| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,0,2,3,4,0,1,1,1,1,1,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.530'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+76t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.530'] |
| 2-strand cable arrow polynomial of the knot is: 1280*K1**4*K2 - 3504*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 + 96*K1**2*K2**2*K4 - 3040*K1**2*K2**2 - 384*K1**2*K2*K4 + 7624*K1**2*K2 - 656*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 32*K1**2*K4*K6 - 5276*K1**2 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 6048*K1*K2*K3 - 160*K1*K2*K4*K5 + 1976*K1*K3*K4 + 480*K1*K4*K5 + 96*K1*K5*K6 - 104*K2**4 - 144*K2**2*K3**2 - 72*K2**2*K4**2 + 1312*K2**2*K4 - 4990*K2**2 - 32*K2*K3**2*K4 + 568*K2*K3*K5 + 152*K2*K4*K6 + 16*K3**2*K6 - 2664*K3**2 - 1250*K4**2 - 308*K5**2 - 58*K6**2 + 5008 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.530'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4878', 'vk6.5221', 'vk6.6468', 'vk6.6887', 'vk6.8429', 'vk6.8848', 'vk6.9773', 'vk6.10064', 'vk6.11672', 'vk6.12025', 'vk6.13014', 'vk6.20495', 'vk6.20774', 'vk6.21856', 'vk6.27899', 'vk6.29401', 'vk6.29737', 'vk6.32657', 'vk6.33000', 'vk6.39332', 'vk6.39814', 'vk6.46374', 'vk6.47600', 'vk6.47949', 'vk6.48836', 'vk6.49105', 'vk6.51355', 'vk6.51566', 'vk6.53275', 'vk6.57356', 'vk6.64336', 'vk6.66909'] |
| 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 | O1O2O3O4O5U2U5U4O6U3U1U6 |
| R3 orbit | {'O1O2O3O4O5U2U5U4O6U3U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U5U3O6U2U1U4 |
| Gauss code of K* | O1O2O3U4O5O6O4U6U1U5U3U2 |
| Gauss code of -K* | O1O2O3U1O4O5O6U5U4U3U6U2 |
| 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 -3 0 1 1 2],[ 1 0 -3 1 1 1 2],[ 3 3 0 3 2 1 1],[ 0 -1 -3 0 0 0 1],[-1 -1 -2 0 0 0 0],[-1 -1 -1 0 0 0 0],[-2 -2 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -1 -2 -1],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[ 0 1 0 0 0 -1 -3],[ 1 2 1 1 1 0 -3],[ 3 1 1 2 3 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,0,1,2,1,0,0,1,1,0,1,2,1,3,3] |
| Phi over symmetry | [-3,-1,0,1,1,2,-1,0,2,3,4,0,1,1,1,1,1,1,0,1,1] |
| Phi of -K | [-3,-1,0,1,1,2,-1,0,2,3,4,0,1,1,1,1,1,1,0,1,1] |
| Phi of K* | [-2,-1,-1,0,1,3,1,1,1,1,4,0,1,1,2,1,1,3,0,0,-1] |
| Phi of -K* | [-3,-1,0,1,1,2,3,3,1,2,1,1,1,1,2,0,0,1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+32t^4+27t^2+1 |
| Outer characteristic polynomial | t^7+48t^5+76t^3+4t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 1280*K1**4*K2 - 3504*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 928*K1**3*K3 + 96*K1**2*K2**2*K4 - 3040*K1**2*K2**2 - 384*K1**2*K2*K4 + 7624*K1**2*K2 - 656*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 32*K1**2*K4*K6 - 5276*K1**2 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 576*K1*K2*K3*K4 + 6048*K1*K2*K3 - 160*K1*K2*K4*K5 + 1976*K1*K3*K4 + 480*K1*K4*K5 + 96*K1*K5*K6 - 104*K2**4 - 144*K2**2*K3**2 - 72*K2**2*K4**2 + 1312*K2**2*K4 - 4990*K2**2 - 32*K2*K3**2*K4 + 568*K2*K3*K5 + 152*K2*K4*K6 + 16*K3**2*K6 - 2664*K3**2 - 1250*K4**2 - 308*K5**2 - 58*K6**2 + 5008 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |