 0 -2],[ 2 2 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,2,2,0,1,2,1,0,1,2,0,1,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,1,0,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,1,0,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,0,2,0,0,0,1,1,1,1,1,1,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,1,1,2,2,0,2,1,2,1,0,1,0,-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+26t^4+29t^2 |
| Outer characteristic polynomial | t^7+34t^5+48t^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 - 2048*K1**4 + 480*K1**3*K2*K3 - 192*K1**3*K3 - 384*K1**2*K2**4 + 1568*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4704*K1**2*K2**2 - 288*K1**2*K2*K4 + 4040*K1**2*K2 - 224*K1**2*K3**2 - 860*K1**2 + 640*K1*K2**3*K3 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2808*K1*K2*K3 + 184*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1224*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 848*K2**2*K4 - 670*K2**2 + 112*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 | [[{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]] |
| If K is slice | False |