| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,3,3,0,1,1,1,1,1,2,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.558'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+65t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.558'] |
| 2-strand cable arrow polynomial of the knot is: 1824*K1**4*K2 - 5664*K1**4 + 1088*K1**3*K2*K3 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 672*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6480*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 9040*K1**2*K2 - 2080*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 2844*K1**2 + 256*K1*K2**3*K3 - 768*K1*K2**2*K3 - 96*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6960*K1*K2*K3 + 1880*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 632*K2**4 - 256*K2**2*K3**2 - 8*K2**2*K4**2 + 664*K2**2*K4 - 3024*K2**2 + 224*K2*K3*K5 - 1740*K3**2 - 454*K4**2 - 48*K5**2 + 3452 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.558'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13910', 'vk6.14005', 'vk6.14188', 'vk6.14429', 'vk6.14981', 'vk6.15102', 'vk6.15660', 'vk6.16116', 'vk6.16700', 'vk6.16724', 'vk6.16842', 'vk6.18817', 'vk6.19291', 'vk6.19585', 'vk6.23137', 'vk6.23225', 'vk6.25411', 'vk6.26478', 'vk6.33721', 'vk6.33796', 'vk6.34275', 'vk6.35133', 'vk6.37536', 'vk6.42726', 'vk6.44700', 'vk6.54123', 'vk6.54917', 'vk6.54943', 'vk6.56403', 'vk6.56616', 'vk6.59344', 'vk6.64602'] |
| 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 | O1O2O3O4O5U4U1U3O6U5U6U2 |
| R3 orbit | {'O1O2O3O4O5U4U1U3O6U5U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U6U1O6U3U5U2 |
| Gauss code of K* | O1O2O3U4O5O4O6U2U6U3U1U5 |
| Gauss code of -K* | O1O2O3U2O4O5O6U3U6U4U1U5 |
| 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 0 -1 2 1],[ 3 0 3 1 0 3 1],[-1 -3 0 -1 -1 1 1],[ 0 -1 1 0 0 2 1],[ 1 0 1 0 0 1 1],[-2 -3 -1 -2 -1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 1 -1 -2 -1 -3],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -3],[ 0 2 1 1 0 0 -1],[ 1 1 1 1 0 0 0],[ 3 3 1 3 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,-1,1,2,1,3,1,1,1,1,1,1,3,0,1,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,1,1,3,3,0,1,1,1,1,1,2,-1,-1,1] |
| Phi of -K | [-3,-1,0,1,1,2,2,2,1,3,2,1,1,1,2,0,0,0,-1,0,2] |
| Phi of K* | [-2,-1,-1,0,1,3,0,2,0,2,2,1,0,1,1,0,1,3,1,2,2] |
| Phi of -K* | [-3,-1,0,1,1,2,0,1,1,3,3,0,1,1,1,1,1,2,-1,-1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 7z^2+28z+29 |
| Enhanced Jones-Krushkal polynomial | 7w^3z^2+28w^2z+29w |
| Inner characteristic polynomial | t^6+32t^4+20t^2+1 |
| Outer characteristic polynomial | t^7+48t^5+65t^3+6t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | 1824*K1**4*K2 - 5664*K1**4 + 1088*K1**3*K2*K3 - 1216*K1**3*K3 - 128*K1**2*K2**4 + 672*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6480*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 9040*K1**2*K2 - 2080*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 2844*K1**2 + 256*K1*K2**3*K3 - 768*K1*K2**2*K3 - 96*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6960*K1*K2*K3 + 1880*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 632*K2**4 - 256*K2**2*K3**2 - 8*K2**2*K4**2 + 664*K2**2*K4 - 3024*K2**2 + 224*K2*K3*K5 - 1740*K3**2 - 454*K4**2 - 48*K5**2 + 3452 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |