| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,2,3,1,1,2,2,-1,-1,-1,1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.748'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.520', '6.682', '6.706', '6.748', '6.1331'] |
| Outer characteristic polynomial of the knot is: t^7+64t^5+99t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.748'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 864*K1**4*K2 - 1536*K1**4 + 192*K1**3*K2*K3 - 256*K1**3*K3 - 192*K1**2*K2**4 + 800*K1**2*K2**3 - 5712*K1**2*K2**2 - 192*K1**2*K2*K4 + 8096*K1**2*K2 - 5320*K1**2 + 288*K1*K2**3*K3 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4936*K1*K2*K3 + 144*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 560*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 576*K2**2*K4 - 3464*K2**2 + 24*K2*K3*K5 - 1096*K3**2 - 124*K4**2 + 3578 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.748'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11429', 'vk6.11724', 'vk6.12739', 'vk6.13082', 'vk6.20335', 'vk6.21677', 'vk6.27636', 'vk6.29181', 'vk6.31180', 'vk6.31521', 'vk6.32344', 'vk6.32761', 'vk6.39070', 'vk6.41328', 'vk6.45822', 'vk6.47494', 'vk6.52194', 'vk6.52451', 'vk6.53021', 'vk6.53337', 'vk6.57194', 'vk6.58410', 'vk6.61805', 'vk6.62930', 'vk6.63764', 'vk6.63874', 'vk6.64188', 'vk6.64374', 'vk6.66809', 'vk6.67678', 'vk6.69446', 'vk6.70169'] |
| 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 | O1O2O3O4U2O5U3U6U1O6U4U5 |
| R3 orbit | {'O1O2O3O4U2O5U3U6U1O6U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1O6U4U6U2O5U3 |
| Gauss code of K* | O1O2O3U2O4O5U3U6U1U4O6U5 |
| Gauss code of -K* | O1O2O3U4O5U6U3U5U1O4O6U2 |
| 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 2 3 -1],[ 1 0 -1 1 2 2 1],[ 2 1 0 1 2 2 2],[ 1 -1 -1 0 1 2 1],[-2 -2 -2 -1 0 1 -2],[-3 -2 -2 -2 -1 0 -3],[ 1 -1 -2 -1 2 3 0]] |
| Primitive based matrix | [[ 0 3 2 -1 -1 -1 -2],[-3 0 -1 -2 -2 -3 -2],[-2 1 0 -1 -2 -2 -2],[ 1 2 1 0 -1 1 -1],[ 1 2 2 1 0 1 -1],[ 1 3 2 -1 -1 0 -2],[ 2 2 2 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,1,1,1,2,1,2,2,3,2,1,2,2,2,1,-1,1,-1,1,2] |
| Phi over symmetry | [-3,-2,1,1,1,2,0,1,2,2,3,1,1,2,2,-1,-1,-1,1,0,0] |
| Phi of -K | [-2,-1,-1,-1,2,3,-1,0,0,2,3,1,1,1,1,-1,1,2,2,2,0] |
| Phi of K* | [-3,-2,1,1,1,2,0,1,2,2,3,1,1,2,2,-1,-1,-1,1,0,0] |
| Phi of -K* | [-2,-1,-1,-1,2,3,1,1,2,2,2,-1,1,1,2,1,2,2,2,3,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+3t |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-2w^3z+25w^2z+31w |
| Inner characteristic polynomial | t^6+44t^4+41t^2+1 |
| Outer characteristic polynomial | t^7+64t^5+99t^3+10t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 2*K1*K2 - 2*K1 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 864*K1**4*K2 - 1536*K1**4 + 192*K1**3*K2*K3 - 256*K1**3*K3 - 192*K1**2*K2**4 + 800*K1**2*K2**3 - 5712*K1**2*K2**2 - 192*K1**2*K2*K4 + 8096*K1**2*K2 - 5320*K1**2 + 288*K1*K2**3*K3 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4936*K1*K2*K3 + 144*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 560*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 576*K2**2*K4 - 3464*K2**2 + 24*K2*K3*K5 - 1096*K3**2 - 124*K4**2 + 3578 |
| 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}, {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}, {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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |