| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,2,3,0,1,1,1,1,1,0,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1282'] |
| Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354'] |
| Outer characteristic polynomial of the knot is: t^7+52t^5+96t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1282'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 880*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 + 512*K1**2*K2**3 - 2688*K1**2*K2**2 - 32*K1**2*K2*K4 + 3136*K1**2*K2 - 176*K1**2*K3**2 - 1668*K1**2 + 224*K1*K2**3*K3 - 672*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 2560*K1*K2*K3 + 360*K1*K3*K4 + 8*K1*K4*K5 - 600*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 624*K2**2*K4 - 1262*K2**2 + 160*K2*K3*K5 + 8*K2*K4*K6 - 680*K3**2 - 202*K4**2 - 20*K5**2 - 2*K6**2 + 1440 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1282'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17082', 'vk6.17324', 'vk6.20054', 'vk6.20254', 'vk6.21183', 'vk6.21559', 'vk6.23463', 'vk6.26900', 'vk6.27115', 'vk6.27487', 'vk6.28653', 'vk6.29082', 'vk6.35600', 'vk6.38324', 'vk6.38512', 'vk6.38906', 'vk6.40463', 'vk6.41105', 'vk6.42973', 'vk6.45200', 'vk6.45408', 'vk6.45659', 'vk6.47023', 'vk6.47390', 'vk6.55221', 'vk6.56735', 'vk6.56868', 'vk6.57835', 'vk6.59618', 'vk6.61163', 'vk6.61393', 'vk6.61623', 'vk6.62403', 'vk6.62801', 'vk6.65025', 'vk6.66575', 'vk6.68293', 'vk6.69086', 'vk6.69223', 'vk6.69867'] |
| 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 | O1O2O3U1O4O5U6U2O6U3U4U5 |
| R3 orbit | {'O1O2O3U1O4O5U3U6U2O6U4U5', 'O1O2O3U1O4O5U6U2O6U3U4U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U5U1O6U2U6O4O5U3 |
| Gauss code of K* | O1O2O3U4U5U1O4U2U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1U2O6U3U5U6 |
| 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 3 -1],[ 2 0 1 2 2 2 1],[ 1 -1 0 0 1 2 1],[ 0 -2 0 0 1 2 0],[-1 -2 -1 -1 0 1 -1],[-3 -2 -2 -2 -1 0 -3],[ 1 -1 -1 0 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -2 -3 -2],[-1 1 0 -1 -1 -1 -2],[ 0 2 1 0 0 0 -2],[ 1 2 1 0 0 1 -1],[ 1 3 1 0 -1 0 -1],[ 2 2 2 2 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,2,2,3,2,1,1,1,2,0,0,2,-1,1,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,1,2,3,0,1,1,1,1,1,0,-1,0,0] |
| Phi of -K | [-2,-1,-1,0,1,3,0,0,0,1,3,-1,1,1,2,1,1,1,0,1,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,1,1,2,3,0,1,1,1,1,1,0,-1,0,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,1,2,2,2,-1,0,1,3,0,1,2,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+36t^4+55t^2+1 |
| Outer characteristic polynomial | t^7+52t^5+96t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 352*K1**4*K2 - 880*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 + 512*K1**2*K2**3 - 2688*K1**2*K2**2 - 32*K1**2*K2*K4 + 3136*K1**2*K2 - 176*K1**2*K3**2 - 1668*K1**2 + 224*K1*K2**3*K3 - 672*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 2560*K1*K2*K3 + 360*K1*K3*K4 + 8*K1*K4*K5 - 600*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 624*K2**2*K4 - 1262*K2**2 + 160*K2*K3*K5 + 8*K2*K4*K6 - 680*K3**2 - 202*K4**2 - 20*K5**2 - 2*K6**2 + 1440 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |