| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,2,4,-1,0,1,1,0,0,1,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1069'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+46t^5+106t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1069'] |
| 2-strand cable arrow polynomial of the knot is: -240*K1**4 + 96*K1**2*K2**3 - 688*K1**2*K2**2 + 1168*K1**2*K2 - 80*K1**2*K3**2 - 1280*K1**2 + 96*K1*K2**3*K3 + 1536*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 840*K2**4 - 240*K2**2*K3**2 - 8*K2**2*K4**2 + 552*K2**2*K4 - 576*K2**2 + 192*K2*K3*K5 - 668*K3**2 - 118*K4**2 - 52*K5**2 + 1116 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1069'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73770', 'vk6.73779', 'vk6.73792', 'vk6.73795', 'vk6.73907', 'vk6.73916', 'vk6.73928', 'vk6.73931', 'vk6.75736', 'vk6.75738', 'vk6.75909', 'vk6.75918', 'vk6.78709', 'vk6.78721', 'vk6.78745', 'vk6.78749', 'vk6.78909', 'vk6.78925', 'vk6.80329', 'vk6.80338', 'vk6.80349', 'vk6.80351', 'vk6.80451', 'vk6.80460', 'vk6.81714', 'vk6.81715', 'vk6.82490', 'vk6.82491', 'vk6.84454', 'vk6.84455', 'vk6.88362', 'vk6.88364'] |
| 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 | O1O2O3O4U5U3O6U2O5U1U6U4 |
| R3 orbit | {'O1O2O3O4U5U3O6U2O5U1U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U4O6U3O5U2U6 |
| Gauss code of K* | O1O2O3U1U4U5U3O6O5U2O4U6 |
| Gauss code of -K* | O1O2O3U4O5U2O6O4U1U6U5U3 |
| 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 -2 -1 0 3 -1 1],[ 2 0 1 1 4 0 1],[ 1 -1 0 0 2 0 0],[ 0 -1 0 0 0 0 -1],[-3 -4 -2 0 0 -2 -1],[ 1 0 0 0 2 0 1],[-1 -1 0 1 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 0 -2 -2 -4],[-1 1 0 1 0 -1 -1],[ 0 0 -1 0 0 0 -1],[ 1 2 0 0 0 0 -1],[ 1 2 1 0 0 0 0],[ 2 4 1 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,0,2,2,4,-1,0,1,1,0,0,1,0,1,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,0,2,2,4,-1,0,1,1,0,0,1,0,1,0] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,1,2,1,0,1,2,2,1,1,2,2,3,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,3,2,2,1,2,1,2,2,1,1,1,0,1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,1,1,4,0,0,1,2,0,0,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | 4w^4z-10w^3z+4w^3+13w^2z+11w |
| Inner characteristic polynomial | t^6+30t^4+41t^2 |
| Outer characteristic polynomial | t^7+46t^5+106t^3 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -240*K1**4 + 96*K1**2*K2**3 - 688*K1**2*K2**2 + 1168*K1**2*K2 - 80*K1**2*K3**2 - 1280*K1**2 + 96*K1*K2**3*K3 + 1536*K1*K2*K3 + 152*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 840*K2**4 - 240*K2**2*K3**2 - 8*K2**2*K4**2 + 552*K2**2*K4 - 576*K2**2 + 192*K2*K3*K5 - 668*K3**2 - 118*K4**2 - 52*K5**2 + 1116 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |