| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,4,2,0,0,2,1,-1,0,1,-1,2,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.1083'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.573', '6.628', '6.703', '6.704', '6.723', '6.796', '6.1083', '6.1099', '6.1315'] |
| Outer characteristic polynomial of the knot is: t^7+52t^5+88t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1083'] |
| 2-strand cable arrow polynomial of the knot is: -752*K1**4 - 1792*K1**2*K2**4 + 4416*K1**2*K2**3 - 8544*K1**2*K2**2 - 32*K1**2*K2*K4 + 7144*K1**2*K2 - 48*K1**2*K3**2 - 4196*K1**2 + 1664*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 192*K1*K2**2*K5 + 5360*K1*K2*K3 + 184*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 3496*K2**4 - 336*K2**2*K3**2 - 8*K2**2*K4**2 + 2088*K2**2*K4 - 1192*K2**2 + 72*K2*K3*K5 - 1064*K3**2 - 230*K4**2 - 4*K5**2 + 2964 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1083'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11410', 'vk6.11691', 'vk6.12708', 'vk6.13067', 'vk6.20268', 'vk6.21587', 'vk6.27532', 'vk6.29106', 'vk6.31143', 'vk6.31466', 'vk6.32289', 'vk6.32736', 'vk6.38935', 'vk6.41160', 'vk6.45697', 'vk6.47411', 'vk6.52161', 'vk6.52388', 'vk6.52974', 'vk6.53308', 'vk6.57101', 'vk6.58273', 'vk6.61680', 'vk6.62839', 'vk6.63739', 'vk6.63837', 'vk6.64157', 'vk6.64355', 'vk6.66736', 'vk6.67610', 'vk6.69388', 'vk6.70120'] |
| 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 | O1O2O3O4U1U2O5U3U4O6U5U6 |
| R3 orbit | {'O1O2O3O4U1U2O5U3U4O6U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U6O5U1U2O6U3U4 |
| Gauss code of K* | O1O2U3O4O5U6O3O6U1U2U4U5 |
| Gauss code of -K* | O1O2U1O3O4U2O5O6U3U4U5U6 |
| 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 -3 -1 0 2 1 1],[ 3 0 1 2 3 2 0],[ 1 -1 0 1 2 2 0],[ 0 -2 -1 0 1 2 1],[-2 -3 -2 -1 0 1 1],[-1 -2 -2 -2 -1 0 1],[-1 0 0 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 1 1 -1 -2 -3],[-1 -1 0 1 -2 -2 -2],[-1 -1 -1 0 -1 0 0],[ 0 1 2 1 0 -1 -2],[ 1 2 2 0 1 0 -1],[ 3 3 2 0 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,-1,-1,1,2,3,-1,2,2,2,1,0,0,1,2,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,2,4,2,0,0,2,1,-1,0,1,-1,2,2] |
| Phi of -K | [-3,-1,0,1,1,2,1,1,2,4,2,0,0,2,1,-1,0,1,-1,2,2] |
| Phi of K* | [-2,-1,-1,0,1,3,2,2,1,1,2,-1,0,2,4,-1,0,2,0,1,1] |
| Phi of -K* | [-3,-1,0,1,1,2,1,2,0,2,3,1,0,2,2,1,2,1,-1,-1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+8w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+36t^4+13t^2+1 |
| Outer characteristic polynomial | t^7+52t^5+88t^3+12t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -752*K1**4 - 1792*K1**2*K2**4 + 4416*K1**2*K2**3 - 8544*K1**2*K2**2 - 32*K1**2*K2*K4 + 7144*K1**2*K2 - 48*K1**2*K3**2 - 4196*K1**2 + 1664*K1*K2**3*K3 - 1888*K1*K2**2*K3 - 192*K1*K2**2*K5 + 5360*K1*K2*K3 + 184*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 160*K2**4*K4 - 3496*K2**4 - 336*K2**2*K3**2 - 8*K2**2*K4**2 + 2088*K2**2*K4 - 1192*K2**2 + 72*K2*K3*K5 - 1064*K3**2 - 230*K4**2 - 4*K5**2 + 2964 |
| 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}], [{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 |