| Min(phi) over symmetries of the knot is: [-3,-3,-1,2,2,3,0,0,2,3,2,0,3,4,3,1,1,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.112'] |
| 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+104t^5+92t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.112'] |
| 2-strand cable arrow polynomial of the knot is: -768*K1**2*K2**2 + 1192*K1**2*K2 - 1176*K1**2 - 96*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 1128*K1*K2*K3 + 224*K1*K3*K4 + 152*K1*K4*K5 + 32*K1*K5*K6 - 128*K2**4 - 288*K2**2*K3**2 - 168*K2**2*K4**2 + 504*K2**2*K4 - 1062*K2**2 + 424*K2*K3*K5 + 176*K2*K4*K6 + 8*K3**2*K6 - 480*K3**2 - 348*K4**2 - 192*K5**2 - 58*K6**2 + 1082 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.112'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11513', 'vk6.11840', 'vk6.12855', 'vk6.13168', 'vk6.18122', 'vk6.18458', 'vk6.22048', 'vk6.22688', 'vk6.22787', 'vk6.24579', 'vk6.28102', 'vk6.29545', 'vk6.31281', 'vk6.31664', 'vk6.34652', 'vk6.34740', 'vk6.36712', 'vk6.39516', 'vk6.41739', 'vk6.42316', 'vk6.42363', 'vk6.43984', 'vk6.46123', 'vk6.47768', 'vk6.52280', 'vk6.52540', 'vk6.54627', 'vk6.54653', 'vk6.55932', 'vk6.58687', 'vk6.59171', 'vk6.60466', 'vk6.62190', 'vk6.63140', 'vk6.63795', 'vk6.63921', 'vk6.64697', 'vk6.65580', 'vk6.68027', 'vk6.68657'] |
| 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 | O1O2O3O4O5O6U3U2U6U5U1U4 |
| R3 orbit | {'O1O2O3O4O5O6U3U2U6U5U1U4', 'O1O2O3O4O5U2U1U5U6U3O6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U3U6U2U1U5U4 |
| Gauss code of K* | O1O2O3O4O5O6U5U2U1U6U4U3 |
| Gauss code of -K* | O1O2O3O4O5O6U4U3U1U6U5U2 |
| 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 -3 -3 3 2 2],[ 1 0 -2 -2 3 2 2],[ 3 2 0 0 4 3 2],[ 3 2 0 0 3 2 1],[-3 -3 -4 -3 0 0 0],[-2 -2 -3 -2 0 0 0],[-2 -2 -2 -1 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 2 -1 -3 -3],[-3 0 0 0 -3 -3 -4],[-2 0 0 0 -2 -1 -2],[-2 0 0 0 -2 -2 -3],[ 1 3 2 2 0 -2 -2],[ 3 3 1 2 2 0 0],[ 3 4 2 3 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-2,1,3,3,0,0,3,3,4,0,2,1,2,2,2,3,2,2,0] |
| Phi over symmetry | [-3,-3,-1,2,2,3,0,0,2,3,2,0,3,4,3,1,1,1,0,1,1] |
| Phi of -K | [-3,-3,-1,2,2,3,0,0,2,3,2,0,3,4,3,1,1,1,0,1,1] |
| Phi of K* | [-3,-2,-2,1,3,3,1,1,1,2,3,0,1,2,3,1,3,4,0,0,0] |
| Phi of -K* | [-3,-3,-1,2,2,3,0,2,1,2,3,2,2,3,4,2,2,3,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+19w^2z+15w |
| Inner characteristic polynomial | t^6+68t^4+40t^2+1 |
| Outer characteristic polynomial | t^7+104t^5+92t^3+4t |
| Flat arrow polynomial | -2*K1*K2 + K1 + K3 + 1 |
| 2-strand cable arrow polynomial | -768*K1**2*K2**2 + 1192*K1**2*K2 - 1176*K1**2 - 96*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 1128*K1*K2*K3 + 224*K1*K3*K4 + 152*K1*K4*K5 + 32*K1*K5*K6 - 128*K2**4 - 288*K2**2*K3**2 - 168*K2**2*K4**2 + 504*K2**2*K4 - 1062*K2**2 + 424*K2*K3*K5 + 176*K2*K4*K6 + 8*K3**2*K6 - 480*K3**2 - 348*K4**2 - 192*K5**2 - 58*K6**2 + 1082 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {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}, {4}, {2, 3}, {1}]] |
| If K is slice | False |