| Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,0,2,3,-1,1,0,0,1,1,1,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.590'] |
| Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182'] |
| Outer characteristic polynomial of the knot is: t^7+34t^5+51t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.590'] |
| 2-strand cable arrow polynomial of the knot is: -784*K1**4 + 320*K1**3*K2*K3 - 544*K1**3*K3 - 528*K1**2*K2**2 - 352*K1**2*K2*K4 + 2488*K1**2*K2 - 1376*K1**2*K3**2 - 3344*K1**2 - 192*K1*K2**2*K3 + 32*K1*K2*K3**3 + 4560*K1*K2*K3 - 32*K1*K3**2*K5 + 1968*K1*K3*K4 + 8*K1*K5*K6 - 16*K2**4 - 80*K2**2*K3**2 - 24*K2**2*K4**2 + 448*K2**2*K4 - 2662*K2**2 + 144*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2236*K3**2 - 776*K4**2 - 52*K5**2 - 18*K6**2 + 3006 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.590'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3624', 'vk6.3705', 'vk6.3896', 'vk6.4007', 'vk6.7050', 'vk6.7101', 'vk6.7276', 'vk6.7381', 'vk6.11408', 'vk6.12591', 'vk6.12702', 'vk6.19112', 'vk6.19159', 'vk6.19804', 'vk6.25725', 'vk6.25786', 'vk6.26239', 'vk6.26682', 'vk6.31008', 'vk6.31135', 'vk6.32188', 'vk6.37840', 'vk6.37897', 'vk6.44960', 'vk6.48260', 'vk6.48439', 'vk6.50018', 'vk6.50162', 'vk6.52155', 'vk6.63731', 'vk6.66205', 'vk6.66234'] |
| 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 | O1O2O3O4U1O5U4O6U5U6U3U2 |
| R3 orbit | {'O1O2O3O4U1O5U4O6U5U6U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3U2U5U6O5U1O6U4 |
| Gauss code of K* | O1O2O3O4U5U4U3U6O5U1O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U4O6U5U2U1U6 |
| 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 1 0 0 1],[ 3 0 3 2 1 1 0],[-1 -3 0 0 -1 0 1],[-1 -2 0 0 -1 0 1],[ 0 -1 1 1 0 1 1],[ 0 -1 0 0 -1 0 1],[-1 0 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 -1 -2],[-1 -1 0 -1 -1 -1 0],[-1 0 1 0 0 -1 -3],[ 0 0 1 0 0 -1 -1],[ 0 1 1 1 1 0 -1],[ 3 2 0 3 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,0,3,-1,0,0,1,2,1,1,1,0,0,1,3,1,1,1] |
| Phi over symmetry | [-3,0,0,1,1,1,1,1,0,2,3,-1,1,0,0,1,1,1,-1,-1,0] |
| Phi of -K | [-3,0,0,1,1,1,2,2,1,2,4,-1,0,0,0,1,1,0,0,-1,-1] |
| Phi of K* | [-1,-1,-1,0,0,3,-1,-1,0,0,4,0,0,1,1,0,1,2,1,2,2] |
| Phi of -K* | [-3,0,0,1,1,1,1,1,0,2,3,-1,1,0,0,1,1,1,-1,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2-8w^3z+23w^2z+23w |
| Inner characteristic polynomial | t^6+22t^4+17t^2 |
| Outer characteristic polynomial | t^7+34t^5+51t^3+7t |
| Flat arrow polynomial | -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3 |
| 2-strand cable arrow polynomial | -784*K1**4 + 320*K1**3*K2*K3 - 544*K1**3*K3 - 528*K1**2*K2**2 - 352*K1**2*K2*K4 + 2488*K1**2*K2 - 1376*K1**2*K3**2 - 3344*K1**2 - 192*K1*K2**2*K3 + 32*K1*K2*K3**3 + 4560*K1*K2*K3 - 32*K1*K3**2*K5 + 1968*K1*K3*K4 + 8*K1*K5*K6 - 16*K2**4 - 80*K2**2*K3**2 - 24*K2**2*K4**2 + 448*K2**2*K4 - 2662*K2**2 + 144*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 16*K3**2*K6 - 2236*K3**2 - 776*K4**2 - 52*K5**2 - 18*K6**2 + 3006 |
| 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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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 |