| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,1,1,0,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1801'] |
| Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+30t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1801'] |
| 2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 960*K1**4*K2**2 + 2304*K1**4*K2 - 3840*K1**4 - 256*K1**3*K2**2*K3 + 576*K1**3*K2*K3 + 32*K1**3*K3*K4 - 608*K1**3*K3 - 448*K1**2*K2**4 + 3104*K1**2*K2**3 - 9536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 10152*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 5116*K1**2 + 768*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6928*K1*K2*K3 + 864*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2136*K2**4 - 400*K2**2*K3**2 - 48*K2**2*K4**2 + 1496*K2**2*K4 - 3166*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1572*K3**2 - 486*K4**2 - 48*K5**2 - 2*K6**2 + 4292 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1801'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16542', 'vk6.16635', 'vk6.17519', 'vk6.17576', 'vk6.18866', 'vk6.18942', 'vk6.19197', 'vk6.19490', 'vk6.23067', 'vk6.24122', 'vk6.25492', 'vk6.25565', 'vk6.26006', 'vk6.26390', 'vk6.34936', 'vk6.35054', 'vk6.36300', 'vk6.36369', 'vk6.37597', 'vk6.37684', 'vk6.42510', 'vk6.42621', 'vk6.43481', 'vk6.44595', 'vk6.54788', 'vk6.54876', 'vk6.56443', 'vk6.56545', 'vk6.59300', 'vk6.60182', 'vk6.66105', 'vk6.66145'] |
| 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 | O1O2O3U4U3O5O6U1U6O4U2U5 |
| R3 orbit | {'O1O2O3U4U3O5O6U1U6O4U2U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O5U6U3O6O4U1U5 |
| Gauss code of K* | O1O2U3O4O5U1U4U6O3O6U5U2 |
| Gauss code of -K* | O1O2U3O4O5U4U1O6O3U6U2U5 |
| 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 1 -1 1 1],[ 2 0 1 0 0 2 1],[ 0 -1 0 1 -1 0 0],[-1 0 -1 0 -1 -1 0],[ 1 0 1 1 0 1 1],[-1 -2 0 1 -1 0 0],[-1 -1 0 0 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 -1 -1 0],[-1 0 0 0 0 -1 -1],[ 0 0 1 0 0 -1 -1],[ 1 1 1 1 1 0 0],[ 2 2 0 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,0,0,1,1,1,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,1,1,0,0,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,1,2,3,0,1,1,1,1,1,0,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,0,1,3,0,1,1,1,1,1,2,0,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,1,1,0,0,0,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+19z+31 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2-4w^3z+23w^2z+31w |
| Inner characteristic polynomial | t^6+12t^4+13t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+30t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8 |
| 2-strand cable arrow polynomial | 256*K1**4*K2**3 - 960*K1**4*K2**2 + 2304*K1**4*K2 - 3840*K1**4 - 256*K1**3*K2**2*K3 + 576*K1**3*K2*K3 + 32*K1**3*K3*K4 - 608*K1**3*K3 - 448*K1**2*K2**4 + 3104*K1**2*K2**3 - 9536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 10152*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 5116*K1**2 + 768*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6928*K1*K2*K3 + 864*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2136*K2**4 - 400*K2**2*K3**2 - 48*K2**2*K4**2 + 1496*K2**2*K4 - 3166*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 1572*K3**2 - 486*K4**2 - 48*K5**2 - 2*K6**2 + 4292 |
| 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}], [{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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |