| Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,0,2,3,3,0,1,2,3,0,1,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.720'] |
| Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384'] |
| Outer characteristic polynomial of the knot is: t^7+81t^5+95t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.720'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 192*K1**4*K2 - 3408*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 352*K1**2*K2**3 - 3968*K1**2*K2**2 - 416*K1**2*K2*K4 + 7816*K1**2*K2 - 1552*K1**2*K3**2 - 128*K1**2*K3*K5 - 112*K1**2*K4**2 - 4684*K1**2 - 704*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 6992*K1*K2*K3 + 2328*K1*K3*K4 + 208*K1*K4*K5 - 640*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 1112*K2**2*K4 - 4110*K2**2 + 200*K2*K3*K5 + 8*K2*K4*K6 - 2552*K3**2 - 936*K4**2 - 116*K5**2 - 2*K6**2 + 4574 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.720'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16952', 'vk6.17195', 'vk6.20536', 'vk6.21935', 'vk6.23348', 'vk6.23643', 'vk6.27990', 'vk6.29455', 'vk6.35400', 'vk6.35821', 'vk6.39398', 'vk6.41589', 'vk6.42873', 'vk6.43152', 'vk6.45974', 'vk6.47648', 'vk6.55103', 'vk6.55360', 'vk6.57412', 'vk6.58585', 'vk6.59501', 'vk6.59797', 'vk6.62079', 'vk6.63059', 'vk6.64952', 'vk6.65160', 'vk6.66956', 'vk6.67815', 'vk6.68241', 'vk6.68384', 'vk6.69567', 'vk6.70262'] |
| 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 | O1O2O3O4U1O5U2U6U4O6U5U3 |
| R3 orbit | {'O1O2O3O4U1O5U2U6U4O6U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U5O6U1U6U3O5U4 |
| Gauss code of K* | O1O2O3U2O4O5U6U1U5U3O6U4 |
| Gauss code of -K* | O1O2O3U4O5U1U6U3U5O6O4U2 |
| 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 2 2 -1],[ 3 0 1 3 2 2 2],[ 2 -1 0 3 1 2 1],[-2 -3 -3 0 0 1 -3],[-2 -2 -1 0 0 0 -2],[-2 -2 -2 -1 0 0 -2],[ 1 -2 -1 3 2 2 0]] |
| Primitive based matrix | [[ 0 2 2 2 -1 -2 -3],[-2 0 1 0 -3 -3 -3],[-2 -1 0 0 -2 -2 -2],[-2 0 0 0 -2 -1 -2],[ 1 3 2 2 0 -1 -2],[ 2 3 2 1 1 0 -1],[ 3 3 2 2 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-2,1,2,3,-1,0,3,3,3,0,2,2,2,2,1,2,1,2,1] |
| Phi over symmetry | [-3,-2,-1,2,2,2,0,0,2,3,3,0,1,2,3,0,1,1,-1,0,0] |
| Phi of -K | [-3,-2,-1,2,2,2,0,0,2,3,3,0,1,2,3,0,1,1,-1,0,0] |
| Phi of K* | [-2,-2,-2,1,2,3,-1,0,1,2,3,0,0,1,2,1,3,3,0,0,0] |
| Phi of -K* | [-3,-2,-1,2,2,2,1,2,2,2,3,1,1,2,3,2,2,3,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+55t^4+45t^2+4 |
| Outer characteristic polynomial | t^7+81t^5+95t^3+10t |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 + 192*K1**4*K2 - 3408*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 + 352*K1**2*K2**3 - 3968*K1**2*K2**2 - 416*K1**2*K2*K4 + 7816*K1**2*K2 - 1552*K1**2*K3**2 - 128*K1**2*K3*K5 - 112*K1**2*K4**2 - 4684*K1**2 - 704*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 6992*K1*K2*K3 + 2328*K1*K3*K4 + 208*K1*K4*K5 - 640*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 1112*K2**2*K4 - 4110*K2**2 + 200*K2*K3*K5 + 8*K2*K4*K6 - 2552*K3**2 - 936*K4**2 - 116*K5**2 - 2*K6**2 + 4574 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |