| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,3,0,1,1,1,0,0,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.554', '7.24817', '7.39787'] |
| 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+50t^5+70t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.554', '7.39787'] |
| 2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 2400*K1**4*K2 - 4352*K1**4 + 800*K1**3*K2*K3 + 128*K1**3*K3*K4 - 512*K1**3*K3 - 960*K1**2*K2**4 + 3328*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 9280*K1**2*K2**2 - 640*K1**2*K2*K4 + 7832*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 1656*K1**2 + 1344*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 192*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6000*K1*K2*K3 + 712*K1*K3*K4 + 120*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2384*K2**4 - 800*K2**2*K3**2 - 224*K2**2*K4**2 + 1696*K2**2*K4 - 1292*K2**2 - 64*K2*K3**2*K4 + 392*K2*K3*K5 + 104*K2*K4*K6 - 884*K3**2 - 268*K4**2 - 36*K5**2 - 4*K6**2 + 2306 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.554'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.339', 'vk6.379', 'vk6.449', 'vk6.733', 'vk6.787', 'vk6.896', 'vk6.1471', 'vk6.1529', 'vk6.1605', 'vk6.1966', 'vk6.2006', 'vk6.2082', 'vk6.2489', 'vk6.2744', 'vk6.3007', 'vk6.3127', 'vk6.3792', 'vk6.3985', 'vk6.7184', 'vk6.7361', 'vk6.18794', 'vk6.19860', 'vk6.24921', 'vk6.25384', 'vk6.25910', 'vk6.26301', 'vk6.26744', 'vk6.37989', 'vk6.38044', 'vk6.45044', 'vk6.50098', 'vk6.60754'] |
| 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 | O1O2O3O4O5U3U6U5O6U4U1U2 |
| R3 orbit | {'O1O2O3O4O5U3U6U5O6U4U1U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U5U2O6U1U6U3 |
| Gauss code of K* | O1O2O3U2O4O5O6U5U6U1U4U3 |
| Gauss code of -K* | O1O2O3U4O5O4O6U5U3U6U1U2 |
| 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 -1 1 -2 1 2 -1],[ 1 0 1 -2 1 2 0],[-1 -1 0 -2 1 2 -2],[ 2 2 2 0 2 1 1],[-1 -1 -1 -2 0 1 -2],[-2 -2 -2 -1 -1 0 -2],[ 1 0 2 -1 2 2 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -2 -2 -2 -1],[-1 1 0 -1 -1 -2 -2],[-1 2 1 0 -1 -2 -2],[ 1 2 1 1 0 0 -2],[ 1 2 2 2 0 0 -1],[ 2 1 2 2 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,1,2,2,2,1,1,1,2,2,1,2,2,0,2,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,1,3,0,1,1,1,0,0,1,-1,-1,0] |
| Phi of -K | [-2,-1,-1,1,1,2,-1,0,1,1,3,0,1,1,1,0,0,1,-1,-1,0] |
| Phi of K* | [-2,-1,-1,1,1,2,-1,0,1,1,3,1,0,1,1,0,1,1,0,0,-1] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,2,2,2,1,0,2,2,2,1,1,2,-1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+38t^4+40t^2+1 |
| Outer characteristic polynomial | t^7+50t^5+70t^3+5t |
| Flat arrow polynomial | 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7 |
| 2-strand cable arrow polynomial | -1024*K1**4*K2**2 + 2400*K1**4*K2 - 4352*K1**4 + 800*K1**3*K2*K3 + 128*K1**3*K3*K4 - 512*K1**3*K3 - 960*K1**2*K2**4 + 3328*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 9280*K1**2*K2**2 - 640*K1**2*K2*K4 + 7832*K1**2*K2 - 672*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 1656*K1**2 + 1344*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 192*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6000*K1*K2*K3 + 712*K1*K3*K4 + 120*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2384*K2**4 - 800*K2**2*K3**2 - 224*K2**2*K4**2 + 1696*K2**2*K4 - 1292*K2**2 - 64*K2*K3**2*K4 + 392*K2*K3*K5 + 104*K2*K4*K6 - 884*K3**2 - 268*K4**2 - 36*K5**2 - 4*K6**2 + 2306 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}]] |
| If K is slice | True |