| Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,-1,2,1,4,4,1,0,3,2,0,1,1,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.817'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.311', '6.528', '6.536', '6.817', '6.982', '6.984', '6.1284'] |
| Outer characteristic polynomial of the knot is: t^7+102t^5+107t^3+15t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.817'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 288*K1**4*K2 - 560*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 608*K1**3*K2*K3 - 544*K1**3*K3 - 320*K1**2*K2**4 + 864*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 4880*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 5592*K1**2*K2 - 1008*K1**2*K3**2 - 5092*K1**2 + 704*K1*K2**3*K3 - 800*K1*K2**2*K3 + 96*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 7440*K1*K2*K3 - 32*K1*K3**2*K5 + 1224*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1016*K2**4 - 640*K2**2*K3**2 - 24*K2**2*K4**2 + 1072*K2**2*K4 - 3492*K2**2 + 320*K2*K3*K5 + 24*K2*K4*K6 - 32*K3**4 + 24*K3**2*K6 - 2496*K3**2 - 586*K4**2 - 68*K5**2 - 12*K6**2 + 4032 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.817'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11448', 'vk6.11745', 'vk6.12758', 'vk6.13103', 'vk6.20321', 'vk6.21663', 'vk6.27622', 'vk6.29167', 'vk6.31199', 'vk6.31540', 'vk6.32363', 'vk6.32780', 'vk6.39043', 'vk6.41303', 'vk6.45795', 'vk6.47472', 'vk6.52201', 'vk6.52464', 'vk6.53028', 'vk6.53350', 'vk6.57192', 'vk6.58405', 'vk6.61803', 'vk6.62925', 'vk6.63767', 'vk6.63879', 'vk6.64191', 'vk6.64379', 'vk6.66799', 'vk6.67668', 'vk6.69436', 'vk6.70159'] |
| 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 | O1O2O3O4U1O5U6U3U5U4O6U2 |
| R3 orbit | {'O1O2O3O4U1O5U6U3U5U4O6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U3O5U1U6U2U5O6U4 |
| Gauss code of K* | O1O2O3O4U1O5U6U5U2U4O6U3 |
| Gauss code of -K* | O1O2O3O4U2O5U1U3U6U5O6U4 |
| 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 -3 1 0 3 2 -3],[ 3 0 3 1 2 1 1],[-1 -3 0 -1 2 2 -4],[ 0 -1 1 0 2 1 -2],[-3 -2 -2 -2 0 0 -4],[-2 -1 -2 -1 0 0 -2],[ 3 -1 4 2 4 2 0]] |
| Primitive based matrix | [[ 0 3 2 1 0 -3 -3],[-3 0 0 -2 -2 -2 -4],[-2 0 0 -2 -1 -1 -2],[-1 2 2 0 -1 -3 -4],[ 0 2 1 1 0 -1 -2],[ 3 2 1 3 1 0 1],[ 3 4 2 4 2 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,0,3,3,0,2,2,2,4,2,1,1,2,1,3,4,1,2,-1] |
| Phi over symmetry | [-3,-3,0,1,2,3,-1,2,1,4,4,1,0,3,2,0,1,1,-1,0,1] |
| Phi of -K | [-3,-3,0,1,2,3,-1,2,1,4,4,1,0,3,2,0,1,1,-1,0,1] |
| Phi of K* | [-3,-2,-1,0,3,3,1,0,1,2,4,-1,1,3,4,0,0,1,1,2,-1] |
| Phi of -K* | [-3,-3,0,1,2,3,-1,2,4,2,4,1,3,1,2,1,1,2,2,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2-8w^3z+22w^2z+25w |
| Inner characteristic polynomial | t^6+70t^4+50t^2+4 |
| Outer characteristic polynomial | t^7+102t^5+107t^3+15t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 6*K1*K2 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 288*K1**4*K2 - 560*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 608*K1**3*K2*K3 - 544*K1**3*K3 - 320*K1**2*K2**4 + 864*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 4880*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 5592*K1**2*K2 - 1008*K1**2*K3**2 - 5092*K1**2 + 704*K1*K2**3*K3 - 800*K1*K2**2*K3 + 96*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 7440*K1*K2*K3 - 32*K1*K3**2*K5 + 1224*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1016*K2**4 - 640*K2**2*K3**2 - 24*K2**2*K4**2 + 1072*K2**2*K4 - 3492*K2**2 + 320*K2*K3*K5 + 24*K2*K4*K6 - 32*K3**4 + 24*K3**2*K6 - 2496*K3**2 - 586*K4**2 - 68*K5**2 - 12*K6**2 + 4032 |
| 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}, {2, 5}, {4}, {3}, {1}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |