| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,-1,0,1,1,1,1,0,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1276'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926'] |
| Outer characteristic polynomial of the knot is: t^7+28t^5+45t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1276'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**6 - 448*K1**4*K2**2 + 2176*K1**4*K2 - 4528*K1**4 + 1568*K1**3*K2*K3 - 1696*K1**3*K3 - 192*K1**2*K2**4 + 1088*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 7824*K1**2*K2**2 - 1600*K1**2*K2*K4 + 12056*K1**2*K2 - 720*K1**2*K3**2 - 80*K1**2*K4**2 - 6676*K1**2 + 864*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 704*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 9608*K1*K2*K3 + 1464*K1*K3*K4 + 264*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1432*K2**4 - 96*K2**3*K6 - 400*K2**2*K3**2 - 128*K2**2*K4**2 + 2224*K2**2*K4 - 5690*K2**2 + 576*K2*K3*K5 + 72*K2*K4*K6 - 2624*K3**2 - 826*K4**2 - 188*K5**2 - 6*K6**2 + 5688 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1276'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4150', 'vk6.4181', 'vk6.5388', 'vk6.5419', 'vk6.7510', 'vk6.7535', 'vk6.9011', 'vk6.9042', 'vk6.12426', 'vk6.12459', 'vk6.13334', 'vk6.13559', 'vk6.13590', 'vk6.14266', 'vk6.14715', 'vk6.14738', 'vk6.15208', 'vk6.15869', 'vk6.15894', 'vk6.30839', 'vk6.30872', 'vk6.32023', 'vk6.32056', 'vk6.33060', 'vk6.33091', 'vk6.33863', 'vk6.34326', 'vk6.48492', 'vk6.50271', 'vk6.53522', 'vk6.53936', 'vk6.54259'] |
| 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 | O1O2O3U1O4O5U3U5O6U4U6U2 |
| R3 orbit | {'O1O2O3U1O4O5U3U5O6U4U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U4U5O4U6U1O6O5U3 |
| Gauss code of K* | O1O2O3U4U3U5O4U1U6O5O6U2 |
| Gauss code of -K* | O1O2O3U2O4O5U4U3O6U5U1U6 |
| 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 -1 0 1 1],[ 2 0 2 1 1 1 0],[-1 -2 0 -2 0 1 1],[ 1 -1 2 0 2 1 1],[ 0 -1 0 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-1 0 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 1 0 -2 -2],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 -1 -1 0],[ 0 0 0 1 0 -2 -1],[ 1 2 1 1 2 0 -1],[ 2 2 1 0 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,-1,0,2,2,0,0,1,1,1,1,0,2,1,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,2,3,-1,0,1,1,1,1,0,-1,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,0,1,1,2,3,-1,0,1,1,1,1,0,-1,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,3,1,1,0,1,1,1,2,-1,1,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,1,0,1,2,2,1,1,2,1,0,0,0,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+20t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+28t^5+45t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6 |
| 2-strand cable arrow polynomial | -448*K1**6 - 448*K1**4*K2**2 + 2176*K1**4*K2 - 4528*K1**4 + 1568*K1**3*K2*K3 - 1696*K1**3*K3 - 192*K1**2*K2**4 + 1088*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 7824*K1**2*K2**2 - 1600*K1**2*K2*K4 + 12056*K1**2*K2 - 720*K1**2*K3**2 - 80*K1**2*K4**2 - 6676*K1**2 + 864*K1*K2**3*K3 - 1568*K1*K2**2*K3 - 704*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 9608*K1*K2*K3 + 1464*K1*K3*K4 + 264*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1432*K2**4 - 96*K2**3*K6 - 400*K2**2*K3**2 - 128*K2**2*K4**2 + 2224*K2**2*K4 - 5690*K2**2 + 576*K2*K3*K5 + 72*K2*K4*K6 - 2624*K3**2 - 826*K4**2 - 188*K5**2 - 6*K6**2 + 5688 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |