| Min(phi) over symmetries of the knot is: [-3,-3,0,1,2,3,-1,1,3,4,3,1,3,3,2,1,1,1,1,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.325'] |
| Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 - 4*K2**2 + K2 + 3*K3 + 2*K4 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.325', '6.928'] |
| Outer characteristic polynomial of the knot is: t^7+78t^5+51t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.325'] |
| 2-strand cable arrow polynomial of the knot is: -1872*K1**4 + 1120*K1**3*K2*K3 + 32*K1**3*K3*K4 - 608*K1**3*K3 - 1616*K1**2*K2**2 - 1216*K1**2*K2*K4 + 3944*K1**2*K2 - 2416*K1**2*K3**2 - 32*K1**2*K3*K5 - 656*K1**2*K4**2 - 4176*K1**2 + 96*K1*K2**3*K3 - 352*K1*K2**2*K3 + 224*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 192*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6112*K1*K2*K3 - 64*K1*K2*K4*K7 + 4528*K1*K3*K4 + 736*K1*K4*K5 + 8*K1*K5*K6 - 72*K2**4 - 384*K2**2*K3**2 - 120*K2**2*K4**2 + 768*K2**2*K4 - 3122*K2**2 - 160*K2*K3**2*K4 + 320*K2*K3*K5 + 168*K2*K4*K6 + 8*K2*K5*K7 - 288*K3**4 - 272*K3**2*K4**2 + 232*K3**2*K6 - 2976*K3**2 + 176*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1706*K4**2 - 236*K5**2 - 62*K6**2 - 20*K7**2 - 4*K8**2 + 4164 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.325'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11012', 'vk6.11093', 'vk6.12178', 'vk6.12287', 'vk6.18204', 'vk6.18539', 'vk6.24662', 'vk6.25086', 'vk6.30577', 'vk6.30674', 'vk6.31847', 'vk6.31896', 'vk6.36792', 'vk6.37244', 'vk6.44033', 'vk6.44373', 'vk6.51811', 'vk6.51880', 'vk6.52675', 'vk6.52771', 'vk6.56010', 'vk6.56283', 'vk6.60549', 'vk6.60889', 'vk6.63491', 'vk6.63537', 'vk6.63969', 'vk6.64015', 'vk6.65669', 'vk6.65953', 'vk6.68715', 'vk6.68923'] |
| 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 | O1O2O3O4O5U2U5O6U1U3U6U4 |
| R3 orbit | {'O1O2O3O4O5U2U5O6U1U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U6U3U5O6U1U4 |
| Gauss code of K* | O1O2O3O4U1U5U2U4U6O5O6U3 |
| Gauss code of -K* | O1O2O3O4U2O5O6U5U1U3U6U4 |
| 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 -3 0 3 1 2],[ 3 0 -1 2 4 1 2],[ 3 1 0 2 3 1 1],[ 0 -2 -2 0 2 0 1],[-3 -4 -3 -2 0 0 0],[-1 -1 -1 0 0 0 0],[-2 -2 -1 -1 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 0 -3 -3],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 -1 -1 -2],[-1 0 0 0 0 -1 -1],[ 0 2 1 0 0 -2 -2],[ 3 3 1 1 2 0 1],[ 3 4 2 1 2 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,0,3,3,0,0,2,3,4,0,1,1,2,0,1,1,2,2,-1] |
| Phi over symmetry | [-3,-3,0,1,2,3,-1,1,3,4,3,1,3,3,2,1,1,1,1,2,1] |
| Phi of -K | [-3,-3,0,1,2,3,-1,1,3,4,3,1,3,3,2,1,1,1,1,2,1] |
| Phi of K* | [-3,-2,-1,0,3,3,1,2,1,2,3,1,1,3,4,1,3,3,1,1,-1] |
| Phi of -K* | [-3,-3,0,1,2,3,-1,2,1,2,4,2,1,1,3,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+46t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+78t^5+51t^3+8t |
| Flat arrow polynomial | -2*K1**2 - 6*K1*K2 + 3*K1 - 4*K2**2 + K2 + 3*K3 + 2*K4 + 4 |
| 2-strand cable arrow polynomial | -1872*K1**4 + 1120*K1**3*K2*K3 + 32*K1**3*K3*K4 - 608*K1**3*K3 - 1616*K1**2*K2**2 - 1216*K1**2*K2*K4 + 3944*K1**2*K2 - 2416*K1**2*K3**2 - 32*K1**2*K3*K5 - 656*K1**2*K4**2 - 4176*K1**2 + 96*K1*K2**3*K3 - 352*K1*K2**2*K3 + 224*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 192*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6112*K1*K2*K3 - 64*K1*K2*K4*K7 + 4528*K1*K3*K4 + 736*K1*K4*K5 + 8*K1*K5*K6 - 72*K2**4 - 384*K2**2*K3**2 - 120*K2**2*K4**2 + 768*K2**2*K4 - 3122*K2**2 - 160*K2*K3**2*K4 + 320*K2*K3*K5 + 168*K2*K4*K6 + 8*K2*K5*K7 - 288*K3**4 - 272*K3**2*K4**2 + 232*K3**2*K6 - 2976*K3**2 + 176*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1706*K4**2 - 236*K5**2 - 62*K6**2 - 20*K7**2 - 4*K8**2 + 4164 |
| 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}], [{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}, {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}, {2, 3}, {1}]] |
| If K is slice | False |