| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,0,1,-1,0,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.921'] |
| 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+41t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.921'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 416*K1**4 - 192*K1**2*K2**2 + 520*K1**2*K2 - 160*K1**2*K3**2 - 48*K1**2*K4**2 - 208*K1**2 + 432*K1*K2*K3 + 248*K1*K3*K4 + 40*K1*K4*K5 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 270*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 208*K3**2 - 94*K4**2 - 16*K5**2 - 2*K6**2 + 340 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.921'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13872', 'vk6.13874', 'vk6.13967', 'vk6.13969', 'vk6.14348', 'vk6.14349', 'vk6.14945', 'vk6.14947', 'vk6.15580', 'vk6.15581', 'vk6.16051', 'vk6.16052', 'vk6.16292', 'vk6.16300', 'vk6.16315', 'vk6.16323', 'vk6.17434', 'vk6.17438', 'vk6.22607', 'vk6.22615', 'vk6.22638', 'vk6.22646', 'vk6.23946', 'vk6.23950', 'vk6.33691', 'vk6.33693', 'vk6.34140', 'vk6.34141', 'vk6.34728', 'vk6.34736', 'vk6.36213', 'vk6.36215', 'vk6.36242', 'vk6.36246', 'vk6.38068', 'vk6.38069', 'vk6.44550', 'vk6.44551', 'vk6.53850', 'vk6.53852', 'vk6.54389', 'vk6.54391', 'vk6.55576', 'vk6.55579', 'vk6.56533', 'vk6.56534', 'vk6.59057', 'vk6.59065'] |
| 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 | O1O2O3O4U3U5O6O5U1U4U2U6 |
| R3 orbit | {'O1O2O3U2O4U5O6O5U1U3U4U6', 'O1O2O3U2U4O5O4O6U1U3U6U5', 'O1O2O3O4U3U5O6O5U1U4U2U6'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4U5U3U1U4O6O5U6U2 |
| Gauss code of K* | O1O2O3O4U1U3U5U2O5O6U4U6 |
| Gauss code of -K* | O1O2O3O4U5U1O5O6U3U6U2U4 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 0 -1 1 1 2],[ 3 0 2 0 2 3 2],[ 0 -2 0 -1 1 0 1],[ 1 0 1 0 1 1 1],[-1 -2 -1 -1 0 -1 0],[-1 -3 0 -1 1 0 2],[-2 -2 -1 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -1 -2],[-1 0 0 -1 -1 -1 -2],[-1 2 1 0 0 -1 -3],[ 0 1 1 0 0 -1 -2],[ 1 1 1 1 1 0 0],[ 3 2 2 3 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,2,1,1,2,1,1,1,2,0,1,3,1,2,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,0,1,-1,0,2] |
| Phi of -K | [-3,-1,0,1,1,2,2,1,1,2,3,0,1,1,2,1,0,1,-1,-1,1] |
| Phi of K* | [-2,-1,-1,0,1,3,-1,1,1,2,3,1,1,1,1,0,1,2,0,1,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,0,1,-1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | 7w^2z+15w |
| Inner characteristic polynomial | t^6+32t^4+12t^2 |
| Outer characteristic polynomial | t^7+48t^5+41t^3 |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -64*K1**6 + 64*K1**4*K2 - 416*K1**4 - 192*K1**2*K2**2 + 520*K1**2*K2 - 160*K1**2*K3**2 - 48*K1**2*K4**2 - 208*K1**2 + 432*K1*K2*K3 + 248*K1*K3*K4 + 40*K1*K4*K5 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 270*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 208*K3**2 - 94*K4**2 - 16*K5**2 - 2*K6**2 + 340 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {2, 5}, {1, 4}], [{6}, {4, 5}, {2, 3}, {1}]] |
| If K is slice | False |