| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,1,2,0,1,1,1,0,1,0,1,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1240'] |
| 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+30t^5+22t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1240'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 864*K1**4*K2 - 2672*K1**4 + 224*K1**3*K2*K3 - 384*K1**3*K3 + 128*K1**2*K2**3 - 2752*K1**2*K2**2 - 160*K1**2*K2*K4 + 4760*K1**2*K2 - 272*K1**2*K3**2 - 1860*K1**2 - 288*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 3112*K1*K2*K3 + 456*K1*K3*K4 + 16*K1*K4*K5 - 360*K2**4 - 208*K2**2*K3**2 - 48*K2**2*K4**2 + 584*K2**2*K4 - 2028*K2**2 + 272*K2*K3*K5 + 32*K2*K4*K6 - 904*K3**2 - 266*K4**2 - 76*K5**2 - 4*K6**2 + 2088 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1240'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13940', 'vk6.13952', 'vk6.14034', 'vk6.14046', 'vk6.15011', 'vk6.15023', 'vk6.15131', 'vk6.15143', 'vk6.16545', 'vk6.16636', 'vk6.17450', 'vk6.17470', 'vk6.17490', 'vk6.23963', 'vk6.23983', 'vk6.23994', 'vk6.24014', 'vk6.24117', 'vk6.25999', 'vk6.26385', 'vk6.33750', 'vk6.33838', 'vk6.34945', 'vk6.35063', 'vk6.36268', 'vk6.36372', 'vk6.37591', 'vk6.37678', 'vk6.43423', 'vk6.44580', 'vk6.53876', 'vk6.54426', 'vk6.54775', 'vk6.54865', 'vk6.55603', 'vk6.56445', 'vk6.56556', 'vk6.60092', 'vk6.60112', 'vk6.60177'] |
| 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 | O1O2O3O4U3U5U1O5O6U4U2U6 |
| R3 orbit | {'O1O2O3O4U3U5U1O5O6U4U2U6', 'O1O2O3U2O4U5U1O5O6U3U4U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3U1O5O6U4U6U2 |
| Gauss code of K* | O1O2O3U2U4O5O6O4U3U6U1U5 |
| Gauss code of -K* | O1O2O3U1U4O5O4O6U3U6U2U5 |
| 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 -1 1 -1 2],[ 1 0 1 0 1 1 2],[ 0 -1 0 -1 1 0 2],[ 1 0 1 0 1 1 1],[-1 -1 -1 -1 0 -1 1],[ 1 -1 0 -1 1 0 2],[-2 -2 -2 -1 -1 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -2 -1 -2 -2],[-1 1 0 -1 -1 -1 -1],[ 0 2 1 0 -1 0 -1],[ 1 1 1 1 0 1 0],[ 1 2 1 0 -1 0 -1],[ 1 2 1 1 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,2,1,2,2,1,1,1,1,1,0,1,-1,0,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,1,2,0,1,1,1,0,1,0,1,0,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,0,1,1,1,1,1,1,0,1,2,0,0,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,0,1,1,2,0,1,1,1,0,1,0,1,0,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,-1,0,1,2,0,1,1,1,1,1,2,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+22t^4+5t^2 |
| Outer characteristic polynomial | t^7+30t^5+22t^3+3t |
| Flat arrow polynomial | -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 864*K1**4*K2 - 2672*K1**4 + 224*K1**3*K2*K3 - 384*K1**3*K3 + 128*K1**2*K2**3 - 2752*K1**2*K2**2 - 160*K1**2*K2*K4 + 4760*K1**2*K2 - 272*K1**2*K3**2 - 1860*K1**2 - 288*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 3112*K1*K2*K3 + 456*K1*K3*K4 + 16*K1*K4*K5 - 360*K2**4 - 208*K2**2*K3**2 - 48*K2**2*K4**2 + 584*K2**2*K4 - 2028*K2**2 + 272*K2*K3*K5 + 32*K2*K4*K6 - 904*K3**2 - 266*K4**2 - 76*K5**2 - 4*K6**2 + 2088 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |