| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,1,4,4,1,0,2,2,0,1,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.599', '7.17814'] |
| 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+78t^5+73t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.599'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 768*K1**4*K2 - 1424*K1**4 + 320*K1**3*K2*K3 - 608*K1**3*K3 - 192*K1**2*K2**4 + 800*K1**2*K2**3 - 5952*K1**2*K2**2 - 320*K1**2*K2*K4 + 8752*K1**2*K2 - 208*K1**2*K3**2 - 6896*K1**2 + 288*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 64*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7920*K1*K2*K3 + 944*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 992*K2**4 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 1408*K2**2*K4 - 4942*K2**2 + 296*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 2588*K3**2 - 648*K4**2 - 116*K5**2 - 10*K6**2 + 5182 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.599'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11446', 'vk6.11743', 'vk6.12760', 'vk6.13105', 'vk6.20320', 'vk6.21662', 'vk6.27623', 'vk6.29168', 'vk6.31197', 'vk6.31538', 'vk6.32365', 'vk6.32782', 'vk6.39041', 'vk6.41301', 'vk6.45797', 'vk6.47473', 'vk6.52199', 'vk6.52462', 'vk6.53030', 'vk6.53352', 'vk6.57191', 'vk6.58403', 'vk6.61804', 'vk6.62927', 'vk6.63765', 'vk6.63877', 'vk6.64193', 'vk6.64381', 'vk6.66798', 'vk6.67667', 'vk6.69437', 'vk6.70160'] |
| 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 | O1O2O3O4U2O5U3O6U1U6U4U5 |
| R3 orbit | {'O1O2O3O4U2O5U3O6U1U6U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U6U4O6U2O5U3 |
| Gauss code of K* | O1O2O3O4U1U5U6U3O5U4O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U1O6U2U5U6U4 |
| 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 -3 -2 -1 2 3 1],[ 3 0 -1 1 4 4 1],[ 2 1 0 1 2 2 0],[ 1 -1 -1 0 1 2 0],[-2 -4 -2 -1 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 0 -2 -2 -4],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 1 2 1 0 0 -1 -1],[ 2 2 2 0 1 0 1],[ 3 4 4 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,1,0,2,2,4,0,1,2,4,0,0,1,1,1,-1] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,1,4,4,1,0,2,2,0,1,2,0,0,1] |
| Phi of -K | [-3,-2,-1,1,2,3,2,1,3,1,2,0,3,2,3,2,2,2,1,2,0] |
| Phi of K* | [-3,-2,-1,1,2,3,0,2,2,3,2,1,2,2,1,2,3,3,0,1,2] |
| Phi of -K* | [-3,-2,-1,1,2,3,-1,1,1,4,4,1,0,2,2,0,1,2,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+25z+31 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+25w^2z+31w |
| Inner characteristic polynomial | t^6+50t^4+27t^2+1 |
| Outer characteristic polynomial | t^7+78t^5+73t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 768*K1**4*K2 - 1424*K1**4 + 320*K1**3*K2*K3 - 608*K1**3*K3 - 192*K1**2*K2**4 + 800*K1**2*K2**3 - 5952*K1**2*K2**2 - 320*K1**2*K2*K4 + 8752*K1**2*K2 - 208*K1**2*K3**2 - 6896*K1**2 + 288*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 64*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7920*K1*K2*K3 + 944*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 992*K2**4 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 1408*K2**2*K4 - 4942*K2**2 + 296*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 2588*K3**2 - 648*K4**2 - 116*K5**2 - 10*K6**2 + 5182 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}]] |
| If K is slice | True |