| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,1,0,2,1,0,1,1,0,0,2,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1526', '7.32240', '7.32491'] |
| 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+114t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1526', '7.32491'] |
| 2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 2400*K1**4*K2 - 3968*K1**4 + 800*K1**3*K2*K3 + 128*K1**3*K3*K4 - 256*K1**3*K3 - 960*K1**2*K2**4 + 3328*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 9600*K1**2*K2**2 - 672*K1**2*K2*K4 + 7576*K1**2*K2 - 544*K1**2*K3**2 - 32*K1**2*K3*K5 - 160*K1**2*K4**2 - 1784*K1**2 + 1344*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6032*K1*K2*K3 - 32*K1*K2*K4*K5 + 744*K1*K3*K4 + 120*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2448*K2**4 - 64*K2**3*K6 - 768*K2**2*K3**2 - 160*K2**2*K4**2 + 1760*K2**2*K4 - 1292*K2**2 - 32*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 :['6.1526'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.328', 'vk6.368', 'vk6.444', 'vk6.726', 'vk6.776', 'vk6.895', 'vk6.1466', 'vk6.1524', 'vk6.1592', 'vk6.1965', 'vk6.2005', 'vk6.2073', 'vk6.2496', 'vk6.2753', 'vk6.3014', 'vk6.3138', 'vk6.3793', 'vk6.3984', 'vk6.7185', 'vk6.7360', 'vk6.18783', 'vk6.19855', 'vk6.24912', 'vk6.25373', 'vk6.25912', 'vk6.26298', 'vk6.26743', 'vk6.37987', 'vk6.38042', 'vk6.45033', 'vk6.50099', 'vk6.60761'] |
| 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 | O1O2O3U2O4U5U6O5O6U1U4U3 |
| R3 orbit | {'O1O2O3U2O4U5U6O5O6U1U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U3O5O6U5U6O4U2 |
| Gauss code of K* | O1O2O3U1U4U3O4U2O5O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O5U2O6U1U6U3 |
| 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 1 3],[ 1 0 0 1 0 1 1],[-2 -3 -1 0 0 -3 -1],[-1 -1 0 0 0 -2 0],[ 1 -1 -1 3 2 0 1],[-1 -3 -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 -3],[ 1 1 0 1 0 1 0],[ 1 3 2 1 -1 0 -1],[ 2 3 1 3 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,0,1,1,3,3,0,0,2,1,1,1,3,-1,0,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,0,1,0,2,1,0,1,1,0,0,2,2,-1,0,1] |
| Phi of -K | [-2,-1,-1,1,1,2,0,1,0,2,1,1,1,0,0,1,2,2,0,0,1] |
| Phi of K* | [-2,-1,-1,1,1,2,0,1,0,2,1,0,1,1,0,0,2,2,-1,0,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,1,1,3,3,1,0,1,1,2,1,3,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+38t^4+84t^2+1 |
| Outer characteristic polynomial | t^7+50t^5+114t^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 - 3968*K1**4 + 800*K1**3*K2*K3 + 128*K1**3*K3*K4 - 256*K1**3*K3 - 960*K1**2*K2**4 + 3328*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 9600*K1**2*K2**2 - 672*K1**2*K2*K4 + 7576*K1**2*K2 - 544*K1**2*K3**2 - 32*K1**2*K3*K5 - 160*K1**2*K4**2 - 1784*K1**2 + 1344*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6032*K1*K2*K3 - 32*K1*K2*K4*K5 + 744*K1*K3*K4 + 120*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2448*K2**4 - 64*K2**3*K6 - 768*K2**2*K3**2 - 160*K2**2*K4**2 + 1760*K2**2*K4 - 1292*K2**2 - 32*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 | [[{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | True |