| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,2,2,3,1,1,1,1,1,0,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1128'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
| Outer characteristic polynomial of the knot is: t^7+29t^5+54t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1008', '6.1128'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 704*K1**4*K2**2 + 1728*K1**4*K2 - 4672*K1**4 + 640*K1**3*K2*K3 - 1056*K1**3*K3 - 192*K1**2*K2**4 + 1632*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 7600*K1**2*K2**2 - 1152*K1**2*K2*K4 + 10128*K1**2*K2 - 320*K1**2*K3**2 - 48*K1**2*K4**2 - 3812*K1**2 + 512*K1*K2**3*K3 - 736*K1*K2**2*K3 - 320*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6752*K1*K2*K3 + 672*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1072*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 1032*K2**2*K4 - 3198*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 1284*K3**2 - 260*K4**2 - 24*K5**2 - 2*K6**2 + 3498 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1128'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4696', 'vk6.4999', 'vk6.6186', 'vk6.6657', 'vk6.8183', 'vk6.8601', 'vk6.9565', 'vk6.9904', 'vk6.17390', 'vk6.20911', 'vk6.20975', 'vk6.22323', 'vk6.22399', 'vk6.23561', 'vk6.23898', 'vk6.28387', 'vk6.36158', 'vk6.40041', 'vk6.40168', 'vk6.42094', 'vk6.43073', 'vk6.43377', 'vk6.46569', 'vk6.46673', 'vk6.48736', 'vk6.49536', 'vk6.49739', 'vk6.51436', 'vk6.55556', 'vk6.58911', 'vk6.65294', 'vk6.69763'] |
| 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 | O1O2O3O4U5U1O5U3U4O6U2U6 |
| R3 orbit | {'O1O2O3O4U5U1O5U3U4O6U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3O5U1U2O6U4U6 |
| Gauss code of K* | O1O2U1O3O4U5O6O5U2U6U3U4 |
| Gauss code of -K* | O1O2U1O3O4U5O6O5U3U4U2U6 |
| 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 -2 0 0 2 -1 1],[ 2 0 2 0 1 2 1],[ 0 -2 0 -1 1 0 1],[ 0 0 1 0 1 0 0],[-2 -1 -1 -1 0 -2 0],[ 1 -2 0 0 2 0 1],[-1 -1 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -2 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 1 0 0],[ 0 1 1 -1 0 0 -2],[ 1 2 1 0 0 0 -2],[ 2 1 1 0 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,1,2,1,0,1,1,1,-1,0,0,0,2,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,0,2,2,3,1,1,1,1,1,0,1,1,1,1] |
| Phi of -K | [-2,-1,0,0,1,2,-1,0,2,2,3,1,1,1,1,1,0,1,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,1,1,3,0,1,1,2,-1,1,0,1,2,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,0,2,1,1,0,0,1,2,1,0,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+23w^2z+31w |
| Inner characteristic polynomial | t^6+19t^4+24t^2+1 |
| Outer characteristic polynomial | t^7+29t^5+54t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -256*K1**6 - 704*K1**4*K2**2 + 1728*K1**4*K2 - 4672*K1**4 + 640*K1**3*K2*K3 - 1056*K1**3*K3 - 192*K1**2*K2**4 + 1632*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 7600*K1**2*K2**2 - 1152*K1**2*K2*K4 + 10128*K1**2*K2 - 320*K1**2*K3**2 - 48*K1**2*K4**2 - 3812*K1**2 + 512*K1*K2**3*K3 - 736*K1*K2**2*K3 - 320*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6752*K1*K2*K3 + 672*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1072*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 1032*K2**2*K4 - 3198*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 1284*K3**2 - 260*K4**2 - 24*K5**2 - 2*K6**2 + 3498 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]] |
| If K is slice | False |