| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,0,1,2,2,0,0,-1,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1591'] |
| Arrow polynomial of the knot is: -14*K1**2 - 8*K1*K2 + 4*K1 + 7*K2 + 4*K3 + 8 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1591', '6.1654'] |
| Outer characteristic polynomial of the knot is: t^7+30t^5+114t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1591'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 864*K1**4*K2 - 3760*K1**4 + 960*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1376*K1**3*K3 + 128*K1**2*K2**3 - 5744*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 11944*K1**2*K2 - 1424*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 8492*K1**2 + 416*K1*K2**3*K3 - 960*K1*K2**2*K3 - 256*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 10448*K1*K2*K3 + 2264*K1*K3*K4 + 320*K1*K4*K5 - 792*K2**4 - 592*K2**2*K3**2 - 128*K2**2*K4**2 + 1560*K2**2*K4 - 6680*K2**2 + 696*K2*K3*K5 + 128*K2*K4*K6 - 3668*K3**2 - 1070*K4**2 - 248*K5**2 - 32*K6**2 + 7012 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1591'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20006', 'vk6.20080', 'vk6.21278', 'vk6.21362', 'vk6.27057', 'vk6.27145', 'vk6.28762', 'vk6.28834', 'vk6.38454', 'vk6.38538', 'vk6.40643', 'vk6.40735', 'vk6.45338', 'vk6.45438', 'vk6.47107', 'vk6.47180', 'vk6.56805', 'vk6.56901', 'vk6.57939', 'vk6.58039', 'vk6.61323', 'vk6.61431', 'vk6.62499', 'vk6.62588', 'vk6.66525', 'vk6.66601', 'vk6.67314', 'vk6.67392', 'vk6.69171', 'vk6.69253', 'vk6.69922', 'vk6.69994'] |
| 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 | O1O2O3U4O5U6U1O6O4U2U5U3 |
| R3 orbit | {'O1O2O3U4O5U6U1O6O4U2U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U2O5O6U3U6O4U5 |
| Gauss code of K* | O1O2O3U4U1U3O5U2O6O4U6U5 |
| Gauss code of -K* | O1O2O3U4U5O6O5U2O4U1U3U6 |
| 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 -1 2 0 1 -1],[ 1 0 -1 1 2 0 1],[ 1 1 0 2 1 0 1],[-2 -1 -2 0 -1 -1 -2],[ 0 -2 -1 1 0 1 -1],[-1 0 0 1 -1 0 -1],[ 1 -1 -1 2 1 1 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 -1 0 0 -1],[ 0 1 1 0 -2 -1 -1],[ 1 1 0 2 0 -1 1],[ 1 2 0 1 1 0 1],[ 1 2 1 1 -1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,1,1,1,2,2,1,0,0,1,2,1,1,1,-1,-1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,2,0,1,2,2,0,0,-1,-1,-1,1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,-1,0,2,1,-1,-1,2,2,0,1,1,0,1,0] |
| Phi of K* | [-2,-1,0,1,1,1,0,1,1,1,2,0,1,2,2,0,0,-1,-1,-1,1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,-1,1,1,2,-1,2,0,1,1,0,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | z^2+22z+41 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+22w^2z+41w |
| Inner characteristic polynomial | t^6+22t^4+69t^2+1 |
| Outer characteristic polynomial | t^7+30t^5+114t^3+8t |
| Flat arrow polynomial | -14*K1**2 - 8*K1*K2 + 4*K1 + 7*K2 + 4*K3 + 8 |
| 2-strand cable arrow polynomial | -192*K1**6 - 192*K1**4*K2**2 + 864*K1**4*K2 - 3760*K1**4 + 960*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1376*K1**3*K3 + 128*K1**2*K2**3 - 5744*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 11944*K1**2*K2 - 1424*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 8492*K1**2 + 416*K1*K2**3*K3 - 960*K1*K2**2*K3 - 256*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 10448*K1*K2*K3 + 2264*K1*K3*K4 + 320*K1*K4*K5 - 792*K2**4 - 592*K2**2*K3**2 - 128*K2**2*K4**2 + 1560*K2**2*K4 - 6680*K2**2 + 696*K2*K3*K5 + 128*K2*K4*K6 - 3668*K3**2 - 1070*K4**2 - 248*K5**2 - 32*K6**2 + 7012 |
| 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}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |