| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,2,2,2,1,1,1,1,0,0,0,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1225'] |
| 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+26t^5+23t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1225', '7.37718'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 + 384*K1**4*K2 - 2304*K1**4 - 96*K1**3*K3 - 1664*K1**2*K2**2 - 96*K1**2*K2*K4 + 3800*K1**2*K2 - 640*K1**2*K3**2 - 112*K1**2*K4**2 - 1664*K1**2 - 320*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 2784*K1*K2*K3 + 1128*K1*K3*K4 + 184*K1*K4*K5 - 88*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 376*K2**2*K4 - 1788*K2**2 + 152*K2*K3*K5 + 32*K2*K4*K6 - 1052*K3**2 - 454*K4**2 - 68*K5**2 - 4*K6**2 + 1972 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1225'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4458', 'vk6.4553', 'vk6.5840', 'vk6.5967', 'vk6.6398', 'vk6.6829', 'vk6.8016', 'vk6.8355', 'vk6.9327', 'vk6.9446', 'vk6.11635', 'vk6.11986', 'vk6.12981', 'vk6.13427', 'vk6.13522', 'vk6.13715', 'vk6.14082', 'vk6.15059', 'vk6.15179', 'vk6.17785', 'vk6.17816', 'vk6.18831', 'vk6.19424', 'vk6.19719', 'vk6.24332', 'vk6.25430', 'vk6.25461', 'vk6.26602', 'vk6.33273', 'vk6.33332', 'vk6.37550', 'vk6.39272', 'vk6.39753', 'vk6.41452', 'vk6.44877', 'vk6.46313', 'vk6.47890', 'vk6.48661', 'vk6.49903', 'vk6.53230'] |
| 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 | O1O2O3O4U3U1U5O6O5U4U6U2 |
| R3 orbit | {'O1O2O3U4U1U5O6O5U3U6O4U2', 'O1O2O3O4U3U1U5O6O5U4U6U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U3U5U1O6O5U6U4U2 |
| Gauss code of K* | O1O2O3U4U3O5O4O6U2U6U1U5 |
| Gauss code of -K* | O1O2O3U4U2O4O5O6U3U6U1U5 |
| 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 1 1 0],[ 2 0 2 0 2 2 1],[-1 -2 0 -1 0 0 0],[ 1 0 1 0 1 1 1],[-1 -2 0 -1 0 -1 0],[-1 -2 0 -1 1 0 0],[ 0 -1 0 -1 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 0 -1 -2],[-1 0 0 0 0 -1 -2],[ 0 0 0 0 0 -1 -1],[ 1 1 1 1 1 0 0],[ 2 2 2 2 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,1,2,0,1,2,1,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,2,2,2,1,1,1,1,0,0,0,-1,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,-1,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,1,1,0,1,1,1,1,1,1,0,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,2,2,2,1,1,1,1,0,0,0,-1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+18t^4+10t^2+1 |
| Outer characteristic polynomial | t^7+26t^5+23t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -128*K1**6 + 384*K1**4*K2 - 2304*K1**4 - 96*K1**3*K3 - 1664*K1**2*K2**2 - 96*K1**2*K2*K4 + 3800*K1**2*K2 - 640*K1**2*K3**2 - 112*K1**2*K4**2 - 1664*K1**2 - 320*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 2784*K1*K2*K3 + 1128*K1*K3*K4 + 184*K1*K4*K5 - 88*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 376*K2**2*K4 - 1788*K2**2 + 152*K2*K3*K5 + 32*K2*K4*K6 - 1052*K3**2 - 454*K4**2 - 68*K5**2 - 4*K6**2 + 1972 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]] |
| If K is slice | False |