| Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,0,1,1,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1752', '7.28624', '7.33032', '7.45964'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923'] |
| Outer characteristic polynomial of the knot is: t^7+31t^5+63t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1752', '7.28624', '7.33032'] |
| 2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 2048*K1**4*K2**2 + 3840*K1**4*K2 - 4640*K1**4 + 1728*K1**3*K2*K3 - 736*K1**3*K3 - 1344*K1**2*K2**4 + 3456*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 8640*K1**2*K2**2 - 1024*K1**2*K2*K4 + 6512*K1**2*K2 - 448*K1**2*K3**2 - 96*K1**2*K4**2 + 72*K1**2 + 1472*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 416*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4576*K1*K2*K3 + 368*K1*K3*K4 + 80*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 1952*K2**4 - 64*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1264*K2**2*K4 - 508*K2**2 + 176*K2*K3*K5 + 48*K2*K4*K6 - 404*K3**2 - 120*K4**2 - 20*K5**2 - 4*K6**2 + 1334 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1752'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72', 'vk6.74', 'vk6.129', 'vk6.133', 'vk6.226', 'vk6.230', 'vk6.273', 'vk6.275', 'vk6.389', 'vk6.393', 'vk6.809', 'vk6.817', 'vk6.1277', 'vk6.1281', 'vk6.1368', 'vk6.1372', 'vk6.1415', 'vk6.1417', 'vk6.1554', 'vk6.1562', 'vk6.2686', 'vk6.2694', 'vk6.2948', 'vk6.2952', 'vk6.14837', 'vk6.14841', 'vk6.15995', 'vk6.15999', 'vk6.25957', 'vk6.25965', 'vk6.33343', 'vk6.33359'] |
| 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 | O1O2O3U4U5O4O5U6U3O6U1U2 |
| R3 orbit | {'O1O2O3U4U5O4O5U6U3O6U1U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U3O4U1U4O5O6U5U6 |
| Gauss code of K* | O1O2U1O3O4U3U4U2O5O6U5U6 |
| Gauss code of -K* | O1O2U3O4O3U5U6O5O6U4U1U2 |
| 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 1 1 -1 1 -1],[ 1 0 1 1 0 2 0],[-1 -1 0 1 -2 0 -2],[-1 -1 -1 0 -2 0 -1],[ 1 0 2 2 0 1 0],[-1 -2 0 0 -1 0 -2],[ 1 0 2 1 0 2 0]] |
| Primitive based matrix | [[ 0 1 1 1 -1 -1 -1],[-1 0 1 0 -1 -2 -2],[-1 -1 0 0 -1 -1 -2],[-1 0 0 0 -2 -2 -1],[ 1 1 1 2 0 0 0],[ 1 2 1 2 0 0 0],[ 1 2 2 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,1,1,1,-1,0,1,2,2,0,1,1,2,2,2,1,0,0,0] |
| Phi over symmetry | [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,0,1,1,0,0,0,0,0] |
| Phi of -K | [-1,-1,-1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,-1,0] |
| Phi of K* | [-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,0,1,1,0,0,0,0,0] |
| Phi of -K* | [-1,-1,-1,1,1,1,0,0,1,1,2,0,1,2,2,2,2,1,-1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+25t^4+49t^2+9 |
| Outer characteristic polynomial | t^7+31t^5+63t^3+13t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -1024*K1**6 - 2048*K1**4*K2**2 + 3840*K1**4*K2 - 4640*K1**4 + 1728*K1**3*K2*K3 - 736*K1**3*K3 - 1344*K1**2*K2**4 + 3456*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 8640*K1**2*K2**2 - 1024*K1**2*K2*K4 + 6512*K1**2*K2 - 448*K1**2*K3**2 - 96*K1**2*K4**2 + 72*K1**2 + 1472*K1*K2**3*K3 - 1376*K1*K2**2*K3 - 416*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4576*K1*K2*K3 + 368*K1*K3*K4 + 80*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 1952*K2**4 - 64*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1264*K2**2*K4 - 508*K2**2 + 176*K2*K3*K5 + 48*K2*K4*K6 - 404*K3**2 - 120*K4**2 - 20*K5**2 - 4*K6**2 + 1334 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | True |