| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,1,0,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1807'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+44t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1807'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 192*K1**4*K2**2 + 2816*K1**4*K2 - 5792*K1**4 - 128*K1**3*K2**2*K3 + 928*K1**3*K2*K3 - 1760*K1**3*K3 - 192*K1**2*K2**4 + 1664*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 9456*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 12592*K1**2*K2 - 992*K1**2*K3**2 - 176*K1**2*K4**2 - 6008*K1**2 + 448*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 352*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 9392*K1*K2*K3 + 1480*K1*K3*K4 + 304*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1800*K2**4 - 32*K2**3*K6 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 1848*K2**2*K4 - 4758*K2**2 + 288*K2*K3*K5 + 16*K2*K4*K6 - 2352*K3**2 - 738*K4**2 - 144*K5**2 - 2*K6**2 + 5448 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1807'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13882', 'vk6.13977', 'vk6.14135', 'vk6.14358', 'vk6.14953', 'vk6.15074', 'vk6.15587', 'vk6.16057', 'vk6.16285', 'vk6.16310', 'vk6.17423', 'vk6.22596', 'vk6.22629', 'vk6.23935', 'vk6.33701', 'vk6.33776', 'vk6.34147', 'vk6.34259', 'vk6.34588', 'vk6.36200', 'vk6.36227', 'vk6.42282', 'vk6.53868', 'vk6.53909', 'vk6.54108', 'vk6.54413', 'vk6.54572', 'vk6.55568', 'vk6.59021', 'vk6.59040', 'vk6.60061', 'vk6.64572'] |
| 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 | O1O2O3U4U3O5O6U5U1O4U2U6 |
| R3 orbit | {'O1O2O3U4U3O5O6U5U1O4U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O5U3U6O4O6U1U5 |
| Gauss code of K* | O1O2U3O4O5U2U4U6O3O6U1U5 |
| Gauss code of -K* | O1O2U3O4O5U1U5O6O3U6U2U4 |
| 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 1 -1 -1 2],[ 1 0 0 0 0 0 2],[ 0 0 0 1 -1 0 1],[-1 0 -1 0 -1 0 -1],[ 1 0 1 1 0 -1 1],[ 1 0 0 0 1 0 1],[-2 -2 -1 1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 1 -1 -1 -1 -2],[-1 -1 0 -1 0 -1 0],[ 0 1 1 0 0 -1 0],[ 1 1 0 0 0 1 0],[ 1 1 1 1 -1 0 0],[ 1 2 0 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,1,0,-1,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,0,0,1,0,-1,0,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,1,2,2,0,0,1,2,1,2,1,0,1,2] |
| Phi of K* | [-2,-1,0,1,1,1,2,1,1,2,2,0,2,1,2,1,0,1,0,0,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,1,1,1,0,0,0,1,0,0,2,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+12t^4+23t^2+4 |
| Outer characteristic polynomial | t^7+20t^5+44t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -64*K1**6 - 192*K1**4*K2**2 + 2816*K1**4*K2 - 5792*K1**4 - 128*K1**3*K2**2*K3 + 928*K1**3*K2*K3 - 1760*K1**3*K3 - 192*K1**2*K2**4 + 1664*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 9456*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 12592*K1**2*K2 - 992*K1**2*K3**2 - 176*K1**2*K4**2 - 6008*K1**2 + 448*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 352*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 9392*K1*K2*K3 + 1480*K1*K3*K4 + 304*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1800*K2**4 - 32*K2**3*K6 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 1848*K2**2*K4 - 4758*K2**2 + 288*K2*K3*K5 + 16*K2*K4*K6 - 2352*K3**2 - 738*K4**2 - 144*K5**2 - 2*K6**2 + 5448 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |