| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,1,2,4,1,1,1,2,1,1,2,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.571', '7.17898'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+92t^5+41t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.571'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 784*K1**4 + 32*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 - 448*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 4680*K1**2*K2 - 848*K1**2*K3**2 - 64*K1**2*K3*K5 - 80*K1**2*K4**2 - 4904*K1**2 + 96*K1*K2**3*K3 - 800*K1*K2**2*K3 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 4792*K1*K2*K3 + 1424*K1*K3*K4 + 96*K1*K4*K5 - 80*K2**4 - 320*K2**2*K3**2 - 16*K2**2*K4**2 + 576*K2**2*K4 - 3412*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 8*K3**2*K6 - 1988*K3**2 - 552*K4**2 - 36*K5**2 - 4*K6**2 + 3470 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.571'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81565', 'vk6.81579', 'vk6.81647', 'vk6.81661', 'vk6.81728', 'vk6.81740', 'vk6.81859', 'vk6.81865', 'vk6.82233', 'vk6.82247', 'vk6.82385', 'vk6.82395', 'vk6.82501', 'vk6.82511', 'vk6.82576', 'vk6.82583', 'vk6.83176', 'vk6.83181', 'vk6.83602', 'vk6.83609', 'vk6.84131', 'vk6.84143', 'vk6.84334', 'vk6.84348', 'vk6.84565', 'vk6.84569', 'vk6.86489', 'vk6.86495', 'vk6.88734', 'vk6.88744', 'vk6.88905', 'vk6.88920'] |
| 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 | O1O2O3O4U1O5U2O6U3U5U4U6 |
| R3 orbit | {'O1O2O3O4U1O5U2O6U3U5U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U6U2O5U3O6U4 |
| Gauss code of K* | O1O2O3O4U5U6U1U3O5U2O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5U3O6U2U4U5U6 |
| 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 1 3],[ 3 0 1 2 3 2 2],[ 2 -1 0 1 3 2 3],[ 1 -2 -1 0 2 1 3],[-2 -3 -3 -2 0 0 2],[-1 -2 -2 -1 0 0 1],[-3 -2 -3 -3 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 -2 -1 -3 -3 -2],[-2 2 0 0 -2 -3 -3],[-1 1 0 0 -1 -2 -2],[ 1 3 2 1 0 -1 -2],[ 2 3 3 2 1 0 -1],[ 3 2 3 2 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,2,1,3,3,2,0,2,3,3,1,2,2,1,2,1] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,1,2,4,1,1,1,2,1,1,2,0,0,0] |
| Phi of -K | [-3,-2,-1,1,2,3,0,0,2,2,4,0,1,1,2,1,1,1,1,1,-1] |
| Phi of K* | [-3,-2,-1,1,2,3,-1,1,1,2,4,1,1,1,2,1,1,2,0,0,0] |
| Phi of -K* | [-3,-2,-1,1,2,3,1,2,2,3,2,1,2,3,3,1,2,3,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+20z+29 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+20w^2z+29w |
| Inner characteristic polynomial | t^6+64t^4+19t^2 |
| Outer characteristic polynomial | t^7+92t^5+41t^3+4t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 784*K1**4 + 32*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 - 448*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 4680*K1**2*K2 - 848*K1**2*K3**2 - 64*K1**2*K3*K5 - 80*K1**2*K4**2 - 4904*K1**2 + 96*K1*K2**3*K3 - 800*K1*K2**2*K3 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 4792*K1*K2*K3 + 1424*K1*K3*K4 + 96*K1*K4*K5 - 80*K2**4 - 320*K2**2*K3**2 - 16*K2**2*K4**2 + 576*K2**2*K4 - 3412*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 8*K3**2*K6 - 1988*K3**2 - 552*K4**2 - 36*K5**2 - 4*K6**2 + 3470 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}]] |
| If K is slice | False |