| Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,0,1,2,4,4,1,2,3,2,0,0,0,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.280'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511'] |
| Outer characteristic polynomial of the knot is: t^7+90t^5+128t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.280'] |
| 2-strand cable arrow polynomial of the knot is: -352*K1**3*K3 + 128*K1**2*K2*K3**2 + 1040*K1**2*K2 - 2176*K1**2*K3**2 - 288*K1**2*K3*K5 - 32*K1**2*K4*K6 - 48*K1**2*K6**2 - 2744*K1**2 - 96*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 3744*K1*K2*K3 - 32*K1*K3*K4*K6 + 2648*K1*K3*K4 + 424*K1*K4*K5 + 96*K1*K5*K6 + 88*K1*K6*K7 + 24*K1*K7*K8 - 320*K2**2*K3**2 - 8*K2**2*K4**2 + 176*K2**2*K4 - 8*K2**2*K6**2 - 1754*K2**2 + 752*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 + 16*K3**2*K6 - 2080*K3**2 + 8*K3*K4*K7 - 870*K4**2 - 380*K5**2 - 102*K6**2 - 44*K7**2 - 18*K8**2 + 2486 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.280'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16988', 'vk6.17231', 'vk6.20216', 'vk6.21508', 'vk6.23390', 'vk6.23700', 'vk6.27410', 'vk6.29025', 'vk6.35449', 'vk6.35891', 'vk6.38822', 'vk6.41009', 'vk6.42883', 'vk6.43186', 'vk6.45581', 'vk6.47351', 'vk6.55145', 'vk6.55395', 'vk6.57055', 'vk6.58178', 'vk6.59517', 'vk6.59869', 'vk6.61570', 'vk6.62741', 'vk6.64961', 'vk6.65168', 'vk6.66671', 'vk6.67508', 'vk6.68251', 'vk6.68409', 'vk6.69322', 'vk6.70077'] |
| 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 | O1O2O3O4O5U1U5O6U2U4U3U6 |
| R3 orbit | {'O1O2O3O4O5U1U5O6U2U4U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U3U2U4O6U1U5 |
| Gauss code of K* | O1O2O3O4U5U1U3U2U6O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U5U3U2U4U6 |
| 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 -4 -2 1 1 1 3],[ 4 0 2 4 3 1 3],[ 2 -2 0 2 1 0 3],[-1 -4 -2 0 0 0 2],[-1 -3 -1 0 0 0 1],[-1 -1 0 0 0 0 0],[-3 -3 -3 -2 -1 0 0]] |
| Primitive based matrix | [[ 0 3 1 1 1 -2 -4],[-3 0 0 -1 -2 -3 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -3],[-1 2 0 0 0 -2 -4],[ 2 3 0 1 2 0 -2],[ 4 3 1 3 4 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,-1,2,4,0,1,2,3,3,0,0,0,1,0,1,3,2,4,2] |
| Phi over symmetry | [-4,-2,1,1,1,3,0,1,2,4,4,1,2,3,2,0,0,0,0,1,2] |
| Phi of -K | [-4,-2,1,1,1,3,0,1,2,4,4,1,2,3,2,0,0,0,0,1,2] |
| Phi of K* | [-3,-1,-1,-1,2,4,0,1,2,2,4,0,0,1,1,0,2,2,3,4,0] |
| Phi of -K* | [-4,-2,1,1,1,3,2,1,3,4,3,0,1,2,3,0,0,0,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3+t^2-3t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w |
| Inner characteristic polynomial | t^6+58t^4+47t^2 |
| Outer characteristic polynomial | t^7+90t^5+128t^3+7t |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -352*K1**3*K3 + 128*K1**2*K2*K3**2 + 1040*K1**2*K2 - 2176*K1**2*K3**2 - 288*K1**2*K3*K5 - 32*K1**2*K4*K6 - 48*K1**2*K6**2 - 2744*K1**2 - 96*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 3744*K1*K2*K3 - 32*K1*K3*K4*K6 + 2648*K1*K3*K4 + 424*K1*K4*K5 + 96*K1*K5*K6 + 88*K1*K6*K7 + 24*K1*K7*K8 - 320*K2**2*K3**2 - 8*K2**2*K4**2 + 176*K2**2*K4 - 8*K2**2*K6**2 - 1754*K2**2 + 752*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 + 16*K3**2*K6 - 2080*K3**2 + 8*K3*K4*K7 - 870*K4**2 - 380*K5**2 - 102*K6**2 - 44*K7**2 - 18*K8**2 + 2486 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | False |