| Min(phi) over symmetries of the knot is: [-3,-2,2,3,-1,2,5,0,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.598', '7.15699'] |
| Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857'] |
| Outer characteristic polynomial of the knot is: t^5+61t^3+19t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.598'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1024*K1**4*K2 - 3360*K1**4 + 704*K1**3*K2*K3 - 832*K1**3*K3 + 960*K1**2*K2**3 - 8896*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 13216*K1**2*K2 - 672*K1**2*K3**2 - 96*K1**2*K4**2 - 8816*K1**2 + 448*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 128*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 10720*K1*K2*K3 + 1392*K1*K3*K4 + 112*K1*K4*K5 + 16*K1*K5*K6 - 1616*K2**4 - 448*K2**2*K3**2 - 16*K2**2*K4**2 + 1888*K2**2*K4 - 6340*K2**2 + 304*K2*K3*K5 + 16*K2*K4*K6 - 3048*K3**2 - 708*K4**2 - 104*K5**2 - 12*K6**2 + 6786 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.598'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71640', 'vk6.71665', 'vk6.71817', 'vk6.71833', 'vk6.72241', 'vk6.72268', 'vk6.72369', 'vk6.77264', 'vk6.77360', 'vk6.77377', 'vk6.77612', 'vk6.77621', 'vk6.77704', 'vk6.77720', 'vk6.81419', 'vk6.81448', 'vk6.86952', 'vk6.87161', 'vk6.88007', 'vk6.89551'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3O4U2O5U3O6U1U5U6U4 |
| R3 orbit | {'O1O2O3O4U2O5U3O6U1U5U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U6U4O5U2O6U3 |
| Gauss code of K* | O1O2O3O4U1U5U6U4O5U2O6U3 |
| Gauss code of -K* | Same |
| Diagrammatic symmetry type | - |
| 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 -1 3 1 2],[ 3 0 -1 1 5 2 2],[ 2 1 0 1 2 1 0],[ 1 -1 -1 0 2 1 1],[-3 -5 -2 -2 0 -1 1],[-1 -2 -1 -1 1 0 1],[-2 -2 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 -2 -3],[-3 0 1 -2 -5],[-2 -1 0 0 -2],[ 2 2 0 0 1],[ 3 5 2 -1 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-3,-2,2,3,-1,2,5,0,2,-1] |
| Phi over symmetry | [-3,-2,2,3,-1,2,5,0,2,-1] |
| Phi of -K | [-3,-2,2,3,2,3,1,4,3,2] |
| Phi of K* | [-3,-2,2,3,2,3,1,4,3,2] |
| Phi of -K* | [-3,-2,2,3,-1,2,5,0,2,-1] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^4+35t^2+9 |
| Outer characteristic polynomial | t^5+61t^3+19t |
| Flat arrow polynomial | -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 1024*K1**4*K2 - 3360*K1**4 + 704*K1**3*K2*K3 - 832*K1**3*K3 + 960*K1**2*K2**3 - 8896*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 13216*K1**2*K2 - 672*K1**2*K3**2 - 96*K1**2*K4**2 - 8816*K1**2 + 448*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 128*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 10720*K1*K2*K3 + 1392*K1*K3*K4 + 112*K1*K4*K5 + 16*K1*K5*K6 - 1616*K2**4 - 448*K2**2*K3**2 - 16*K2**2*K4**2 + 1888*K2**2*K4 - 6340*K2**2 + 304*K2*K3*K5 + 16*K2*K4*K6 - 3048*K3**2 - 708*K4**2 - 104*K5**2 - 12*K6**2 + 6786 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | True |