| Min(phi) over symmetries of the knot is: [-4,-3,0,1,3,3,0,1,3,3,4,1,3,2,3,1,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.147'] |
| Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + 2*K2 + 3*K3 + K4 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.147', '6.270'] |
| Outer characteristic polynomial of the knot is: t^7+124t^5+90t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.147'] |
| 2-strand cable arrow polynomial of the knot is: -1616*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 480*K1**3*K3 - 1328*K1**2*K2**2 - 448*K1**2*K2*K4 + 4336*K1**2*K2 - 2256*K1**2*K3**2 - 160*K1**2*K3*K5 - 240*K1**2*K4**2 - 4952*K1**2 + 96*K1*K2**3*K3 - 448*K1*K2**2*K3 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 6744*K1*K2*K3 - 32*K1*K2*K4*K5 + 32*K1*K3**3*K4 + 3784*K1*K3*K4 + 408*K1*K4*K5 + 8*K1*K5*K6 - 112*K2**4 - 496*K2**2*K3**2 - 56*K2**2*K4**2 + 608*K2**2*K4 - 3706*K2**2 - 128*K2*K3**2*K4 + 512*K2*K3*K5 + 96*K2*K4*K6 - 288*K3**4 - 128*K3**2*K4**2 + 208*K3**2*K6 - 3256*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1396*K4**2 - 224*K5**2 - 38*K6**2 - 8*K7**2 - 2*K8**2 + 4604 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.147'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81568', 'vk6.81582', 'vk6.81648', 'vk6.81662', 'vk6.81727', 'vk6.81739', 'vk6.81860', 'vk6.81867', 'vk6.82232', 'vk6.82246', 'vk6.82386', 'vk6.82396', 'vk6.82500', 'vk6.82510', 'vk6.82575', 'vk6.82581', 'vk6.83168', 'vk6.83185', 'vk6.83590', 'vk6.83597', 'vk6.84134', 'vk6.84146', 'vk6.84335', 'vk6.84349', 'vk6.84566', 'vk6.84571', 'vk6.86473', 'vk6.86479', 'vk6.88735', 'vk6.88745', 'vk6.88903', 'vk6.88918'] |
| 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 | O1O2O3O4O5U1O6U2U5U3U6U4 |
| R3 orbit | {'O1O2O3O4O5U1O6U2U5U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U6U3U1U4O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U1U3U5U2O6U4 |
| Gauss code of -K* | O1O2O3O4O5U2O6U4U1U3U5U6 |
| 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 -3 0 3 1 3],[ 4 0 1 3 4 2 3],[ 3 -1 0 2 4 1 3],[ 0 -3 -2 0 2 0 2],[-3 -4 -4 -2 0 -1 1],[-1 -2 -1 0 1 0 1],[-3 -3 -3 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 3 1 0 -3 -4],[-3 0 1 -1 -2 -4 -4],[-3 -1 0 -1 -2 -3 -3],[-1 1 1 0 0 -1 -2],[ 0 2 2 0 0 -2 -3],[ 3 4 3 1 2 0 -1],[ 4 4 3 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,-1,0,3,4,-1,1,2,4,4,1,2,3,3,0,1,2,2,3,1] |
| Phi over symmetry | [-4,-3,0,1,3,3,0,1,3,3,4,1,3,2,3,1,1,1,1,1,-1] |
| Phi of -K | [-4,-3,0,1,3,3,0,1,3,3,4,1,3,2,3,1,1,1,1,1,-1] |
| Phi of K* | [-3,-3,-1,0,3,4,-1,1,1,3,4,1,1,2,3,1,3,3,1,1,0] |
| Phi of -K* | [-4,-3,0,1,3,3,1,3,2,3,4,2,1,3,4,0,2,2,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+80t^4+24t^2 |
| Outer characteristic polynomial | t^7+124t^5+90t^3+7t |
| Flat arrow polynomial | -4*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + 2*K2 + 3*K3 + K4 + 4 |
| 2-strand cable arrow polynomial | -1616*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 480*K1**3*K3 - 1328*K1**2*K2**2 - 448*K1**2*K2*K4 + 4336*K1**2*K2 - 2256*K1**2*K3**2 - 160*K1**2*K3*K5 - 240*K1**2*K4**2 - 4952*K1**2 + 96*K1*K2**3*K3 - 448*K1*K2**2*K3 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 6744*K1*K2*K3 - 32*K1*K2*K4*K5 + 32*K1*K3**3*K4 + 3784*K1*K3*K4 + 408*K1*K4*K5 + 8*K1*K5*K6 - 112*K2**4 - 496*K2**2*K3**2 - 56*K2**2*K4**2 + 608*K2**2*K4 - 3706*K2**2 - 128*K2*K3**2*K4 + 512*K2*K3*K5 + 96*K2*K4*K6 - 288*K3**4 - 128*K3**2*K4**2 + 208*K3**2*K6 - 3256*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1396*K4**2 - 224*K5**2 - 38*K6**2 - 8*K7**2 - 2*K8**2 + 4604 |
| 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}, {1, 5}, {4}, {3}, {2}], [{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}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |