| Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,1,2,2,3,1,1,1,1,0,1,0,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.238'] |
| Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376'] |
| Outer characteristic polynomial of the knot is: t^7+45t^5+58t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.238'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 608*K1**4*K2 - 1888*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 512*K1**3*K3 - 448*K1**2*K2**4 + 2432*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 - 10560*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 10944*K1**2*K2 - 576*K1**2*K3**2 - 32*K1**2*K4**2 - 7200*K1**2 + 2304*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 - 96*K1*K2**2*K5 + 128*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 9336*K1*K2*K3 + 952*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 2544*K2**4 - 1536*K2**2*K3**2 - 200*K2**2*K4**2 + 1768*K2**2*K4 - 3926*K2**2 + 488*K2*K3*K5 + 48*K2*K4*K6 - 2388*K3**2 - 528*K4**2 - 68*K5**2 - 2*K6**2 + 5350 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.238'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3610', 'vk6.3677', 'vk6.3868', 'vk6.3993', 'vk6.7032', 'vk6.7065', 'vk6.7240', 'vk6.7363', 'vk6.17706', 'vk6.17755', 'vk6.24253', 'vk6.24314', 'vk6.36556', 'vk6.36633', 'vk6.43662', 'vk6.43769', 'vk6.48246', 'vk6.48317', 'vk6.48400', 'vk6.48425', 'vk6.50002', 'vk6.50035', 'vk6.50118', 'vk6.50145', 'vk6.55738', 'vk6.55795', 'vk6.60310', 'vk6.60393', 'vk6.65438', 'vk6.65467', 'vk6.68566', 'vk6.68595'] |
| 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 | O1O2O3O4O5U3O6U5U6U1U2U4 |
| R3 orbit | {'O1O2O3O4O5U3O6U5U6U1U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U4U5U6U1O6U3 |
| Gauss code of K* | O1O2O3O4O5U3U4U6U5U1O6U2 |
| Gauss code of -K* | O1O2O3O4O5U4O6U5U1U6U2U3 |
| 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 -2 0 -2 3 0 1],[ 2 0 1 -1 3 0 1],[ 0 -1 0 -1 2 0 1],[ 2 1 1 0 2 1 1],[-3 -3 -2 -2 0 -1 1],[ 0 0 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -2 -2],[-3 0 1 -1 -2 -2 -3],[-1 -1 0 -1 -1 -1 -1],[ 0 1 1 0 0 -1 0],[ 0 2 1 0 0 -1 -1],[ 2 2 1 1 1 0 1],[ 2 3 1 0 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,2,2,-1,1,2,2,3,1,1,1,1,0,1,0,1,1,-1] |
| Phi over symmetry | [-3,-1,0,0,2,2,-1,1,2,2,3,1,1,1,1,0,1,0,1,1,-1] |
| Phi of -K | [-2,-2,0,0,1,3,-1,1,1,2,3,1,2,2,2,0,0,1,0,2,3] |
| Phi of K* | [-3,-1,0,0,2,2,3,1,2,2,3,0,0,2,2,0,1,1,2,1,-1] |
| Phi of -K* | [-2,-2,0,0,1,3,-1,0,1,1,3,1,1,1,2,0,1,1,1,2,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+27t^4+20t^2 |
| Outer characteristic polynomial | t^7+45t^5+58t^3+7t |
| Flat arrow polynomial | 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 608*K1**4*K2 - 1888*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 1088*K1**3*K2*K3 - 512*K1**3*K3 - 448*K1**2*K2**4 + 2432*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 - 10560*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 10944*K1**2*K2 - 576*K1**2*K3**2 - 32*K1**2*K4**2 - 7200*K1**2 + 2304*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 - 96*K1*K2**2*K5 + 128*K1*K2*K3**3 - 160*K1*K2*K3*K4 + 9336*K1*K2*K3 + 952*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 96*K2**4*K4 - 2544*K2**4 - 1536*K2**2*K3**2 - 200*K2**2*K4**2 + 1768*K2**2*K4 - 3926*K2**2 + 488*K2*K3*K5 + 48*K2*K4*K6 - 2388*K3**2 - 528*K4**2 - 68*K5**2 - 2*K6**2 + 5350 |
| 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}, {3, 5}, {4}, {2}, {1}], [{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}]] |
| If K is slice | False |