| Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,1,3,4,3,0,2,2,2,1,0,1,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.155'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.155', '6.319'] |
| Outer characteristic polynomial of the knot is: t^7+97t^5+76t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.155'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 64*K1**3*K2*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 544*K1**2*K2**2 + 584*K1**2*K2 - 160*K1**2*K3**2 - 564*K1**2 + 192*K1*K2**3*K3 + 96*K1*K2*K3**3 + 888*K1*K2*K3 + 112*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 280*K2**4 + 32*K2**3*K3*K5 - 368*K2**2*K3**2 - 8*K2**2*K4**2 + 136*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 324*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 16*K3**2*K6 - 304*K3**2 - 68*K4**2 - 44*K5**2 - 12*K6**2 - 2*K8**2 + 548 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.155'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71344', 'vk6.71395', 'vk6.71405', 'vk6.71854', 'vk6.71866', 'vk6.71917', 'vk6.71927', 'vk6.73279', 'vk6.73422', 'vk6.74330', 'vk6.74345', 'vk6.74538', 'vk6.74975', 'vk6.74988', 'vk6.75195', 'vk6.75614', 'vk6.76016', 'vk6.76375', 'vk6.76542', 'vk6.76557', 'vk6.76948', 'vk6.76997', 'vk6.77001', 'vk6.77060', 'vk6.78156', 'vk6.78597', 'vk6.78987', 'vk6.79231', 'vk6.79375', 'vk6.79800', 'vk6.79813', 'vk6.79981', 'vk6.80243', 'vk6.80506', 'vk6.80715', 'vk6.81270', 'vk6.82152', 'vk6.84035', 'vk6.84060', 'vk6.84600', 'vk6.85927', 'vk6.86735', 'vk6.87061', 'vk6.87065', 'vk6.87750', 'vk6.88041', 'vk6.88203', 'vk6.89979'] |
| 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 | O1O2O3O4O5U1O6U3U5U6U2U4 |
| R3 orbit | {'O1O2O3O4O5U1U2U4O6U3U5U6', 'O1O2O3O4O5U1U2O6U5U3U6U4', 'O1O2O3O4O5U1O6U3U5U6U2U4'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4O5U2U4U6U1U3O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U4U1U5U2O6U3 |
| Gauss code of -K* | O1O2O3O4O5U3O6U4U1U5U2U6 |
| 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 -4 0 -2 3 1 2],[ 4 0 3 1 4 2 2],[ 0 -3 0 -2 2 0 2],[ 2 -1 2 0 3 1 2],[-3 -4 -2 -3 0 -1 1],[-1 -2 0 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 0 -2 -4],[-3 0 1 -1 -2 -3 -4],[-2 -1 0 -1 -2 -2 -2],[-1 1 1 0 0 -1 -2],[ 0 2 2 0 0 -2 -3],[ 2 3 2 1 2 0 -1],[ 4 4 2 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,0,2,4,-1,1,2,3,4,1,2,2,2,0,1,2,2,3,1] |
| Phi over symmetry | [-4,-2,0,1,2,3,1,1,3,4,3,0,2,2,2,1,0,1,0,1,2] |
| Phi of -K | [-4,-2,0,1,2,3,1,1,3,4,3,0,2,2,2,1,0,1,0,1,2] |
| Phi of K* | [-3,-2,-1,0,2,4,2,1,1,2,3,0,0,2,4,1,2,3,0,1,1] |
| Phi of -K* | [-4,-2,0,1,2,3,1,3,2,2,4,2,1,2,3,0,2,2,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+9w^2z+15w |
| Inner characteristic polynomial | t^6+63t^4+8t^2 |
| Outer characteristic polynomial | t^7+97t^5+76t^3 |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + K4 + 2 |
| 2-strand cable arrow polynomial | -144*K1**4 + 64*K1**3*K2*K3 + 64*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 544*K1**2*K2**2 + 584*K1**2*K2 - 160*K1**2*K3**2 - 564*K1**2 + 192*K1*K2**3*K3 + 96*K1*K2*K3**3 + 888*K1*K2*K3 + 112*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 280*K2**4 + 32*K2**3*K3*K5 - 368*K2**2*K3**2 - 8*K2**2*K4**2 + 136*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 324*K2**2 + 192*K2*K3*K5 + 8*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 48*K3**4 + 16*K3**2*K6 - 304*K3**2 - 68*K4**2 - 44*K5**2 - 12*K6**2 - 2*K8**2 + 548 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]] |
| If K is slice | False |