| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,1,2,4,0,1,0,2,0,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.896'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147'] |
| Outer characteristic polynomial of the knot is: t^7+48t^5+63t^3+13t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.896'] |
| 2-strand cable arrow polynomial of the knot is: -592*K1**4 + 544*K1**3*K2*K3 - 256*K1**3*K3 - 2656*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 3776*K1**2*K2 - 816*K1**2*K3**2 - 4060*K1**2 + 256*K1*K2**3*K3 - 608*K1*K2**2*K3 - 384*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6488*K1*K2*K3 - 64*K1*K2*K4*K5 + 1352*K1*K3*K4 + 256*K1*K4*K5 + 168*K1*K5*K6 - 640*K2**4 - 32*K2**3*K6 - 1008*K2**2*K3**2 - 8*K2**2*K4**2 + 1336*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 3974*K2**2 - 160*K2*K3**2*K4 + 1592*K2*K3*K5 + 288*K2*K4*K6 + 32*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 + 264*K3**2*K6 - 2900*K3**2 - 876*K4**2 - 596*K5**2 - 234*K6**2 - 4*K7**2 - 4*K8**2 + 4158 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.896'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3629', 'vk6.3716', 'vk6.3909', 'vk6.4014', 'vk6.7051', 'vk6.7104', 'vk6.7281', 'vk6.7384', 'vk6.11407', 'vk6.12592', 'vk6.12705', 'vk6.19111', 'vk6.19156', 'vk6.19798', 'vk6.25720', 'vk6.25779', 'vk6.26237', 'vk6.26680', 'vk6.31005', 'vk6.31134', 'vk6.32189', 'vk6.37839', 'vk6.37894', 'vk6.44966', 'vk6.48257', 'vk6.48438', 'vk6.50013', 'vk6.50158', 'vk6.52154', 'vk6.63730', 'vk6.66212', 'vk6.66239'] |
| 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 | O1O2O3O4U2U4O5O6U1U6U5U3 |
| R3 orbit | {'O1O2O3O4U2U4O5O6U1U6U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U5U6U4O6O5U1U3 |
| Gauss code of K* | O1O2O3O4U1U5U4U6O5O6U3U2 |
| Gauss code of -K* | O1O2O3O4U3U2O5O6U5U1U6U4 |
| 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 1 1 1],[ 3 0 -1 4 1 2 1],[ 2 1 0 2 1 0 0],[-2 -4 -2 0 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 1 -2 -3],[-2 0 0 0 0 -2 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[-1 0 0 0 0 -1 -1],[ 2 2 0 0 1 0 1],[ 3 4 1 2 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,-1,2,3,0,0,0,2,4,0,0,0,1,0,0,2,1,1,-1] |
| Phi over symmetry | [-3,-2,1,1,1,2,-1,1,1,2,4,0,1,0,2,0,0,0,0,0,0] |
| Phi of -K | [-3,-2,1,1,1,2,2,2,3,3,1,3,2,3,2,0,0,1,0,1,1] |
| Phi of K* | [-2,-1,-1,-1,2,3,1,1,1,2,1,0,0,2,3,0,3,2,3,3,2] |
| Phi of -K* | [-3,-2,1,1,1,2,-1,1,1,2,4,0,1,0,2,0,0,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+28t^4+29t^2 |
| Outer characteristic polynomial | t^7+48t^5+63t^3+13t |
| Flat arrow polynomial | -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1 |
| 2-strand cable arrow polynomial | -592*K1**4 + 544*K1**3*K2*K3 - 256*K1**3*K3 - 2656*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 3776*K1**2*K2 - 816*K1**2*K3**2 - 4060*K1**2 + 256*K1*K2**3*K3 - 608*K1*K2**2*K3 - 384*K1*K2**2*K5 + 32*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6488*K1*K2*K3 - 64*K1*K2*K4*K5 + 1352*K1*K3*K4 + 256*K1*K4*K5 + 168*K1*K5*K6 - 640*K2**4 - 32*K2**3*K6 - 1008*K2**2*K3**2 - 8*K2**2*K4**2 + 1336*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 3974*K2**2 - 160*K2*K3**2*K4 + 1592*K2*K3*K5 + 288*K2*K4*K6 + 32*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 + 264*K3**2*K6 - 2900*K3**2 - 876*K4**2 - 596*K5**2 - 234*K6**2 - 4*K7**2 - 4*K8**2 + 4158 |
| 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}, {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 |