| Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,1,0,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1841', '7.44957'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 8*K1*K2 + K1 + 3*K2 + 3*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1080', '6.1837', '6.1841', '6.1865'] |
| Outer characteristic polynomial of the knot is: t^5+11t^3+11t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1841'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 128*K1**4*K2**2 + 640*K1**4*K2 - 1440*K1**4 + 32*K1**3*K2*K3 + 32*K1**3*K3*K4 + 32*K1**3*K4*K5 + 256*K1**2*K2**3 - 1488*K1**2*K2**2 + 2416*K1**2*K2 - 416*K1**2*K3**2 - 208*K1**2*K4**2 - 64*K1**2*K5**2 - 1532*K1**2 + 64*K1*K2**3*K3 + 1616*K1*K2*K3 + 760*K1*K3*K4 + 320*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 312*K2**4 - 144*K2**2*K3**2 - 64*K2**2*K4**2 + 384*K2**2*K4 - 1370*K2**2 + 272*K2*K3*K5 + 56*K2*K4*K6 - 780*K3**2 - 506*K4**2 - 224*K5**2 - 30*K6**2 + 1808 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1841'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4219', 'vk6.4298', 'vk6.5480', 'vk6.5591', 'vk6.7586', 'vk6.7680', 'vk6.9088', 'vk6.9167', 'vk6.11181', 'vk6.12269', 'vk6.12376', 'vk6.19373', 'vk6.19666', 'vk6.19771', 'vk6.26157', 'vk6.26208', 'vk6.26573', 'vk6.26653', 'vk6.30771', 'vk6.31318', 'vk6.31713', 'vk6.31976', 'vk6.32476', 'vk6.32891', 'vk6.38153', 'vk6.38184', 'vk6.39080', 'vk6.41336', 'vk6.44814', 'vk6.44929', 'vk6.45836', 'vk6.48529', 'vk6.49333', 'vk6.52310', 'vk6.53154', 'vk6.58428', 'vk6.62952', 'vk6.63599', 'vk6.66309', 'vk6.66332'] |
| 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 | O1O2O3U1U4O5U3O6O4U6U5U2 |
| R3 orbit | {'O1O2O3U1U4O5U3O6O4U6U5U2', 'O1O2O3U1U4U2O5O6O4U6U3U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U2U4U5O6O5U1O4U6U3 |
| Gauss code of K* | O1O2O3U4U3U5O4O6U2O5U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5U2O4O6U5U1U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 1 1 1 0 -1],[ 2 0 2 1 1 1 0],[-1 -2 0 0 0 0 -1],[-1 -1 0 0 -1 0 -1],[-1 -1 0 1 0 -1 -1],[ 0 -1 0 0 1 0 0],[ 1 0 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 1 1 0 -2],[-1 0 1 -1 -1],[-1 -1 0 0 -1],[ 0 1 0 0 -1],[ 2 1 1 1 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-1,-1,0,2,-1,1,1,0,1,1] |
| Phi over symmetry | [-2,0,1,1,1,1,1,0,1,-1] |
| Phi of -K | [-2,0,1,1,1,2,2,0,1,-1] |
| Phi of K* | [-1,-1,0,2,-1,1,2,0,2,1] |
| Phi of -K* | [-2,0,1,1,1,1,1,0,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 11z+23 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+13w^2z+23w |
| Inner characteristic polynomial | t^4+5t^2+4 |
| Outer characteristic polynomial | t^5+11t^3+11t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 8*K1*K2 + K1 + 3*K2 + 3*K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 - 128*K1**4*K2**2 + 640*K1**4*K2 - 1440*K1**4 + 32*K1**3*K2*K3 + 32*K1**3*K3*K4 + 32*K1**3*K4*K5 + 256*K1**2*K2**3 - 1488*K1**2*K2**2 + 2416*K1**2*K2 - 416*K1**2*K3**2 - 208*K1**2*K4**2 - 64*K1**2*K5**2 - 1532*K1**2 + 64*K1*K2**3*K3 + 1616*K1*K2*K3 + 760*K1*K3*K4 + 320*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 312*K2**4 - 144*K2**2*K3**2 - 64*K2**2*K4**2 + 384*K2**2*K4 - 1370*K2**2 + 272*K2*K3*K5 + 56*K2*K4*K6 - 780*K3**2 - 506*K4**2 - 224*K5**2 - 30*K6**2 + 1808 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{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}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}]] |
| If K is slice | False |