| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,1,1,3,4,1,0,1,2,0,1,1,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.603'] |
| Arrow polynomial of the knot is: 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.212', '6.358', '6.588', '6.589', '6.603', '6.990'] |
| Outer characteristic polynomial of the knot is: t^7+59t^5+33t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.603'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 736*K1**2*K2**3 - 3008*K1**2*K2**2 - 96*K1**2*K2*K4 + 3432*K1**2*K2 - 48*K1**2*K3**2 - 2368*K1**2 + 352*K1*K2**3*K3 - 832*K1*K2**2*K3 - 64*K1*K2**2*K5 + 2840*K1*K2*K3 + 232*K1*K3*K4 + 24*K1*K4*K5 - 96*K2**6 + 160*K2**4*K4 - 1112*K2**4 - 224*K2**2*K3**2 - 72*K2**2*K4**2 + 1112*K2**2*K4 - 1604*K2**2 + 112*K2*K3*K5 + 32*K2*K4*K6 - 744*K3**2 - 298*K4**2 - 40*K5**2 - 12*K6**2 + 1888 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.603'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11297', 'vk6.11377', 'vk6.12560', 'vk6.12673', 'vk6.18349', 'vk6.18686', 'vk6.24792', 'vk6.25249', 'vk6.30974', 'vk6.31102', 'vk6.31215', 'vk6.31564', 'vk6.32157', 'vk6.32278', 'vk6.32393', 'vk6.32796', 'vk6.36970', 'vk6.37426', 'vk6.39616', 'vk6.41855', 'vk6.44158', 'vk6.44478', 'vk6.46228', 'vk6.47833', 'vk6.52053', 'vk6.52494', 'vk6.52894', 'vk6.53372', 'vk6.56358', 'vk6.57602', 'vk6.60998', 'vk6.62258', 'vk6.63664', 'vk6.63711', 'vk6.64094', 'vk6.64141', 'vk6.65784', 'vk6.66042', 'vk6.68788', 'vk6.68996'] |
| 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 | O1O2O3O4U2O5U4O6U1U6U5U3 |
| R3 orbit | {'O1O2O3O4U2U3O5O6U1U6U4U5', 'O1O2O3O4U2O5U4O6U1U6U5U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U5U6U4O6U1O5U3 |
| Gauss code of K* | O1O2O3O4U1U5U4U6O5U3O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U2O6U5U1U6U4 |
| 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 -2 2 0 2 1],[ 3 0 -1 4 1 3 1],[ 2 1 0 2 1 1 0],[-2 -4 -2 0 -1 1 0],[ 0 -1 -1 1 0 1 0],[-2 -3 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 1 0 -1 -2 -4],[-2 -1 0 0 -1 -1 -3],[-1 0 0 0 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 2 2 1 0 1 0 1],[ 3 4 3 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,-1,0,1,2,4,0,1,1,3,0,0,1,1,1,-1] |
| Phi over symmetry | [-3,-2,0,1,2,2,-1,1,1,3,4,1,0,1,2,0,1,1,0,0,-1] |
| Phi of -K | [-3,-2,0,1,2,2,2,2,3,1,2,1,3,2,3,1,1,1,1,1,-1] |
| Phi of K* | [-2,-2,-1,0,2,3,-1,1,1,3,2,1,1,2,1,1,3,3,1,2,2] |
| Phi of -K* | [-3,-2,0,1,2,2,-1,1,1,3,4,1,0,1,2,0,1,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+37t^4+12t^2 |
| Outer characteristic polynomial | t^7+59t^5+33t^3+3t |
| Flat arrow polynomial | 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 352*K1**4*K2 - 528*K1**4 + 736*K1**2*K2**3 - 3008*K1**2*K2**2 - 96*K1**2*K2*K4 + 3432*K1**2*K2 - 48*K1**2*K3**2 - 2368*K1**2 + 352*K1*K2**3*K3 - 832*K1*K2**2*K3 - 64*K1*K2**2*K5 + 2840*K1*K2*K3 + 232*K1*K3*K4 + 24*K1*K4*K5 - 96*K2**6 + 160*K2**4*K4 - 1112*K2**4 - 224*K2**2*K3**2 - 72*K2**2*K4**2 + 1112*K2**2*K4 - 1604*K2**2 + 112*K2*K3*K5 + 32*K2*K4*K6 - 744*K3**2 - 298*K4**2 - 40*K5**2 - 12*K6**2 + 1888 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |