| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,1,2,4,0,0,0,1,0,0,1,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.733'] |
| 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+64t^5+96t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.733'] |
| 2-strand cable arrow polynomial of the knot is: 1152*K1**4*K2 - 2224*K1**4 + 416*K1**3*K2*K3 - 768*K1**3*K3 + 128*K1**2*K2**2*K4 - 2784*K1**2*K2**2 - 128*K1**2*K2*K4 + 5544*K1**2*K2 - 496*K1**2*K3**2 - 96*K1**2*K4**2 - 3956*K1**2 - 256*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4416*K1*K2*K3 + 656*K1*K3*K4 + 136*K1*K4*K5 - 72*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 576*K2**2*K4 - 3430*K2**2 + 88*K2*K3*K5 + 8*K2*K4*K6 - 1568*K3**2 - 434*K4**2 - 52*K5**2 - 2*K6**2 + 3360 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.733'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3637', 'vk6.3732', 'vk6.3925', 'vk6.4022', 'vk6.7059', 'vk6.7120', 'vk6.7297', 'vk6.7392', 'vk6.11409', 'vk6.12590', 'vk6.12703', 'vk6.19103', 'vk6.19148', 'vk6.19800', 'vk6.25712', 'vk6.25771', 'vk6.26235', 'vk6.26678', 'vk6.31007', 'vk6.31136', 'vk6.32187', 'vk6.37823', 'vk6.37878', 'vk6.44964', 'vk6.48265', 'vk6.48446', 'vk6.50019', 'vk6.50164', 'vk6.52156', 'vk6.63732', 'vk6.66204', 'vk6.66231'] |
| 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 | O1O2O3O4U1O5U4U6U5O6U3U2 |
| R3 orbit | {'O1O2O3O4U1O5U4U6U5O6U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U2O5U6U5U1O6U4 |
| Gauss code of K* | O1O2O3U2O4O5U6U5U4U1O6U3 |
| Gauss code of -K* | O1O2O3U1O4U3U5U6U4O6O5U2 |
| 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 1 1 0 2 -1],[ 3 0 3 2 1 1 3],[-1 -3 0 0 -1 2 -2],[-1 -2 0 0 -1 2 -2],[ 0 -1 1 1 0 1 -1],[-2 -1 -2 -2 -1 0 -2],[ 1 -3 2 2 1 2 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 -2 -2 -1 -2 -1],[-1 2 0 0 -1 -2 -2],[-1 2 0 0 -1 -2 -3],[ 0 1 1 1 0 -1 -1],[ 1 2 2 2 1 0 -3],[ 3 1 2 3 1 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,2,2,1,2,1,0,1,2,2,1,2,3,1,1,3] |
| Phi over symmetry | [-3,-1,0,1,1,2,-1,2,1,2,4,0,0,0,1,0,0,1,0,-1,-1] |
| Phi of -K | [-3,-1,0,1,1,2,-1,2,1,2,4,0,0,0,1,0,0,1,0,-1,-1] |
| Phi of K* | [-2,-1,-1,0,1,3,-1,-1,1,1,4,0,0,0,1,0,0,2,0,2,-1] |
| Phi of -K* | [-3,-1,0,1,1,2,3,1,2,3,1,1,2,2,2,1,1,1,0,2,2] |
| 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+48t^4+67t^2+4 |
| Outer characteristic polynomial | t^7+64t^5+96t^3+6t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | 1152*K1**4*K2 - 2224*K1**4 + 416*K1**3*K2*K3 - 768*K1**3*K3 + 128*K1**2*K2**2*K4 - 2784*K1**2*K2**2 - 128*K1**2*K2*K4 + 5544*K1**2*K2 - 496*K1**2*K3**2 - 96*K1**2*K4**2 - 3956*K1**2 - 256*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4416*K1*K2*K3 + 656*K1*K3*K4 + 136*K1*K4*K5 - 72*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 576*K2**2*K4 - 3430*K2**2 + 88*K2*K3*K5 + 8*K2*K4*K6 - 1568*K3**2 - 434*K4**2 - 52*K5**2 - 2*K6**2 + 3360 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |