| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,1,3,2,3,1,3,2,2,1,1,1,2,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.758'] |
| Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 6*K1*K2 - 3*K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.677', '6.696', '6.738', '6.758'] |
| Outer characteristic polynomial of the knot is: t^7+71t^5+78t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.758'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 176*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 1600*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2368*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6608*K1**2*K2**2 - 640*K1**2*K2*K4 + 4264*K1**2*K2 - 272*K1**2*K3**2 - 2620*K1**2 + 2016*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 + 32*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5336*K1*K2*K3 + 744*K1*K3*K4 + 8*K1*K4*K5 - 448*K2**6 + 448*K2**4*K4 - 2616*K2**4 - 704*K2**2*K3**2 - 152*K2**2*K4**2 + 1944*K2**2*K4 - 886*K2**2 + 128*K2*K3*K5 + 24*K2*K4*K6 - 1280*K3**2 - 534*K4**2 - 4*K5**2 - 2*K6**2 + 2220 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.758'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17500', 'vk6.17505', 'vk6.17557', 'vk6.17560', 'vk6.24024', 'vk6.24033', 'vk6.24095', 'vk6.24100', 'vk6.36280', 'vk6.36285', 'vk6.36346', 'vk6.36348', 'vk6.43433', 'vk6.43438', 'vk6.43463', 'vk6.43465', 'vk6.55614', 'vk6.55627', 'vk6.55645', 'vk6.55654', 'vk6.60124', 'vk6.60144', 'vk6.60161', 'vk6.60173', 'vk6.65317', 'vk6.65327', 'vk6.65350', 'vk6.65360', 'vk6.68489', 'vk6.68498', 'vk6.68513', 'vk6.68521'] |
| 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 | O1O2O3O4U2O5U6U3U4O6U1U5 |
| R3 orbit | {'O1O2O3O4U2O5U6U3U4O6U1U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4O6U1U2U6O5U3 |
| Gauss code of K* | O1O2O3U1O4O5U4U6U2U3O6U5 |
| Gauss code of -K* | O1O2O3U4O5U1U2U5U6O4O6U3 |
| 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 -2 0 2 3 -2],[ 1 0 -2 1 3 3 -1],[ 2 2 0 1 2 2 0],[ 0 -1 -1 0 1 1 -1],[-2 -3 -2 -1 0 0 -2],[-3 -3 -2 -1 0 0 -3],[ 2 1 0 1 2 3 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-2 0 0 -1 -3 -2 -2],[ 0 1 1 0 -1 -1 -1],[ 1 3 3 1 0 -2 -1],[ 2 2 2 1 2 0 0],[ 2 3 2 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,2,2,0,1,3,2,3,1,3,2,2,1,1,1,2,1,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,1,3,2,3,1,3,2,2,1,1,1,2,1,0] |
| Phi of -K | [-2,-2,-1,0,2,3,0,-1,1,2,3,0,1,2,2,0,0,1,1,2,1] |
| Phi of K* | [-3,-2,0,1,2,2,1,2,1,2,3,1,0,2,2,0,1,1,0,-1,0] |
| Phi of -K* | [-2,-2,-1,0,2,3,0,1,1,2,3,2,1,2,2,1,3,3,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | z^2+6z+9 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w |
| Inner characteristic polynomial | t^6+49t^4+39t^2+1 |
| Outer characteristic polynomial | t^7+71t^5+78t^3+11t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 6*K1*K2 - 3*K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 96*K1**4*K2 - 176*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 1600*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2368*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6608*K1**2*K2**2 - 640*K1**2*K2*K4 + 4264*K1**2*K2 - 272*K1**2*K3**2 - 2620*K1**2 + 2016*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 + 32*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5336*K1*K2*K3 + 744*K1*K3*K4 + 8*K1*K4*K5 - 448*K2**6 + 448*K2**4*K4 - 2616*K2**4 - 704*K2**2*K3**2 - 152*K2**2*K4**2 + 1944*K2**2*K4 - 886*K2**2 + 128*K2*K3*K5 + 24*K2*K4*K6 - 1280*K3**2 - 534*K4**2 - 4*K5**2 - 2*K6**2 + 2220 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |