| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,2,3,-1,1,1,1,1,1,0,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.825'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+58t^5+124t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.825'] |
| 2-strand cable arrow polynomial of the knot is: -576*K1**4*K2**2 + 1152*K1**4*K2 - 1152*K1**4 - 128*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 320*K1**3*K3 - 192*K1**2*K2**4 + 2144*K1**2*K2**3 - 8720*K1**2*K2**2 - 320*K1**2*K2*K4 + 7072*K1**2*K2 - 3928*K1**2 + 832*K1*K2**3*K3 - 736*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 5904*K1*K2*K3 + 40*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1496*K2**4 - 400*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1880*K2**2 + 64*K2*K3*K5 - 1088*K3**2 - 50*K4**2 + 2744 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.825'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4714', 'vk6.5027', 'vk6.6235', 'vk6.6689', 'vk6.8211', 'vk6.8645', 'vk6.9587', 'vk6.9918', 'vk6.20305', 'vk6.21638', 'vk6.27601', 'vk6.29153', 'vk6.39023', 'vk6.41271', 'vk6.45791', 'vk6.47468', 'vk6.48754', 'vk6.48953', 'vk6.49551', 'vk6.49767', 'vk6.50764', 'vk6.50966', 'vk6.51239', 'vk6.51448', 'vk6.57160', 'vk6.58348', 'vk6.61786', 'vk6.62905', 'vk6.66777', 'vk6.67653', 'vk6.69425', 'vk6.70147'] |
| 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 | O1O2O3O4U2O5U3U4U6U1O6U5 |
| R3 orbit | {'O1O2O3O4U2O5U3U4U6U1O6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5O6U4U6U1U2O5U3 |
| Gauss code of K* | O1O2O3O4U3O5U4U6U1U2O6U5 |
| Gauss code of -K* | O1O2O3O4U5O6U3U4U6U1O5U2 |
| 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 0 -2 -1 1 3 -1],[ 0 0 -2 0 2 2 0],[ 2 2 0 1 2 2 2],[ 1 0 -1 0 1 2 1],[-1 -2 -2 -1 0 1 -1],[-3 -2 -2 -2 -1 0 -3],[ 1 0 -2 -1 1 3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -2 -3 -2],[-1 1 0 -2 -1 -1 -2],[ 0 2 2 0 0 0 -2],[ 1 2 1 0 0 1 -1],[ 1 3 1 0 -1 0 -2],[ 2 2 2 2 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,2,2,3,2,2,1,1,2,0,0,2,-1,1,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,1,2,3,-1,1,1,1,1,1,0,-1,-1,0] |
| Phi of -K | [-2,-1,-1,0,1,3,-1,0,0,1,3,1,1,1,1,1,1,2,-1,1,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,1,1,2,3,-1,1,1,1,1,1,0,-1,-1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,1,2,2,2,2,1,0,1,2,0,1,3,2,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+42t^4+55t^2 |
| Outer characteristic polynomial | t^7+58t^5+124t^3+13t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -576*K1**4*K2**2 + 1152*K1**4*K2 - 1152*K1**4 - 128*K1**3*K2**2*K3 + 416*K1**3*K2*K3 - 320*K1**3*K3 - 192*K1**2*K2**4 + 2144*K1**2*K2**3 - 8720*K1**2*K2**2 - 320*K1**2*K2*K4 + 7072*K1**2*K2 - 3928*K1**2 + 832*K1*K2**3*K3 - 736*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 5904*K1*K2*K3 + 40*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1496*K2**4 - 400*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1880*K2**2 + 64*K2*K3*K5 - 1088*K3**2 - 50*K4**2 + 2744 |
| 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}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |