| Min(phi) over symmetries of the knot is: [-4,-4,1,1,2,4,0,1,2,4,3,2,3,5,4,0,1,1,1,2,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.64'] |
| Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1*K3 - K2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500'] |
| Outer characteristic polynomial of the knot is: t^7+149t^5+246t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.64'] |
| 2-strand cable arrow polynomial of the knot is: -928*K1**2*K2**2 - 288*K1**2*K2*K4 + 1752*K1**2*K2 - 128*K1**2*K3**2 - 3040*K1**2 + 256*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4688*K1*K2*K3 - 32*K1*K2*K4*K5 + 1296*K1*K3*K4 + 176*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1952*K2**2*K3**2 - 304*K2**2*K4**2 + 1008*K2**2*K4 - 72*K2**2*K6**2 - 2680*K2**2 - 320*K2*K3**2*K4 + 1512*K2*K3*K5 + 504*K2*K4*K6 + 56*K2*K6*K8 + 144*K3**2*K6 - 2272*K3**2 + 8*K4**2*K8 - 938*K4**2 - 352*K5**2 - 192*K6**2 - 16*K8**2 + 3072 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.64'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74155', 'vk6.74167', 'vk6.74751', 'vk6.74766', 'vk6.76283', 'vk6.76307', 'vk6.76821', 'vk6.76836', 'vk6.79179', 'vk6.79197', 'vk6.79644', 'vk6.79664', 'vk6.80661', 'vk6.80679', 'vk6.81039', 'vk6.81049', 'vk6.82888', 'vk6.82912', 'vk6.83105', 'vk6.83354', 'vk6.83397', 'vk6.84173', 'vk6.84279', 'vk6.85561', 'vk6.85620', 'vk6.85878', 'vk6.86201', 'vk6.86627', 'vk6.87494', 'vk6.88148', 'vk6.88583', 'vk6.89386'] |
| 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 | O1O2O3O4O5O6U2U1U5U4U6U3 |
| R3 orbit | {'O1O2O3O4O5O6U2U1U5U4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5O6U4U1U3U2U6U5 |
| Gauss code of K* | O1O2O3O4O5O6U2U1U6U4U3U5 |
| Gauss code of -K* | O1O2O3O4O5O6U2U4U3U1U6U5 |
| 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 -4 2 1 1 4],[ 4 0 0 5 3 2 4],[ 4 0 0 4 2 1 3],[-2 -5 -4 0 -1 -1 2],[-1 -3 -2 1 0 0 2],[-1 -2 -1 1 0 0 1],[-4 -4 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 4 2 1 1 -4 -4],[-4 0 -2 -1 -2 -3 -4],[-2 2 0 -1 -1 -4 -5],[-1 1 1 0 0 -1 -2],[-1 2 1 0 0 -2 -3],[ 4 3 4 1 2 0 0],[ 4 4 5 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-2,-1,-1,4,4,2,1,2,3,4,1,1,4,5,0,1,2,2,3,0] |
| Phi over symmetry | [-4,-4,1,1,2,4,0,1,2,4,3,2,3,5,4,0,1,1,1,2,2] |
| Phi of -K | [-4,-4,1,1,2,4,0,2,3,1,4,3,4,2,5,0,0,1,0,2,0] |
| Phi of K* | [-4,-2,-1,-1,4,4,0,1,2,4,5,0,0,1,2,0,2,3,3,4,0] |
| Phi of -K* | [-4,-4,1,1,2,4,0,1,2,4,3,2,3,5,4,0,1,1,1,2,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^2-2t |
| Normalized Jones-Krushkal polynomial | 9z^2+28z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+11w^3z^2+28w^2z+21w |
| Inner characteristic polynomial | t^6+95t^4+35t^2+1 |
| Outer characteristic polynomial | t^7+149t^5+246t^3+10t |
| Flat arrow polynomial | 4*K1**2*K2 - 2*K1*K3 - K2 |
| 2-strand cable arrow polynomial | -928*K1**2*K2**2 - 288*K1**2*K2*K4 + 1752*K1**2*K2 - 128*K1**2*K3**2 - 3040*K1**2 + 256*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4688*K1*K2*K3 - 32*K1*K2*K4*K5 + 1296*K1*K3*K4 + 176*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1952*K2**2*K3**2 - 304*K2**2*K4**2 + 1008*K2**2*K4 - 72*K2**2*K6**2 - 2680*K2**2 - 320*K2*K3**2*K4 + 1512*K2*K3*K5 + 504*K2*K4*K6 + 56*K2*K6*K8 + 144*K3**2*K6 - 2272*K3**2 + 8*K4**2*K8 - 938*K4**2 - 352*K5**2 - 192*K6**2 - 16*K8**2 + 3072 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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 |