| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,2,0,1,0,1,1,0,1,0,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1133'] |
| 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+37t^5+70t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1133'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 704*K1**4*K2 - 3264*K1**4 + 288*K1**3*K2*K3 - 384*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5648*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 9680*K1**2*K2 - 352*K1**2*K3**2 - 48*K1**2*K4**2 - 5704*K1**2 + 608*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 448*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7200*K1*K2*K3 + 864*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 784*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 16*K2**2*K4**2 + 1272*K2**2*K4 - 4742*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 2024*K3**2 - 416*K4**2 - 32*K5**2 - 2*K6**2 + 4670 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1133'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16949', 'vk6.17190', 'vk6.20543', 'vk6.21944', 'vk6.23345', 'vk6.23638', 'vk6.27997', 'vk6.29464', 'vk6.35389', 'vk6.35808', 'vk6.39401', 'vk6.41594', 'vk6.42862', 'vk6.43139', 'vk6.45977', 'vk6.47653', 'vk6.55112', 'vk6.55371', 'vk6.57407', 'vk6.58582', 'vk6.59510', 'vk6.59808', 'vk6.62074', 'vk6.63056', 'vk6.64957', 'vk6.65163', 'vk6.66947', 'vk6.67808', 'vk6.68246', 'vk6.68387', 'vk6.69558', 'vk6.70255'] |
| 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 | O1O2O3O4U5U3O6U4U2O5U1U6 |
| R3 orbit | {'O1O2O3O4U5U3O6U4U2O5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4O6U3U1O5U2U6 |
| Gauss code of K* | O1O2U3O4O5U1O6O3U6U5U2U4 |
| Gauss code of -K* | O1O2U3O4O5U1O6O3U5U6U4U2 |
| 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 0 0 1 -2 2],[ 1 0 1 0 2 -2 2],[ 0 -1 0 -1 1 -2 1],[ 0 0 1 0 1 -1 0],[-1 -2 -1 -1 0 -1 0],[ 2 2 2 1 1 0 2],[-2 -2 -1 0 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -2 -2],[-1 0 0 -1 -1 -2 -1],[ 0 0 1 0 1 0 -1],[ 0 1 1 -1 0 -1 -2],[ 1 2 2 0 1 0 -2],[ 2 2 1 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,2,2,1,1,2,1,-1,0,1,1,2,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,0,1,2,2,0,1,0,1,1,0,1,0,2,1] |
| Phi of -K | [-2,-1,0,0,1,2,-1,0,1,2,2,0,1,0,1,1,0,1,0,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,1,2,0,0,0,2,-1,0,0,1,1,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,1,2,1,2,0,1,2,2,1,1,0,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+22z+33 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+22w^2z+33w |
| Inner characteristic polynomial | t^6+27t^4+34t^2+4 |
| Outer characteristic polynomial | t^7+37t^5+70t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 + 704*K1**4*K2 - 3264*K1**4 + 288*K1**3*K2*K3 - 384*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5648*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 9680*K1**2*K2 - 352*K1**2*K3**2 - 48*K1**2*K4**2 - 5704*K1**2 + 608*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 448*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7200*K1*K2*K3 + 864*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 784*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 16*K2**2*K4**2 + 1272*K2**2*K4 - 4742*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 2024*K3**2 - 416*K4**2 - 32*K5**2 - 2*K6**2 + 4670 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]] |
| If K is slice | False |