| Min(phi) over symmetries of the knot is: [-4,0,1,1,1,1,1,1,2,3,4,1,1,1,1,-1,-1,-1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.184'] |
| Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506'] |
| Outer characteristic polynomial of the knot is: t^7+58t^5+73t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.184'] |
| 2-strand cable arrow polynomial of the knot is: -240*K1**4 + 96*K1**3*K3*K4 + 104*K1**2*K2 - 336*K1**2*K3**2 - 256*K1**2*K4**2 - 596*K1**2 + 600*K1*K2*K3 + 1120*K1*K3*K4 + 248*K1*K4*K5 - 16*K2**2*K4**2 + 104*K2**2*K4 - 364*K2**2 + 40*K2*K3*K5 + 32*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 - 688*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 572*K4**2 - 96*K5**2 - 12*K6**2 - 12*K7**2 - 2*K8**2 + 828 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.184'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4704', 'vk6.5009', 'vk6.6204', 'vk6.6673', 'vk6.8195', 'vk6.8615', 'vk6.9571', 'vk6.9910', 'vk6.16698', 'vk6.19080', 'vk6.19127', 'vk6.19254', 'vk6.19547', 'vk6.22519', 'vk6.23013', 'vk6.23130', 'vk6.23579', 'vk6.23916', 'vk6.25709', 'vk6.26064', 'vk6.26441', 'vk6.28408', 'vk6.29924', 'vk6.29975', 'vk6.30082', 'vk6.35125', 'vk6.37809', 'vk6.40078', 'vk6.40343', 'vk6.44653', 'vk6.46801', 'vk6.48058', 'vk6.48742', 'vk6.49751', 'vk6.51629', 'vk6.51734', 'vk6.56591', 'vk6.59333', 'vk6.64874', 'vk6.66189'] |
| 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 | O1O2O3O4O5U1O6U5U6U4U3U2 |
| R3 orbit | {'O1O2O3O4O5U1O6U5U6U4U3U2', 'O1O2O3O4O5U1U4O6U5U6U3U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U3U2U6U1O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U5U4U3U1O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U5U3U2U1U6 |
| 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 -4 1 1 1 0 1],[ 4 0 4 3 2 1 1],[-1 -4 0 0 0 -1 1],[-1 -3 0 0 0 -1 1],[-1 -2 0 0 0 -1 1],[ 0 -1 1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 1 0 -4],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 -1 -1 -1],[-1 0 1 0 0 -1 -3],[-1 0 1 0 0 -1 -4],[ 0 1 1 1 1 0 -1],[ 4 2 1 3 4 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,-1,0,4,-1,0,0,1,2,1,1,1,1,0,1,3,1,4,1] |
| Phi over symmetry | [-4,0,1,1,1,1,1,1,2,3,4,1,1,1,1,-1,-1,-1,0,0,0] |
| Phi of -K | [-4,0,1,1,1,1,3,1,2,3,4,0,0,0,0,0,0,-1,0,-1,-1] |
| Phi of K* | [-1,-1,-1,-1,0,4,-1,-1,-1,0,4,0,0,0,1,0,0,2,0,3,3] |
| Phi of -K* | [-4,0,1,1,1,1,1,1,2,3,4,1,1,1,1,-1,-1,-1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-4t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -8w^3z+13w^2z+11w |
| Inner characteristic polynomial | t^6+38t^4+17t^2 |
| Outer characteristic polynomial | t^7+58t^5+73t^3 |
| Flat arrow polynomial | -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -240*K1**4 + 96*K1**3*K3*K4 + 104*K1**2*K2 - 336*K1**2*K3**2 - 256*K1**2*K4**2 - 596*K1**2 + 600*K1*K2*K3 + 1120*K1*K3*K4 + 248*K1*K4*K5 - 16*K2**2*K4**2 + 104*K2**2*K4 - 364*K2**2 + 40*K2*K3*K5 + 32*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 - 688*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 572*K4**2 - 96*K5**2 - 12*K6**2 - 12*K7**2 - 2*K8**2 + 828 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |