| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,2,1,2,-1,-1,1,1,0,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1786'] |
| Arrow polynomial of the knot is: -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.425', '6.655', '6.755', '6.769', '6.792', '6.1240', '6.1494', '6.1522', '6.1534', '6.1587', '6.1707', '6.1746', '6.1747', '6.1786', '6.1814', '6.1828', '6.1835', '6.1854', '6.1870'] |
| Outer characteristic polynomial of the knot is: t^7+25t^5+68t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1786', '6.1918'] |
| 2-strand cable arrow polynomial of the knot is: -640*K1**6 - 256*K1**4*K2**2 + 1664*K1**4*K2 - 5584*K1**4 + 992*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1600*K1**3*K3 - 4112*K1**2*K2**2 - 1248*K1**2*K2*K4 + 11560*K1**2*K2 - 1808*K1**2*K3**2 - 560*K1**2*K4**2 - 7060*K1**2 - 384*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 8776*K1*K2*K3 + 3176*K1*K3*K4 + 536*K1*K4*K5 - 72*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 888*K2**2*K4 - 5948*K2**2 + 96*K2*K3*K5 + 32*K2*K4*K6 - 3264*K3**2 - 1230*K4**2 - 124*K5**2 - 4*K6**2 + 6380 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1786'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4776', 'vk6.4790', 'vk6.5111', 'vk6.5125', 'vk6.6345', 'vk6.6773', 'vk6.6783', 'vk6.8303', 'vk6.8309', 'vk6.8753', 'vk6.9677', 'vk6.9679', 'vk6.9986', 'vk6.9988', 'vk6.21012', 'vk6.21023', 'vk6.22434', 'vk6.22447', 'vk6.28463', 'vk6.40240', 'vk6.40243', 'vk6.42168', 'vk6.46738', 'vk6.46741', 'vk6.48800', 'vk6.49015', 'vk6.49045', 'vk6.49835', 'vk6.49861', 'vk6.51502', 'vk6.58958', 'vk6.69798'] |
| 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 | O1O2O3U4U5O6O4U6U2O5U1U3 |
| R3 orbit | {'O1O2O3U4U5O6O4U6U2O5U1U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U3O4U2U5O6O5U4U6 |
| Gauss code of K* | O1O2U3O4O5U4U2U5O6O3U1U6 |
| Gauss code of -K* | O1O2U3O4O5U6U5O3O6U1U4U2 |
| 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 0 2 0 0 -1],[ 1 0 1 2 1 1 -1],[ 0 -1 0 0 1 1 -1],[-2 -2 0 0 -2 -1 -1],[ 0 -1 -1 2 0 0 -1],[ 0 -1 -1 1 0 0 -1],[ 1 1 1 1 1 1 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -2 -1 -2],[ 0 0 0 1 1 -1 -1],[ 0 1 -1 0 0 -1 -1],[ 0 2 -1 0 0 -1 -1],[ 1 1 1 1 1 0 1],[ 1 2 1 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,1,2,1,2,-1,-1,1,1,0,1,1,1,1,-1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,2,1,2,-1,-1,1,1,0,1,1,1,1,-1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,0,0,2,0,0,0,1,-1,-1,2,0,0,1] |
| Phi of K* | [-2,0,0,0,1,1,0,1,2,1,2,0,-1,0,0,-1,0,0,0,0,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,1,1,1,2,1,1,1,1,-1,0,1,1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 21z+43 |
| Enhanced Jones-Krushkal polynomial | 21w^2z+43w |
| Inner characteristic polynomial | t^6+19t^4+45t^2+1 |
| Outer characteristic polynomial | t^7+25t^5+68t^3+5t |
| Flat arrow polynomial | -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| 2-strand cable arrow polynomial | -640*K1**6 - 256*K1**4*K2**2 + 1664*K1**4*K2 - 5584*K1**4 + 992*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1600*K1**3*K3 - 4112*K1**2*K2**2 - 1248*K1**2*K2*K4 + 11560*K1**2*K2 - 1808*K1**2*K3**2 - 560*K1**2*K4**2 - 7060*K1**2 - 384*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 8776*K1*K2*K3 + 3176*K1*K3*K4 + 536*K1*K4*K5 - 72*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 888*K2**2*K4 - 5948*K2**2 + 96*K2*K3*K5 + 32*K2*K4*K6 - 3264*K3**2 - 1230*K4**2 - 124*K5**2 - 4*K6**2 + 6380 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]] |
| If K is slice | False |