| Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,3,2,4,1,1,1,1,1,1,2,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.80'] |
| Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.80', '6.163', '6.227', '6.447'] |
| Outer characteristic polynomial of the knot is: t^7+100t^5+37t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.80'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 - 128*K1**2*K2**4 + 192*K1**2*K2**3 - 688*K1**2*K2**2 + 584*K1**2*K2 - 16*K1**2*K3**2 - 468*K1**2 + 448*K1*K2**3*K3 + 704*K1*K2*K3 + 64*K1*K3*K4 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 368*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 352*K2**2*K3**2 - 72*K2**2*K4**2 + 136*K2**2*K4 - 8*K2**2*K6**2 - 140*K2**2 + 112*K2*K3*K5 + 32*K2*K4*K6 + 8*K3**2*K6 - 244*K3**2 - 74*K4**2 - 8*K5**2 - 12*K6**2 + 440 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.80'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4634', 'vk6.4901', 'vk6.6052', 'vk6.6565', 'vk6.8087', 'vk6.8463', 'vk6.9463', 'vk6.9840', 'vk6.20287', 'vk6.21616', 'vk6.27567', 'vk6.29129', 'vk6.38984', 'vk6.41229', 'vk6.45751', 'vk6.47444', 'vk6.48668', 'vk6.48849', 'vk6.49386', 'vk6.49635', 'vk6.50672', 'vk6.50841', 'vk6.51145', 'vk6.51370', 'vk6.57132', 'vk6.58322', 'vk6.61738', 'vk6.62877', 'vk6.66757', 'vk6.67639', 'vk6.69411', 'vk6.70133', 'vk6.81596', 'vk6.81771', 'vk6.82283', 'vk6.82447', 'vk6.82748', 'vk6.84192', 'vk6.84379', 'vk6.84387', 'vk6.85439', 'vk6.85979', 'vk6.86921', 'vk6.87133', 'vk6.88178', 'vk6.88677', 'vk6.88787', 'vk6.89115'] |
| 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 | O1O2O3O4O5O6U2U4U5U6U1U3 |
| R3 orbit | {'O1O2O3O4O5U1O6U4U5U2U6U3', 'O1O2O3O4O5U1U3O6U5U2U4U6', 'O1O2O3O4O5O6U2U4U5U6U1U3'} |
| R3 orbit length | 3 |
| Gauss code of -K | O1O2O3O4O5O6U4U6U1U2U3U5 |
| Gauss code of K* | O1O2O3O4O5O6U5U1U6U2U3U4 |
| Gauss code of -K* | O1O2O3O4O5O6U3U4U5U1U6U2 |
| 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 -1 -4 2 -1 1 3],[ 1 0 -3 2 -1 1 3],[ 4 3 0 4 1 2 3],[-2 -2 -4 0 -2 0 2],[ 1 1 -1 2 0 1 2],[-1 -1 -2 0 -1 0 1],[-3 -3 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -1 -4],[-3 0 -2 -1 -2 -3 -3],[-2 2 0 0 -2 -2 -4],[-1 1 0 0 -1 -1 -2],[ 1 2 2 1 0 1 -1],[ 1 3 2 1 -1 0 -3],[ 4 3 4 2 1 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,1,4,2,1,2,3,3,0,2,2,4,1,1,2,-1,1,3] |
| Phi over symmetry | [-4,-1,-1,1,2,3,0,2,3,2,4,1,1,1,1,1,1,2,1,1,-1] |
| Phi of -K | [-4,-1,-1,1,2,3,0,2,3,2,4,1,1,1,1,1,1,2,1,1,-1] |
| Phi of K* | [-3,-2,-1,1,1,4,-1,1,1,2,4,1,1,1,2,1,1,3,-1,0,2] |
| Phi of -K* | [-4,-1,-1,1,2,3,1,3,2,4,3,1,1,2,2,1,2,3,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -4w^3z+9w^2z+11w |
| Inner characteristic polynomial | t^6+68t^4 |
| Outer characteristic polynomial | t^7+100t^5+37t^3 |
| Flat arrow polynomial | 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -144*K1**4 - 128*K1**2*K2**4 + 192*K1**2*K2**3 - 688*K1**2*K2**2 + 584*K1**2*K2 - 16*K1**2*K3**2 - 468*K1**2 + 448*K1*K2**3*K3 + 704*K1*K2*K3 + 64*K1*K3*K4 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 368*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 352*K2**2*K3**2 - 72*K2**2*K4**2 + 136*K2**2*K4 - 8*K2**2*K6**2 - 140*K2**2 + 112*K2*K3*K5 + 32*K2*K4*K6 + 8*K3**2*K6 - 244*K3**2 - 74*K4**2 - 8*K5**2 - 12*K6**2 + 440 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |