| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,3,1,2,4,1,0,2,2,0,2,1,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.654'] |
| 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+67t^5+132t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.654'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 416*K1**4 + 192*K1**3*K2*K3 - 128*K1**3*K3 - 576*K1**2*K2**4 + 1568*K1**2*K2**3 - 6720*K1**2*K2**2 - 256*K1**2*K2*K4 + 5136*K1**2*K2 - 2836*K1**2 + 672*K1*K2**3*K3 - 416*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4568*K1*K2*K3 + 8*K1*K3*K4 - 288*K2**6 + 160*K2**4*K4 - 2472*K2**4 - 240*K2**2*K3**2 - 8*K2**2*K4**2 + 1392*K2**2*K4 - 648*K2**2 + 40*K2*K3*K5 - 756*K3**2 - 90*K4**2 + 1952 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.654'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17496', 'vk6.17504', 'vk6.17553', 'vk6.17561', 'vk6.24020', 'vk6.24032', 'vk6.24093', 'vk6.24101', 'vk6.36274', 'vk6.36284', 'vk6.36342', 'vk6.36349', 'vk6.43429', 'vk6.43437', 'vk6.43461', 'vk6.43466', 'vk6.55618', 'vk6.55626', 'vk6.55647', 'vk6.55655', 'vk6.60132', 'vk6.60143', 'vk6.60164', 'vk6.60174', 'vk6.65321', 'vk6.65326', 'vk6.65352', 'vk6.65361', 'vk6.68493', 'vk6.68497', 'vk6.68515', 'vk6.68522'] |
| 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 | O1O2O3O4U2O5U1U5O6U3U4U6 |
| R3 orbit | {'O1O2O3O4U2O5U1U5O6U3U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U2O5U6U4O6U3 |
| Gauss code of K* | O1O2O3U4U5U1U2O5U6O4O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U4O6U2U3U6U5 |
| 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 -2 0 2 1 2],[ 3 0 0 3 4 1 2],[ 2 0 0 1 2 0 2],[ 0 -3 -1 0 1 0 2],[-2 -4 -2 -1 0 0 1],[-1 -1 0 0 0 0 0],[-2 -2 -2 -2 -1 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 1 0 -1 -2 -4],[-2 -1 0 0 -2 -2 -2],[-1 0 0 0 0 0 -1],[ 0 1 2 0 0 -1 -3],[ 2 2 2 0 1 0 0],[ 3 4 2 1 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,-1,0,1,2,4,0,2,2,2,0,0,1,1,3,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,3,1,2,4,1,0,2,2,0,2,1,0,0,-1] |
| Phi of -K | [-3,-2,0,1,2,2,1,0,3,1,3,1,3,2,2,1,1,0,1,1,-1] |
| Phi of K* | [-2,-2,-1,0,2,3,-1,1,0,2,3,1,1,2,1,1,3,3,1,0,1] |
| Phi of -K* | [-3,-2,0,1,2,2,0,3,1,2,4,1,0,2,2,0,2,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+9w^3z^2-8w^3z+26w^2z+17w |
| Inner characteristic polynomial | t^6+45t^4+61t^2+1 |
| Outer characteristic polynomial | t^7+67t^5+132t^3+16t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 448*K1**4*K2 - 416*K1**4 + 192*K1**3*K2*K3 - 128*K1**3*K3 - 576*K1**2*K2**4 + 1568*K1**2*K2**3 - 6720*K1**2*K2**2 - 256*K1**2*K2*K4 + 5136*K1**2*K2 - 2836*K1**2 + 672*K1*K2**3*K3 - 416*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4568*K1*K2*K3 + 8*K1*K3*K4 - 288*K2**6 + 160*K2**4*K4 - 2472*K2**4 - 240*K2**2*K3**2 - 8*K2**2*K4**2 + 1392*K2**2*K4 - 648*K2**2 + 40*K2*K3*K5 - 756*K3**2 - 90*K4**2 + 1952 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |