| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,0,1,1,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1041'] |
| 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+30t^5+54t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1041'] |
| 2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 2272*K1**4*K2 - 5280*K1**4 + 768*K1**3*K2*K3 - 1248*K1**3*K3 - 256*K1**2*K2**4 + 1184*K1**2*K2**3 - 7888*K1**2*K2**2 - 800*K1**2*K2*K4 + 11536*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 5376*K1**2 + 480*K1*K2**3*K3 - 960*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7536*K1*K2*K3 + 864*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 856*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 1072*K2**2*K4 - 4486*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 1860*K3**2 - 466*K4**2 - 52*K5**2 - 2*K6**2 + 4736 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1041'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4851', 'vk6.4861', 'vk6.5195', 'vk6.5204', 'vk6.6426', 'vk6.6438', 'vk6.6854', 'vk6.8391', 'vk6.8395', 'vk6.8813', 'vk6.8816', 'vk6.9753', 'vk6.9756', 'vk6.10049', 'vk6.20781', 'vk6.20790', 'vk6.22184', 'vk6.29750', 'vk6.39815', 'vk6.39825', 'vk6.46379', 'vk6.46385', 'vk6.47956', 'vk6.47962', 'vk6.49086', 'vk6.49089', 'vk6.49922', 'vk6.51336', 'vk6.51345', 'vk6.51553', 'vk6.58807', 'vk6.63271'] |
| 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 | O1O2O3O4U5U3O5U2O6U4U1U6 |
| R3 orbit | {'O1O2O3O4U5U3O5U2O6U4U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4U1O5U3O6U2U6 |
| Gauss code of K* | O1O2O3U2U4U5U1O6O5U6O4U3 |
| Gauss code of -K* | O1O2O3U1O4U5O6O5U3U6U4U2 |
| 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 0 1 -1 2],[ 1 0 -1 0 2 0 2],[ 1 1 0 1 2 0 1],[ 0 0 -1 0 0 0 1],[-1 -2 -2 0 0 -1 1],[ 1 0 0 0 1 0 2],[-2 -2 -1 -1 -1 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -2 -1 -2],[ 0 1 0 0 -1 0 0],[ 1 1 2 1 0 0 1],[ 1 2 1 0 0 0 0],[ 1 2 2 0 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,1,1,2,2,0,2,1,2,1,0,0,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,0,1,1,0,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,0,2,0,1,0,1,1,1,1,1,1,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,0,1,1,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,2,2,0,1,2,1,0,1,2,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+22t^4+35t^2+9 |
| Outer characteristic polynomial | t^7+30t^5+54t^3+16t |
| 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 + 2272*K1**4*K2 - 5280*K1**4 + 768*K1**3*K2*K3 - 1248*K1**3*K3 - 256*K1**2*K2**4 + 1184*K1**2*K2**3 - 7888*K1**2*K2**2 - 800*K1**2*K2*K4 + 11536*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 5376*K1**2 + 480*K1*K2**3*K3 - 960*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7536*K1*K2*K3 + 864*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 856*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 1072*K2**2*K4 - 4486*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 1860*K3**2 - 466*K4**2 - 52*K5**2 - 2*K6**2 + 4736 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |