| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,0,3,2,3,0,1,1,1,2,2,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.795'] |
| Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369'] |
| Outer characteristic polynomial of the knot is: t^7+56t^5+77t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.795'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**6 + 864*K1**4*K2 - 3648*K1**4 - 576*K1**3*K3 + 736*K1**2*K2**3 - 4016*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 7032*K1**2*K2 - 1056*K1**2*K3**2 - 160*K1**2*K3*K5 - 144*K1**2*K4**2 - 3356*K1**2 - 608*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 5464*K1*K2*K3 + 1960*K1*K3*K4 + 368*K1*K4*K5 + 24*K1*K5*K6 - 688*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 1040*K2**2*K4 - 3094*K2**2 + 360*K2*K3*K5 + 16*K2*K4*K6 - 1884*K3**2 - 872*K4**2 - 224*K5**2 - 18*K6**2 + 3622 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.795'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11472', 'vk6.11776', 'vk6.12793', 'vk6.13129', 'vk6.17037', 'vk6.17278', 'vk6.20868', 'vk6.20954', 'vk6.22275', 'vk6.22364', 'vk6.23760', 'vk6.28338', 'vk6.31226', 'vk6.31575', 'vk6.32801', 'vk6.35540', 'vk6.35989', 'vk6.39970', 'vk6.40118', 'vk6.42041', 'vk6.42949', 'vk6.43242', 'vk6.46507', 'vk6.46634', 'vk6.52225', 'vk6.53064', 'vk6.53379', 'vk6.55456', 'vk6.58857', 'vk6.59937', 'vk6.64399', 'vk6.69729'] |
| 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 | O1O2O3O4U5O6U2U6U3O5U1U4 |
| R3 orbit | {'O1O2O3O4U5O6U2U6U3O5U1U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U4O5U2U6U3O6U5 |
| Gauss code of K* | O1O2O3U4O5O6U5U1U3U6O4U2 |
| Gauss code of -K* | O1O2O3U2O4U5U1U3U6O5O6U4 |
| 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 -2 1 3 -2 1],[ 1 0 -1 2 3 -1 1],[ 2 1 0 2 2 0 1],[-1 -2 -2 0 0 -1 0],[-3 -3 -2 0 0 -3 0],[ 2 1 0 1 3 0 1],[-1 -1 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 -1 -2 -2],[-3 0 0 0 -3 -2 -3],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -2 -2 -1],[ 1 3 1 2 0 -1 -1],[ 2 2 1 2 1 0 0],[ 2 3 1 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,1,2,2,0,0,3,2,3,0,1,1,1,2,2,1,1,1,0] |
| Phi over symmetry | [-3,-1,-1,1,2,2,0,0,3,2,3,0,1,1,1,2,2,1,1,1,0] |
| Phi of -K | [-2,-2,-1,1,1,3,0,0,1,2,3,0,2,2,2,0,1,1,0,2,2] |
| Phi of K* | [-3,-1,-1,1,2,2,2,2,1,2,3,0,0,2,1,1,2,2,0,0,0] |
| Phi of -K* | [-2,-2,-1,1,1,3,0,1,1,1,3,1,1,2,2,1,2,3,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+36t^4+35t^2+1 |
| Outer characteristic polynomial | t^7+56t^5+77t^3+6t |
| Flat arrow polynomial | -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -320*K1**6 + 864*K1**4*K2 - 3648*K1**4 - 576*K1**3*K3 + 736*K1**2*K2**3 - 4016*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 7032*K1**2*K2 - 1056*K1**2*K3**2 - 160*K1**2*K3*K5 - 144*K1**2*K4**2 - 3356*K1**2 - 608*K1*K2**2*K3 - 192*K1*K2*K3*K4 + 5464*K1*K2*K3 + 1960*K1*K3*K4 + 368*K1*K4*K5 + 24*K1*K5*K6 - 688*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 1040*K2**2*K4 - 3094*K2**2 + 360*K2*K3*K5 + 16*K2*K4*K6 - 1884*K3**2 - 872*K4**2 - 224*K5**2 - 18*K6**2 + 3622 |
| 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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |