| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,0,0,0,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1513', '7.39366'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923'] |
| Outer characteristic polynomial of the knot is: t^7+26t^5+34t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1513', '7.39366'] |
| 2-strand cable arrow polynomial of the knot is: -768*K1**6 - 1344*K1**4*K2**2 + 3040*K1**4*K2 - 4144*K1**4 + 1248*K1**3*K2*K3 - 768*K1**3*K3 - 768*K1**2*K2**4 + 2656*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 7664*K1**2*K2**2 - 960*K1**2*K2*K4 + 7008*K1**2*K2 - 496*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 1296*K1**2 + 1024*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 - 320*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4992*K1*K2*K3 - 32*K1*K2*K4*K5 + 656*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 1552*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 96*K2**2*K4**2 + 1224*K2**2*K4 - 1444*K2**2 + 288*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 768*K3**2 - 252*K4**2 - 56*K5**2 - 12*K6**2 + 2034 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1513'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.57', 'vk6.112', 'vk6.211', 'vk6.258', 'vk6.297', 'vk6.683', 'vk6.1217', 'vk6.1264', 'vk6.1357', 'vk6.1404', 'vk6.1447', 'vk6.1935', 'vk6.2381', 'vk6.2445', 'vk6.2935', 'vk6.2995', 'vk6.5753', 'vk6.5786', 'vk6.7822', 'vk6.7855', 'vk6.13275', 'vk6.13308', 'vk6.14772', 'vk6.14793', 'vk6.15932', 'vk6.15951', 'vk6.18053', 'vk6.24495', 'vk6.33024', 'vk6.33385', 'vk6.43923', 'vk6.50502'] |
| 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 | O1O2O3U2O4U1U4O5O6U5U6U3 |
| R3 orbit | {'O1O2O3U2O4U1U4O5O6U5U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U5O4O5U6U3O6U2 |
| Gauss code of K* | O1O2O3U4U5U3O5U6O4O6U1U2 |
| Gauss code of -K* | O1O2O3U2U3O4O5U4O6U1U6U5 |
| 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 0 0],[ 1 0 0 1 0 0 0],[-2 -3 -1 0 0 -1 1],[-1 -1 0 0 0 0 0],[ 1 0 0 1 0 0 1],[-1 0 0 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -1 -3],[-1 -1 0 0 0 -1 0],[-1 0 0 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 1 1 0 0 0 0],[ 2 3 0 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,0,0,0,1,0,0,0] |
| Phi over symmetry | [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,0,0,0,1,0,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,1,1,2,3,1,0,2,1,2,2,2,2,0,1,2] |
| Phi of K* | [-2,-1,-1,1,1,2,1,2,2,2,1,0,2,2,2,1,2,3,0,1,1] |
| Phi of -K* | [-2,-1,-1,1,1,2,0,0,0,1,3,0,0,0,1,1,0,1,0,-1,0] |
| 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+14t^4+14t^2+1 |
| Outer characteristic polynomial | t^7+26t^5+34t^3+5t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -768*K1**6 - 1344*K1**4*K2**2 + 3040*K1**4*K2 - 4144*K1**4 + 1248*K1**3*K2*K3 - 768*K1**3*K3 - 768*K1**2*K2**4 + 2656*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 7664*K1**2*K2**2 - 960*K1**2*K2*K4 + 7008*K1**2*K2 - 496*K1**2*K3**2 - 32*K1**2*K3*K5 - 80*K1**2*K4**2 - 1296*K1**2 + 1024*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 - 320*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4992*K1*K2*K3 - 32*K1*K2*K4*K5 + 656*K1*K3*K4 + 120*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 128*K2**4*K4 - 1552*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 96*K2**2*K4**2 + 1224*K2**2*K4 - 1444*K2**2 + 288*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 768*K3**2 - 252*K4**2 - 56*K5**2 - 12*K6**2 + 2034 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | True |