| Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,0,2,3,4,0,0,1,2,1,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.368'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354'] |
| Outer characteristic polynomial of the knot is: t^7+65t^5+90t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.368'] |
| 2-strand cable arrow polynomial of the knot is: -1632*K1**2*K2**2 + 1984*K1**2*K2 - 64*K1**2*K3**2 - 2608*K1**2 + 288*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 4408*K1*K2*K3 - 64*K1*K2*K4*K5 + 592*K1*K3*K4 + 88*K1*K4*K5 + 24*K1*K5*K6 - 672*K2**4 - 1568*K2**2*K3**2 - 168*K2**2*K4**2 + 896*K2**2*K4 - 2366*K2**2 - 256*K2*K3**2*K4 + 1368*K2*K3*K5 + 296*K2*K4*K6 - 64*K3**4 + 184*K3**2*K6 - 1960*K3**2 - 524*K4**2 - 336*K5**2 - 130*K6**2 + 2666 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.368'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4229', 'vk6.4310', 'vk6.5496', 'vk6.5612', 'vk6.7597', 'vk6.7690', 'vk6.9094', 'vk6.9175', 'vk6.18379', 'vk6.18717', 'vk6.24832', 'vk6.25289', 'vk6.37020', 'vk6.37468', 'vk6.44189', 'vk6.44508', 'vk6.48541', 'vk6.48598', 'vk6.49240', 'vk6.49354', 'vk6.50324', 'vk6.50385', 'vk6.51067', 'vk6.51100', 'vk6.56160', 'vk6.56387', 'vk6.60689', 'vk6.61040', 'vk6.65828', 'vk6.66080', 'vk6.68817', 'vk6.69025'] |
| 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 | O1O2O3O4O5U3U5O6U2U1U6U4 |
| R3 orbit | {'O1O2O3O4O5U3U5O6U2U1U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U6U5U4O6U1U3 |
| Gauss code of K* | O1O2O3O4U2U1U5U4U6O5O6U3 |
| Gauss code of -K* | O1O2O3O4U2O5O6U5U1U6U4U3 |
| 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 -2 -2 3 1 2],[ 2 0 0 -1 4 1 2],[ 2 0 0 -1 3 1 1],[ 2 1 1 0 2 1 0],[-3 -4 -3 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-2 -2 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 -2 -2 -2],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 2 2 0 1 0 1 1],[ 2 3 1 1 -1 0 0],[ 2 4 2 1 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,2,2,2,0,0,2,3,4,0,0,1,2,1,1,1,-1,-1,0] |
| Phi over symmetry | [-3,-2,-1,2,2,2,0,0,2,3,4,0,0,1,2,1,1,1,-1,-1,0] |
| Phi of -K | [-2,-2,-2,1,2,3,-1,-1,2,4,3,0,2,2,1,2,3,2,1,2,1] |
| Phi of K* | [-3,-2,-1,2,2,2,1,2,1,2,3,1,2,3,4,2,2,2,0,-1,-1] |
| Phi of -K* | [-2,-2,-2,1,2,3,-1,0,1,1,3,1,1,0,2,1,2,4,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 8z^2+27z+23 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2-2w^3z+29w^2z+23w |
| Inner characteristic polynomial | t^6+39t^4+38t^2 |
| Outer characteristic polynomial | t^7+65t^5+90t^3+9t |
| Flat arrow polynomial | -2*K1*K2 + K1 + K3 + 1 |
| 2-strand cable arrow polynomial | -1632*K1**2*K2**2 + 1984*K1**2*K2 - 64*K1**2*K3**2 - 2608*K1**2 + 288*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 4408*K1*K2*K3 - 64*K1*K2*K4*K5 + 592*K1*K3*K4 + 88*K1*K4*K5 + 24*K1*K5*K6 - 672*K2**4 - 1568*K2**2*K3**2 - 168*K2**2*K4**2 + 896*K2**2*K4 - 2366*K2**2 - 256*K2*K3**2*K4 + 1368*K2*K3*K5 + 296*K2*K4*K6 - 64*K3**4 + 184*K3**2*K6 - 1960*K3**2 - 524*K4**2 - 336*K5**2 - 130*K6**2 + 2666 |
| 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}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |