| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,2,1,2,0,0,1,0,0,1,1,1,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.878'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160'] |
| Outer characteristic polynomial of the knot is: t^7+55t^5+44t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.878'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 96*K1**3*K3*K4 - 384*K1**3*K3 + 1288*K1**2*K2 - 1072*K1**2*K3**2 - 192*K1**2*K4**2 - 32*K1**2*K4*K6 - 2256*K1**2 - 992*K1*K2*K3*K4 + 2824*K1*K2*K3 - 32*K1*K3*K4*K6 + 1896*K1*K3*K4 + 512*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 496*K2**2*K3**2 - 8*K2**2*K4**2 + 568*K2**2*K4 - 8*K2**2*K6**2 - 1778*K2**2 + 944*K2*K3*K5 - 32*K2*K4**2*K6 + 72*K2*K4*K6 + 16*K2*K6*K8 - 16*K3**2*K4**2 + 16*K3**2*K6 - 1696*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 922*K4**2 - 316*K5**2 - 38*K6**2 - 4*K7**2 - 4*K8**2 + 2124 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.878'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17121', 'vk6.17362', 'vk6.20258', 'vk6.21567', 'vk6.23521', 'vk6.23854', 'vk6.27500', 'vk6.29090', 'vk6.35678', 'vk6.36105', 'vk6.38915', 'vk6.41122', 'vk6.43029', 'vk6.43335', 'vk6.45666', 'vk6.47395', 'vk6.55274', 'vk6.55520', 'vk6.57083', 'vk6.58239', 'vk6.59695', 'vk6.60032', 'vk6.61634', 'vk6.62815', 'vk6.65079', 'vk6.65266', 'vk6.66720', 'vk6.67580', 'vk6.68333', 'vk6.68479', 'vk6.69366', 'vk6.70108'] |
| 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 | O1O2O3O4U1U5O6O5U4U3U2U6 |
| R3 orbit | {'O1O2O3O4U1U5O6O5U4U3U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U3U2U1O6O5U6U4 |
| Gauss code of K* | O1O2O3O4U5U3U2U1O5O6U4U6 |
| Gauss code of -K* | O1O2O3O4U5U1O5O6U4U3U2U6 |
| 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 0 0 0 1 2],[ 3 0 3 2 1 3 3],[ 0 -3 0 0 0 0 2],[ 0 -2 0 0 0 0 1],[ 0 -1 0 0 0 0 0],[-1 -3 0 0 0 0 2],[-2 -3 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 0 -3],[-2 0 -2 0 -1 -2 -3],[-1 2 0 0 0 0 -3],[ 0 0 0 0 0 0 -1],[ 0 1 0 0 0 0 -2],[ 0 2 0 0 0 0 -3],[ 3 3 3 1 2 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,0,3,2,0,1,2,3,0,0,0,3,0,0,1,0,2,3] |
| Phi over symmetry | [-3,0,0,0,1,2,0,1,2,1,2,0,0,1,0,0,1,1,1,2,-1] |
| Phi of -K | [-3,0,0,0,1,2,0,1,2,1,2,0,0,1,0,0,1,1,1,2,-1] |
| Phi of K* | [-2,-1,0,0,0,3,-1,0,1,2,2,1,1,1,1,0,0,0,0,1,2] |
| Phi of -K* | [-3,0,0,0,1,2,1,2,3,3,3,0,0,0,0,0,0,1,0,2,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+4w^3z^2-12w^3z+17w^2z+11w |
| Inner characteristic polynomial | t^6+41t^4+11t^2 |
| Outer characteristic polynomial | t^7+55t^5+44t^3+6t |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2 |
| 2-strand cable arrow polynomial | -144*K1**4 + 96*K1**3*K3*K4 - 384*K1**3*K3 + 1288*K1**2*K2 - 1072*K1**2*K3**2 - 192*K1**2*K4**2 - 32*K1**2*K4*K6 - 2256*K1**2 - 992*K1*K2*K3*K4 + 2824*K1*K2*K3 - 32*K1*K3*K4*K6 + 1896*K1*K3*K4 + 512*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 496*K2**2*K3**2 - 8*K2**2*K4**2 + 568*K2**2*K4 - 8*K2**2*K6**2 - 1778*K2**2 + 944*K2*K3*K5 - 32*K2*K4**2*K6 + 72*K2*K4*K6 + 16*K2*K6*K8 - 16*K3**2*K4**2 + 16*K3**2*K6 - 1696*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 922*K4**2 - 316*K5**2 - 38*K6**2 - 4*K7**2 - 4*K8**2 + 2124 |
| 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}], [{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}, {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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |