| Min(phi) over symmetries of the knot is: [-4,-3,-1,2,3,3,0,2,2,4,5,1,1,2,3,1,2,3,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.78'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511'] |
| Outer characteristic polynomial of the knot is: t^7+132t^5+163t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.78'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**3*K3 - 480*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 1664*K1**2*K2 - 160*K1**2*K3**2 - 1988*K1**2 - 544*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 2144*K1*K2*K3 + 696*K1*K3*K4 + 72*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4 - 32*K2**3*K6 + 96*K2**2*K3**2*K4 - 400*K2**2*K3**2 - 64*K2**2*K3*K7 - 72*K2**2*K4**2 + 776*K2**2*K4 - 8*K2**2*K6**2 - 1938*K2**2 + 504*K2*K3*K5 + 128*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**2*K4**2 + 16*K3**2*K6 - 1048*K3**2 + 32*K3*K4*K7 - 518*K4**2 - 140*K5**2 - 62*K6**2 - 8*K7**2 - 2*K8**2 + 1750 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.78'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73325', 'vk6.73467', 'vk6.73739', 'vk6.73856', 'vk6.74571', 'vk6.74838', 'vk6.75219', 'vk6.75648', 'vk6.76047', 'vk6.76396', 'vk6.76771', 'vk6.76887', 'vk6.78210', 'vk6.78659', 'vk6.79004', 'vk6.79244', 'vk6.79571', 'vk6.79726', 'vk6.80023', 'vk6.80281', 'vk6.80536', 'vk6.80738', 'vk6.80994', 'vk6.81082', 'vk6.81623', 'vk6.81805', 'vk6.82359', 'vk6.82366', 'vk6.82737', 'vk6.84235', 'vk6.84308', 'vk6.84317', 'vk6.84377', 'vk6.84405', 'vk6.84488', 'vk6.85223', 'vk6.86756', 'vk6.87574', 'vk6.88263', 'vk6.88712'] |
| 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 | O1O2O3O4O5O6U2U4U1U6U5U3 |
| R3 orbit | {'O1O2O3O4O5U1U6U2U5U4O6U3', 'O1O2O3O4O5O6U2U4U1U6U5U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U4U2U1U6U3U5 |
| Gauss code of K* | O1O2O3O4O5O6U3U1U6U2U5U4 |
| Gauss code of -K* | O1O2O3O4O5O6U3U2U5U1U6U4 |
| 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 -4 2 -1 3 3],[ 3 0 -1 4 1 4 3],[ 4 1 0 4 1 3 2],[-2 -4 -4 0 -2 1 1],[ 1 -1 -1 2 0 2 1],[-3 -4 -3 -1 -2 0 0],[-3 -3 -2 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 3 3 2 -1 -3 -4],[-3 0 0 -1 -1 -3 -2],[-3 0 0 -1 -2 -4 -3],[-2 1 1 0 -2 -4 -4],[ 1 1 2 2 0 -1 -1],[ 3 3 4 4 1 0 -1],[ 4 2 3 4 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,-2,1,3,4,0,1,1,3,2,1,2,4,3,2,4,4,1,1,1] |
| Phi over symmetry | [-4,-3,-1,2,3,3,0,2,2,4,5,1,1,2,3,1,2,3,0,0,0] |
| Phi of -K | [-4,-3,-1,2,3,3,0,2,2,4,5,1,1,2,3,1,2,3,0,0,0] |
| Phi of K* | [-3,-3,-2,1,3,4,0,0,2,2,4,0,3,3,5,1,1,2,1,2,0] |
| Phi of -K* | [-4,-3,-1,2,3,3,1,1,4,2,3,1,4,3,4,2,1,2,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+84t^4+44t^2+1 |
| Outer characteristic polynomial | t^7+132t^5+163t^3+4t |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -128*K1**3*K3 - 480*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 1664*K1**2*K2 - 160*K1**2*K3**2 - 1988*K1**2 - 544*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 2144*K1*K2*K3 + 696*K1*K3*K4 + 72*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4 - 32*K2**3*K6 + 96*K2**2*K3**2*K4 - 400*K2**2*K3**2 - 64*K2**2*K3*K7 - 72*K2**2*K4**2 + 776*K2**2*K4 - 8*K2**2*K6**2 - 1938*K2**2 + 504*K2*K3*K5 + 128*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**2*K4**2 + 16*K3**2*K6 - 1048*K3**2 + 32*K3*K4*K7 - 518*K4**2 - 140*K5**2 - 62*K6**2 - 8*K7**2 - 2*K8**2 + 1750 |
| 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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {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 |