| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,2,4,0,0,0,1,0,0,0,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.579'] |
| 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+60t^5+51t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.579'] |
| 2-strand cable arrow polynomial of the knot is: 640*K1**4*K2 - 2992*K1**4 + 320*K1**3*K2*K3 - 448*K1**3*K3 - 4480*K1**2*K2**2 - 512*K1**2*K2*K4 + 9256*K1**2*K2 - 272*K1**2*K3**2 - 32*K1**2*K4**2 - 6244*K1**2 - 672*K1*K2**2*K3 - 32*K1*K2**2*K5 + 6720*K1*K2*K3 + 936*K1*K3*K4 + 56*K1*K4*K5 - 248*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 832*K2**2*K4 - 5070*K2**2 + 120*K2*K3*K5 + 8*K2*K4*K6 - 2232*K3**2 - 502*K4**2 - 52*K5**2 - 2*K6**2 + 4988 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.579'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11025', 'vk6.11103', 'vk6.12191', 'vk6.12298', 'vk6.18198', 'vk6.18533', 'vk6.24654', 'vk6.25080', 'vk6.30594', 'vk6.30689', 'vk6.31860', 'vk6.31906', 'vk6.36786', 'vk6.37236', 'vk6.44027', 'vk6.44367', 'vk6.51824', 'vk6.51891', 'vk6.52692', 'vk6.52786', 'vk6.55993', 'vk6.56265', 'vk6.60525', 'vk6.60867', 'vk6.63504', 'vk6.63548', 'vk6.63982', 'vk6.64026', 'vk6.65652', 'vk6.65931', 'vk6.68697', 'vk6.68906'] |
| 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 | O1O2O3O4U1O5U3O6U4U5U6U2 |
| R3 orbit | {'O1O2O3O4U1O5U3O6U4U5U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U5U6U1O5U2O6U4 |
| Gauss code of K* | O1O2O3O4U5U4U6U1O5U2O6U3 |
| Gauss code of -K* | O1O2O3O4U2O5U3O6U4U5U1U6 |
| 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 -3 1 -1 0 1 2],[ 3 0 3 1 2 2 1],[-1 -3 0 -2 -1 1 2],[ 1 -1 2 0 1 2 2],[ 0 -2 1 -1 0 1 2],[-1 -2 -1 -2 -1 0 1],[-2 -1 -2 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -1 -2 -2 -2 -1],[-1 1 0 -1 -1 -2 -2],[-1 2 1 0 -1 -2 -3],[ 0 2 1 1 0 -1 -2],[ 1 2 2 2 1 0 -1],[ 3 1 2 3 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,1,2,2,2,1,1,1,2,2,1,2,3,1,2,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,1,2,4,0,0,0,1,0,0,0,-1,-1,0] |
| Phi of -K | [-3,-1,0,1,1,2,1,1,1,2,4,0,0,0,1,0,0,0,-1,-1,0] |
| Phi of K* | [-2,-1,-1,0,1,3,-1,0,0,1,4,1,0,0,1,0,0,2,0,1,1] |
| Phi of -K* | [-3,-1,0,1,1,2,1,2,2,3,1,1,2,2,2,1,1,2,-1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+44t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+60t^5+51t^3+5t |
| Flat arrow polynomial | -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | 640*K1**4*K2 - 2992*K1**4 + 320*K1**3*K2*K3 - 448*K1**3*K3 - 4480*K1**2*K2**2 - 512*K1**2*K2*K4 + 9256*K1**2*K2 - 272*K1**2*K3**2 - 32*K1**2*K4**2 - 6244*K1**2 - 672*K1*K2**2*K3 - 32*K1*K2**2*K5 + 6720*K1*K2*K3 + 936*K1*K3*K4 + 56*K1*K4*K5 - 248*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 832*K2**2*K4 - 5070*K2**2 + 120*K2*K3*K5 + 8*K2*K4*K6 - 2232*K3**2 - 502*K4**2 - 52*K5**2 - 2*K6**2 + 4988 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}]] |
| If K is slice | False |