| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,1,1,2,1,1,1,3,1,1,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1320'] |
| Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866'] |
| Outer characteristic polynomial of the knot is: t^7+37t^5+49t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1320'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 288*K1**4*K2 - 2576*K1**4 + 224*K1**3*K2*K3 - 96*K1**3*K3 + 64*K1**2*K2**2*K4 - 2592*K1**2*K2**2 + 96*K1**2*K2*K4**2 - 416*K1**2*K2*K4 + 6112*K1**2*K2 - 752*K1**2*K3**2 - 192*K1**2*K3*K5 - 320*K1**2*K4**2 - 96*K1**2*K4*K6 - 3988*K1**2 - 512*K1*K2**2*K3 - 480*K1*K2*K3*K4 + 4472*K1*K2*K3 + 2056*K1*K3*K4 + 616*K1*K4*K5 + 24*K1*K5*K6 - 152*K2**4 - 112*K2**2*K3**2 - 112*K2**2*K4**2 + 992*K2**2*K4 - 3556*K2**2 + 384*K2*K3*K5 + 112*K2*K4*K6 - 1920*K3**2 - 1038*K4**2 - 220*K5**2 - 28*K6**2 + 3780 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1320'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17100', 'vk6.17341', 'vk6.20586', 'vk6.21993', 'vk6.23487', 'vk6.23824', 'vk6.28048', 'vk6.29505', 'vk6.35640', 'vk6.36079', 'vk6.39458', 'vk6.41657', 'vk6.43004', 'vk6.43314', 'vk6.46042', 'vk6.47708', 'vk6.55239', 'vk6.55489', 'vk6.57472', 'vk6.58633', 'vk6.59637', 'vk6.59983', 'vk6.62143', 'vk6.63103', 'vk6.65041', 'vk6.65238', 'vk6.66996', 'vk6.67859', 'vk6.68306', 'vk6.68454', 'vk6.69611', 'vk6.70302'] |
| 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 | O1O2O3U4O5O6U2U6O4U1U5U3 |
| R3 orbit | {'O1O2O3U4O5O6U2U6O4U1U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U3O5U6U2O6O4U5 |
| Gauss code of K* | O1O2O3U1U4U3O5U2U6O4O6U5 |
| Gauss code of -K* | O1O2O3U4O5O6U5U2O4U1U6U3 |
| 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 -2 2 0 1 1],[ 2 0 0 3 1 1 1],[ 2 0 0 2 1 1 1],[-2 -3 -2 0 -1 -1 0],[ 0 -1 -1 1 0 1 1],[-1 -1 -1 1 -1 0 0],[-1 -1 -1 0 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 0 -1 -1 -2 -3],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 -1 -1 -1],[ 0 1 1 1 0 -1 -1],[ 2 2 1 1 1 0 0],[ 2 3 1 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,0,1,1,2,3,0,1,1,1,1,1,1,1,1,0] |
| Phi over symmetry | [-2,-2,0,1,1,2,0,1,1,1,2,1,1,1,3,1,1,1,0,0,1] |
| Phi of -K | [-2,-2,0,1,1,2,0,1,2,2,1,1,2,2,2,0,0,1,0,0,1] |
| Phi of K* | [-2,-1,-1,0,2,2,0,1,1,1,2,0,0,2,2,0,2,2,1,1,0] |
| Phi of -K* | [-2,-2,0,1,1,2,0,1,1,1,2,1,1,1,3,1,1,1,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | z^2+18z+33 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+18w^2z+33w |
| Inner characteristic polynomial | t^6+23t^4+24t^2 |
| Outer characteristic polynomial | t^7+37t^5+49t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**4*K2**2 + 288*K1**4*K2 - 2576*K1**4 + 224*K1**3*K2*K3 - 96*K1**3*K3 + 64*K1**2*K2**2*K4 - 2592*K1**2*K2**2 + 96*K1**2*K2*K4**2 - 416*K1**2*K2*K4 + 6112*K1**2*K2 - 752*K1**2*K3**2 - 192*K1**2*K3*K5 - 320*K1**2*K4**2 - 96*K1**2*K4*K6 - 3988*K1**2 - 512*K1*K2**2*K3 - 480*K1*K2*K3*K4 + 4472*K1*K2*K3 + 2056*K1*K3*K4 + 616*K1*K4*K5 + 24*K1*K5*K6 - 152*K2**4 - 112*K2**2*K3**2 - 112*K2**2*K4**2 + 992*K2**2*K4 - 3556*K2**2 + 384*K2*K3*K5 + 112*K2*K4*K6 - 1920*K3**2 - 1038*K4**2 - 220*K5**2 - 28*K6**2 + 3780 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |