| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,0,0,1,2,2,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1664', '7.15570', '7.37515'] |
| Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+60t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1664', '7.37515'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 704*K1**4*K2**2 + 2784*K1**4*K2 - 6352*K1**4 + 1120*K1**3*K2*K3 - 1216*K1**3*K3 - 576*K1**2*K2**4 + 2976*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11792*K1**2*K2**2 - 1216*K1**2*K2*K4 + 11560*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 2488*K1**2 + 1120*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8456*K1*K2*K3 - 32*K1*K2*K4*K5 + 1024*K1*K3*K4 + 152*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 2720*K2**4 - 32*K2**3*K6 - 656*K2**2*K3**2 - 96*K2**2*K4**2 + 2096*K2**2*K4 - 2284*K2**2 + 432*K2*K3*K5 + 64*K2*K4*K6 - 1300*K3**2 - 408*K4**2 - 76*K5**2 - 12*K6**2 + 3326 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1664'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.499', 'vk6.591', 'vk6.626', 'vk6.996', 'vk6.1094', 'vk6.1131', 'vk6.1677', 'vk6.1849', 'vk6.2169', 'vk6.2189', 'vk6.2276', 'vk6.2318', 'vk6.2790', 'vk6.2889', 'vk6.3064', 'vk6.3199', 'vk6.5265', 'vk6.6520', 'vk6.8902', 'vk6.9817', 'vk6.20810', 'vk6.21052', 'vk6.22206', 'vk6.22476', 'vk6.28499', 'vk6.29769', 'vk6.39870', 'vk6.40287', 'vk6.46423', 'vk6.46912', 'vk6.49137', 'vk6.58833'] |
| 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 | O1O2O3U2O4U1U5U6O5O6U4U3 |
| R3 orbit | {'O1O2O3U2O4U1U5U6O5O6U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5O6U5U6U3O4U2 |
| Gauss code of K* | O1O2O3U2U3O4O5U1U6U5O6U4 |
| Gauss code of -K* | O1O2O3U4O5U6U5U3O6O4U1U2 |
| 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 -2 -1 2 1 -1 1],[ 2 0 0 3 1 2 2],[ 1 0 0 1 0 1 1],[-2 -3 -1 0 0 -3 -1],[-1 -1 0 0 0 -2 0],[ 1 -2 -1 3 2 0 1],[-1 -2 -1 1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -3 -3],[-1 0 0 0 0 -2 -1],[-1 1 0 0 -1 -1 -2],[ 1 1 0 1 0 1 0],[ 1 3 2 1 -1 0 -2],[ 2 3 1 2 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,1,3,3,0,0,2,1,1,1,2,-1,0,2] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,0,0,1,2,2,0,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,1,1,2,1,1,1,0,0,1,2,2,0,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,0,2,1,0,1,1,1,0,2,2,-1,-1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,2,1,2,3,1,0,1,1,2,1,3,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+24z+33 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+24w^2z+33w |
| Inner characteristic polynomial | t^6+36t^4+38t^2+1 |
| Outer characteristic polynomial | t^7+48t^5+60t^3+7t |
| Flat arrow polynomial | 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7 |
| 2-strand cable arrow polynomial | -256*K1**6 - 704*K1**4*K2**2 + 2784*K1**4*K2 - 6352*K1**4 + 1120*K1**3*K2*K3 - 1216*K1**3*K3 - 576*K1**2*K2**4 + 2976*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11792*K1**2*K2**2 - 1216*K1**2*K2*K4 + 11560*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 2488*K1**2 + 1120*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8456*K1*K2*K3 - 32*K1*K2*K4*K5 + 1024*K1*K3*K4 + 152*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 2720*K2**4 - 32*K2**3*K6 - 656*K2**2*K3**2 - 96*K2**2*K4**2 + 2096*K2**2*K4 - 2284*K2**2 + 432*K2*K3*K5 + 64*K2*K4*K6 - 1300*K3**2 - 408*K4**2 - 76*K5**2 - 12*K6**2 + 3326 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | True |