| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,2,1,1,1,1,1,1,1,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.137', '7.10422'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+44t^5+43t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.137'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**6 - 448*K1**4*K2**2 + 640*K1**4*K2 - 880*K1**4 + 256*K1**3*K2*K3 - 1152*K1**2*K2**2 + 1280*K1**2*K2 - 464*K1**2*K3**2 - 224*K1**2*K4**2 + 144*K1**2 + 1152*K1*K2*K3 + 416*K1*K3*K4 + 128*K1*K4*K5 - 288*K2**4 - 352*K2**2*K3**2 - 176*K2**2*K4**2 + 272*K2**2*K4 - 156*K2**2 + 192*K2*K3*K5 + 96*K2*K4*K6 - 160*K3**2 - 72*K4**2 - 16*K5**2 - 4*K6**2 + 326 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.137'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.321', 'vk6.359', 'vk6.425', 'vk6.713', 'vk6.758', 'vk6.831', 'vk6.870', 'vk6.1134', 'vk6.1503', 'vk6.1575', 'vk6.1674', 'vk6.1948', 'vk6.1986', 'vk6.2048', 'vk6.2173', 'vk6.2280', 'vk6.2645', 'vk6.2722', 'vk6.2786', 'vk6.3109', 'vk6.5253', 'vk6.6508', 'vk6.8882', 'vk6.9797', 'vk6.18320', 'vk6.18658', 'vk6.19411', 'vk6.19704', 'vk6.25208', 'vk6.25871', 'vk6.26191', 'vk6.28504', 'vk6.36933', 'vk6.37398', 'vk6.37978', 'vk6.39874', 'vk6.40284', 'vk6.44860', 'vk6.46936', 'vk6.49125'] |
| 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 | O1O2O3O4O5O6U4U6U3U5U1U2 |
| R3 orbit | {'O1O2O3O4O5O6U4U6U3U5U1U2', 'O1O2O3O4O5U3O6U4U6U5U1U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U5U6U2U4U1U3 |
| Gauss code of K* | O1O2O3O4O5O6U5U6U3U1U4U2 |
| Gauss code of -K* | O1O2O3O4O5O6U5U3U6U4U1U2 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 1 -1 -2 2 1],[ 1 0 1 -1 -2 2 1],[-1 -1 0 -1 -2 2 1],[ 1 1 1 0 -1 2 1],[ 2 2 2 1 0 2 1],[-2 -2 -2 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -2 -2 -2],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -1 -2],[ 1 2 1 1 0 1 -1],[ 1 2 1 1 -1 0 -2],[ 2 2 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,2,2,2,2,1,1,1,1,1,1,2,-1,1,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,2,2,1,1,1,1,1,1,1,-1,-1,1] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,0,1,2,2,1,1,1,1,1,1,1,-1,-1,1] |
| Phi of K* | [-2,-1,-1,1,1,2,-1,1,1,1,2,1,1,1,1,1,1,2,-1,-1,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,2,1,2,2,1,1,1,2,1,1,2,-1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | 7w^2z+15w |
| Inner characteristic polynomial | t^6+32t^4+11t^2 |
| Outer characteristic polynomial | t^7+44t^5+43t^3 |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -512*K1**6 - 448*K1**4*K2**2 + 640*K1**4*K2 - 880*K1**4 + 256*K1**3*K2*K3 - 1152*K1**2*K2**2 + 1280*K1**2*K2 - 464*K1**2*K3**2 - 224*K1**2*K4**2 + 144*K1**2 + 1152*K1*K2*K3 + 416*K1*K3*K4 + 128*K1*K4*K5 - 288*K2**4 - 352*K2**2*K3**2 - 176*K2**2*K4**2 + 272*K2**2*K4 - 156*K2**2 + 192*K2*K3*K5 + 96*K2*K4*K6 - 160*K3**2 - 72*K4**2 - 16*K5**2 - 4*K6**2 + 326 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | True |