| Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,0,1,3,3,0,0,1,2,1,1,2,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.540'] |
| Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922'] |
| Outer characteristic polynomial of the knot is: t^7+63t^5+51t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.540'] |
| 2-strand cable arrow polynomial of the knot is: -1152*K1**4 + 544*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2144*K1**3*K3 - 896*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 5992*K1**2*K2 - 2016*K1**2*K3**2 - 384*K1**2*K3*K5 - 288*K1**2*K4**2 - 96*K1**2*K4*K6 - 5960*K1**2 + 64*K1*K2**3*K3 - 576*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 6624*K1*K2*K3 + 2856*K1*K3*K4 + 600*K1*K4*K5 + 72*K1*K5*K6 - 64*K2**4 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 576*K2**2*K4 - 3684*K2**2 + 288*K2*K3*K5 + 72*K2*K4*K6 - 2640*K3**2 - 1000*K4**2 - 216*K5**2 - 44*K6**2 + 4158 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.540'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4654', 'vk6.4941', 'vk6.6100', 'vk6.6589', 'vk6.8111', 'vk6.8511', 'vk6.9499', 'vk6.9858', 'vk6.20628', 'vk6.22055', 'vk6.28110', 'vk6.29551', 'vk6.39530', 'vk6.41753', 'vk6.46137', 'vk6.47779', 'vk6.48686', 'vk6.48885', 'vk6.49434', 'vk6.49659', 'vk6.50696', 'vk6.50889', 'vk6.51177', 'vk6.51386', 'vk6.57512', 'vk6.58700', 'vk6.62204', 'vk6.63150', 'vk6.67022', 'vk6.67895', 'vk6.69647', 'vk6.70328'] |
| 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 | O1O2O3O4O5U3U2U4O6U5U1U6 |
| R3 orbit | {'O1O2O3O4O5U3U2U4O6U5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U5U1O6U2U4U3 |
| Gauss code of K* | O1O2O3U4O5O6O4U6U2U1U3U5 |
| Gauss code of -K* | O1O2O3U1O4O5O6U3U4U6U5U2 |
| 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 -2 -2 1 2 2],[ 1 0 -2 -2 1 3 2],[ 2 2 0 0 2 3 1],[ 2 2 0 0 1 2 1],[-1 -1 -2 -1 0 1 1],[-2 -3 -3 -2 -1 0 1],[-2 -2 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -2 -2],[-2 0 1 -1 -3 -2 -3],[-2 -1 0 -1 -2 -1 -1],[-1 1 1 0 -1 -1 -2],[ 1 3 2 1 0 -2 -2],[ 2 2 1 1 2 0 0],[ 2 3 1 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,2,2,-1,1,3,2,3,1,2,1,1,1,1,2,2,2,0] |
| Phi over symmetry | [-2,-2,-1,1,2,2,-1,0,1,3,3,0,0,1,2,1,1,2,-1,-1,0] |
| Phi of -K | [-2,-2,-1,1,2,2,0,-1,1,1,3,-1,2,2,3,1,0,1,0,0,-1] |
| Phi of K* | [-2,-2,-1,1,2,2,-1,0,1,3,3,0,0,1,2,1,1,2,-1,-1,0] |
| Phi of -K* | [-2,-2,-1,1,2,2,0,2,1,1,2,2,2,1,3,1,2,3,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+25z+31 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+25w^2z+31w |
| Inner characteristic polynomial | t^6+45t^4+17t^2 |
| Outer characteristic polynomial | t^7+63t^5+51t^3+6t |
| Flat arrow polynomial | -4*K1*K2 + 2*K1 + 2*K3 + 1 |
| 2-strand cable arrow polynomial | -1152*K1**4 + 544*K1**3*K2*K3 + 64*K1**3*K3*K4 - 2144*K1**3*K3 - 896*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 5992*K1**2*K2 - 2016*K1**2*K3**2 - 384*K1**2*K3*K5 - 288*K1**2*K4**2 - 96*K1**2*K4*K6 - 5960*K1**2 + 64*K1*K2**3*K3 - 576*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 6624*K1*K2*K3 + 2856*K1*K3*K4 + 600*K1*K4*K5 + 72*K1*K5*K6 - 64*K2**4 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 576*K2**2*K4 - 3684*K2**2 + 288*K2*K3*K5 + 72*K2*K4*K6 - 2640*K3**2 - 1000*K4**2 - 216*K5**2 - 44*K6**2 + 4158 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | ? |