| Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,0,1,3,4,4,1,2,2,1,0,0,0,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.498'] |
| Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.166', '6.225', '6.296', '6.498'] |
| Outer characteristic polynomial of the knot is: t^7+64t^5+124t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.498'] |
| 2-strand cable arrow polynomial of the knot is: -768*K1**2*K2**4 + 2144*K1**2*K2**3 - 5728*K1**2*K2**2 - 384*K1**2*K2*K4 + 4704*K1**2*K2 - 3512*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 1792*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1408*K1*K2**2*K3 + 256*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 5080*K1*K2*K3 + 480*K1*K3*K4 + 264*K1*K4*K5 - 128*K2**6 - 32*K2**4*K4**2 + 544*K2**4*K4 - 2544*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 864*K2**2*K3**2 - 464*K2**2*K4**2 + 2152*K2**2*K4 - 160*K2**2*K5**2 - 8*K2**2*K6**2 - 1712*K2**2 - 32*K2*K3**2*K4 + 720*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 1312*K3**2 - 526*K4**2 - 192*K5**2 - 8*K6**2 + 2660 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.498'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17025', 'vk6.17267', 'vk6.20567', 'vk6.21974', 'vk6.23448', 'vk6.23746', 'vk6.28028', 'vk6.29488', 'vk6.35520', 'vk6.35968', 'vk6.39438', 'vk6.41638', 'vk6.42936', 'vk6.43231', 'vk6.46022', 'vk6.47690', 'vk6.55207', 'vk6.55442', 'vk6.57438', 'vk6.58608', 'vk6.59602', 'vk6.59922', 'vk6.62108', 'vk6.63080', 'vk6.65009', 'vk6.65214', 'vk6.66974', 'vk6.67838', 'vk6.68284', 'vk6.68436', 'vk6.69589', 'vk6.70282'] |
| 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 | O1O2O3O4O5U1U5U4O6U2U3U6 |
| R3 orbit | {'O1O2O3O4O5U1U5U4O6U2U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U3U4O6U2U1U5 |
| Gauss code of K* | O1O2O3U4O5O6O4U1U5U6U3U2 |
| Gauss code of -K* | O1O2O3U1O4O5O6U5U4U2U3U6 |
| 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 -4 -1 1 1 1 2],[ 4 0 3 4 2 1 2],[ 1 -3 0 1 0 0 2],[-1 -4 -1 0 0 0 1],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[-2 -2 -2 -1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 1 -1 -4],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[-1 1 0 0 0 -1 -4],[ 1 2 0 0 1 0 -3],[ 4 2 1 2 4 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,-1,1,4,0,0,1,2,2,0,0,0,1,0,0,2,1,4,3] |
| Phi over symmetry | [-4,-1,1,1,1,2,0,1,3,4,4,1,2,2,1,0,0,0,0,1,1] |
| Phi of -K | [-4,-1,1,1,1,2,0,1,3,4,4,1,2,2,1,0,0,0,0,1,1] |
| Phi of K* | [-2,-1,-1,-1,1,4,0,1,1,1,4,0,0,1,1,0,2,3,2,4,0] |
| Phi of -K* | [-4,-1,1,1,1,2,3,1,2,4,2,0,0,1,2,0,0,0,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+10w^3z^2-8w^3z+27w^2z+15w |
| Inner characteristic polynomial | t^6+40t^4+39t^2 |
| Outer characteristic polynomial | t^7+64t^5+124t^3+11t |
| Flat arrow polynomial | 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 + K2 + 2 |
| 2-strand cable arrow polynomial | -768*K1**2*K2**4 + 2144*K1**2*K2**3 - 5728*K1**2*K2**2 - 384*K1**2*K2*K4 + 4704*K1**2*K2 - 3512*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 + 1792*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1408*K1*K2**2*K3 + 256*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 5080*K1*K2*K3 + 480*K1*K3*K4 + 264*K1*K4*K5 - 128*K2**6 - 32*K2**4*K4**2 + 544*K2**4*K4 - 2544*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 864*K2**2*K3**2 - 464*K2**2*K4**2 + 2152*K2**2*K4 - 160*K2**2*K5**2 - 8*K2**2*K6**2 - 1712*K2**2 - 32*K2*K3**2*K4 + 720*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 1312*K3**2 - 526*K4**2 - 192*K5**2 - 8*K6**2 + 2660 |
| 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}]] |
| If K is slice | False |