| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,2,3,0,2,1,1,0,1,1,0,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1489'] |
| 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+27t^5+61t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1489'] |
| 2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 896*K1**4*K2**2 + 1504*K1**4*K2 - 2528*K1**4 - 128*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 288*K1**3*K3 - 384*K1**2*K2**4 + 2176*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8592*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 9928*K1**2*K2 - 288*K1**2*K3**2 - 5616*K1**2 + 576*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 224*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6880*K1*K2*K3 + 440*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 1360*K2**2*K4 - 3854*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1476*K3**2 - 252*K4**2 - 28*K5**2 - 2*K6**2 + 4218 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1489'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3609', 'vk6.3676', 'vk6.3869', 'vk6.3994', 'vk6.7031', 'vk6.7064', 'vk6.7241', 'vk6.7364', 'vk6.17702', 'vk6.17751', 'vk6.24249', 'vk6.24310', 'vk6.36552', 'vk6.36629', 'vk6.43658', 'vk6.43765', 'vk6.48245', 'vk6.48316', 'vk6.48401', 'vk6.48426', 'vk6.50001', 'vk6.50034', 'vk6.50119', 'vk6.50146', 'vk6.55742', 'vk6.55799', 'vk6.60314', 'vk6.60397', 'vk6.65442', 'vk6.65471', 'vk6.68570', 'vk6.68599'] |
| 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 | O1O2O3U1O4U3U4O5O6U2U5U6 |
| R3 orbit | {'O1O2O3U1O4U3U4O5O6U2U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U2O4O5U6U1O6U3 |
| Gauss code of K* | O1O2O3U4U1U5O4U6O5O6U2U3 |
| Gauss code of -K* | O1O2O3U1U2O4O5U4O6U5U3U6 |
| 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 -1 0 1 0 2],[ 2 0 2 1 1 1 1],[ 1 -2 0 -1 1 1 2],[ 0 -1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 0 -1 -1 0 0 0 1],[-2 -1 -2 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -2 -1],[-1 0 0 -1 0 -1 -1],[ 0 0 1 0 0 1 -1],[ 0 1 0 0 0 -1 -1],[ 1 2 1 -1 1 0 -2],[ 2 1 1 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,0,1,2,1,1,0,1,1,0,-1,1,1,1,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,1,1,2,3,0,2,1,1,0,1,1,0,2,1] |
| Phi of -K | [-2,-1,0,0,1,2,-1,1,1,2,3,0,2,1,1,0,1,1,0,2,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,2,1,3,1,0,1,2,0,0,1,2,1,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,1,1,1,1,-1,1,1,2,0,1,0,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+17t^4+27t^2+1 |
| Outer characteristic polynomial | t^7+27t^5+61t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | 128*K1**4*K2**3 - 896*K1**4*K2**2 + 1504*K1**4*K2 - 2528*K1**4 - 128*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 288*K1**3*K3 - 384*K1**2*K2**4 + 2176*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8592*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 9928*K1**2*K2 - 288*K1**2*K3**2 - 5616*K1**2 + 576*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 224*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6880*K1*K2*K3 + 440*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 16*K2**2*K4**2 + 1360*K2**2*K4 - 3854*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1476*K3**2 - 252*K4**2 - 28*K5**2 - 2*K6**2 + 4218 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | False |