| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,0,3,3,3,0,1,2,1,0,1,0,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.860'] |
| 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+46t^5+40t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.219', '6.860'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**6 + 288*K1**4*K2 - 704*K1**4 - 256*K1**2*K2**2 + 736*K1**2*K2 - 224*K1**2*K3**2 - 112*K1**2*K4**2 - 208*K1**2 + 504*K1*K2*K3 + 448*K1*K3*K4 + 200*K1*K4*K5 + 24*K1*K5*K6 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 318*K2**2 + 32*K2*K3*K5 + 16*K2*K4*K6 - 296*K3**2 - 230*K4**2 - 96*K5**2 - 18*K6**2 + 532 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.860'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11255', 'vk6.11335', 'vk6.12520', 'vk6.12633', 'vk6.17050', 'vk6.17293', 'vk6.17628', 'vk6.18914', 'vk6.18990', 'vk6.19359', 'vk6.19652', 'vk6.22260', 'vk6.24090', 'vk6.24184', 'vk6.25512', 'vk6.26131', 'vk6.26549', 'vk6.28319', 'vk6.30929', 'vk6.31054', 'vk6.31225', 'vk6.31576', 'vk6.32111', 'vk6.32232', 'vk6.32802', 'vk6.35565', 'vk6.36016', 'vk6.36425', 'vk6.37651', 'vk6.39939', 'vk6.40116', 'vk6.43526', 'vk6.44792', 'vk6.46486', 'vk6.52005', 'vk6.53380', 'vk6.55459', 'vk6.56671', 'vk6.65393', 'vk6.66109'] |
| 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 | O1O2O3O4U1U4O5O6U3U6U2U5 |
| R3 orbit | {'O1O2O3O4U1U4O5O6U3U6U2U5', 'O1O2O3O4U1U4O5U2O6U3U6U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U6U2O6O5U1U4 |
| Gauss code of K* | O1O2O3O4U5U3U1U6O5O6U4U2 |
| Gauss code of -K* | O1O2O3O4U3U1O5O6U5U4U2U6 |
| 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 2 1],[ 3 0 3 2 1 2 1],[ 0 -3 0 -1 0 2 1],[ 1 -2 1 0 0 2 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -2 0 0 0],[-1 -1 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -2 -2 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 2 0 1 0 -1 -3],[ 1 2 0 1 1 0 -2],[ 3 2 1 1 3 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,0,2,2,2,0,0,0,1,1,1,1,1,3,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,0,3,3,3,0,1,2,1,0,1,0,0,1,1] |
| Phi of -K | [-3,-1,0,1,1,2,0,0,3,3,3,0,1,2,1,0,1,0,0,1,1] |
| Phi of K* | [-2,-1,-1,0,1,3,1,1,0,1,3,0,0,1,3,1,2,3,0,0,0] |
| Phi of -K* | [-3,-1,0,1,1,2,2,3,1,1,2,1,0,1,2,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 9z+19 |
| Enhanced Jones-Krushkal polynomial | 9w^2z+19w |
| Inner characteristic polynomial | t^6+30t^4+11t^2 |
| Outer characteristic polynomial | t^7+46t^5+40t^3 |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -320*K1**6 + 288*K1**4*K2 - 704*K1**4 - 256*K1**2*K2**2 + 736*K1**2*K2 - 224*K1**2*K3**2 - 112*K1**2*K4**2 - 208*K1**2 + 504*K1*K2*K3 + 448*K1*K3*K4 + 200*K1*K4*K5 + 24*K1*K5*K6 - 8*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 318*K2**2 + 32*K2*K3*K5 + 16*K2*K4*K6 - 296*K3**2 - 230*K4**2 - 96*K5**2 - 18*K6**2 + 532 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | False |