| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,1,2,3,3,1,1,1,1,1,1,2,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.315'] |
| Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863'] |
| Outer characteristic polynomial of the knot is: t^7+61t^5+38t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.315'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 - 1408*K1**4*K2**2 + 3200*K1**4*K2 - 4704*K1**4 - 384*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 736*K1**3*K3 - 320*K1**2*K2**4 + 2752*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 10608*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 10688*K1**2*K2 - 256*K1**2*K3**2 - 144*K1**2*K4**2 - 4720*K1**2 + 1248*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 7016*K1*K2*K3 + 840*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1616*K2**4 - 528*K2**2*K3**2 - 48*K2**2*K4**2 + 872*K2**2*K4 - 2990*K2**2 + 120*K2*K3*K5 + 8*K2*K4*K6 - 1468*K3**2 - 432*K4**2 - 60*K5**2 - 2*K6**2 + 4206 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.315'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10915', 'vk6.10924', 'vk6.10928', 'vk6.12075', 'vk6.12079', 'vk6.12090', 'vk6.12094', 'vk6.14497', 'vk6.14498', 'vk6.15719', 'vk6.15720', 'vk6.16151', 'vk6.16152', 'vk6.30513', 'vk6.30517', 'vk6.30541', 'vk6.30545', 'vk6.31793', 'vk6.34073', 'vk6.34176', 'vk6.34177', 'vk6.34510', 'vk6.51750', 'vk6.51754', 'vk6.52624', 'vk6.54145', 'vk6.54146', 'vk6.54337', 'vk6.54541', 'vk6.63466', 'vk6.63475', 'vk6.63479'] |
| 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 | O1O2O3O4O5U2U3O6U5U6U1U4 |
| R3 orbit | {'O1O2O3O4O5U2U3O6U5U6U1U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U5U6U1O6U3U4 |
| Gauss code of K* | O1O2O3O4U3U5U6U4U1O5O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U4U1U5U6U2 |
| 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 -3 -1 3 1 1],[ 1 0 -2 0 3 1 1],[ 3 2 0 1 3 2 1],[ 1 0 -1 0 2 1 1],[-3 -3 -3 -2 0 -1 1],[-1 -1 -2 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -1 -3],[-3 0 1 -1 -2 -3 -3],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -2],[ 1 2 1 1 0 0 -1],[ 1 3 1 1 0 0 -2],[ 3 3 1 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,1,3,-1,1,2,3,3,1,1,1,1,1,1,2,0,1,2] |
| Phi over symmetry | [-3,-1,-1,1,1,3,-1,1,2,3,3,1,1,1,1,1,1,2,0,1,2] |
| Phi of -K | [-3,-1,-1,1,1,3,0,1,2,3,3,0,1,1,1,1,1,2,-1,1,3] |
| Phi of K* | [-3,-1,-1,1,1,3,1,3,1,2,3,1,1,1,2,1,1,3,0,0,1] |
| Phi of -K* | [-3,-1,-1,1,1,3,1,2,1,2,3,0,1,1,2,1,1,3,-1,-1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+39t^4+12t^2 |
| Outer characteristic polynomial | t^7+61t^5+38t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -128*K1**6 - 1408*K1**4*K2**2 + 3200*K1**4*K2 - 4704*K1**4 - 384*K1**3*K2**2*K3 + 896*K1**3*K2*K3 - 736*K1**3*K3 - 320*K1**2*K2**4 + 2752*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 10608*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 10688*K1**2*K2 - 256*K1**2*K3**2 - 144*K1**2*K4**2 - 4720*K1**2 + 1248*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 7016*K1*K2*K3 + 840*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1616*K2**4 - 528*K2**2*K3**2 - 48*K2**2*K4**2 + 872*K2**2*K4 - 2990*K2**2 + 120*K2*K3*K5 + 8*K2*K4*K6 - 1468*K3**2 - 432*K4**2 - 60*K5**2 - 2*K6**2 + 4206 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}]] |
| If K is slice | False |