| Min(phi) over symmetries of the knot is: [-4,0,0,0,1,3,0,1,3,3,4,0,1,0,0,1,0,1,0,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.276'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.95', '6.107', '6.276', '6.292', '6.394', '6.429', '6.463'] |
| Outer characteristic polynomial of the knot is: t^7+85t^5+94t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.276'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 96*K1**3*K3*K4 - 320*K1**2*K2*K4 + 344*K1**2*K2 - 1168*K1**2*K3**2 - 64*K1**2*K3*K5 - 240*K1**2*K4**2 - 2264*K1**2 + 96*K1*K2*K3*K4**2 - 192*K1*K2*K3*K4 + 2648*K1*K2*K3 + 2984*K1*K3*K4 + 392*K1*K4*K5 - 72*K2**2*K4**2 + 432*K2**2*K4 - 1426*K2**2 - 160*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 176*K2*K3*K5 + 104*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 96*K3**2*K4**2 + 88*K3**2*K6 - 2076*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1344*K4**2 - 172*K5**2 - 46*K6**2 - 16*K7**2 - 2*K8**2 + 2352 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.276'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71630', 'vk6.71799', 'vk6.72219', 'vk6.72357', 'vk6.73401', 'vk6.73596', 'vk6.73875', 'vk6.74278', 'vk6.74903', 'vk6.75367', 'vk6.75678', 'vk6.75880', 'vk6.76454', 'vk6.77252', 'vk6.77342', 'vk6.77590', 'vk6.77690', 'vk6.78328', 'vk6.78872', 'vk6.79322', 'vk6.80113', 'vk6.80297', 'vk6.80424', 'vk6.80784', 'vk6.82016', 'vk6.82754', 'vk6.85353', 'vk6.86696', 'vk6.86931', 'vk6.87026', 'vk6.87590', 'vk6.89468'] |
| 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 | O1O2O3O4O5U1U4O6U5U3U2U6 |
| R3 orbit | {'O1O2O3O4O5U1U4O6U5U3U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U4U3U1O6U2U5 |
| Gauss code of K* | O1O2O3O4U5U3U2U6U1O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U4U5U3U2U6 |
| 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 -4 0 0 0 1 3],[ 4 0 4 3 1 2 3],[ 0 -4 0 0 -1 1 3],[ 0 -3 0 0 -1 1 2],[ 0 -1 1 1 0 1 1],[-1 -2 -1 -1 -1 0 1],[-3 -3 -3 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 0 -4],[-3 0 -1 -1 -2 -3 -3],[-1 1 0 -1 -1 -1 -2],[ 0 1 1 0 1 1 -1],[ 0 2 1 -1 0 0 -3],[ 0 3 1 -1 0 0 -4],[ 4 3 2 1 3 4 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,0,4,1,1,2,3,3,1,1,1,2,-1,-1,1,0,3,4] |
| Phi over symmetry | [-4,0,0,0,1,3,0,1,3,3,4,0,1,0,0,1,0,1,0,2,1] |
| Phi of -K | [-4,0,0,0,1,3,0,1,3,3,4,0,1,0,0,1,0,1,0,2,1] |
| Phi of K* | [-3,-1,0,0,0,4,1,0,1,2,4,0,0,0,3,0,-1,0,-1,1,3] |
| Phi of -K* | [-4,0,0,0,1,3,1,3,4,2,3,1,1,1,1,0,1,2,1,3,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t |
| Normalized Jones-Krushkal polynomial | 5z^2+22z+25 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2-4w^3z+26w^2z+25w |
| Inner characteristic polynomial | t^6+59t^4+24t^2 |
| Outer characteristic polynomial | t^7+85t^5+94t^3+8t |
| Flat arrow polynomial | -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 96*K1**3*K3*K4 - 320*K1**2*K2*K4 + 344*K1**2*K2 - 1168*K1**2*K3**2 - 64*K1**2*K3*K5 - 240*K1**2*K4**2 - 2264*K1**2 + 96*K1*K2*K3*K4**2 - 192*K1*K2*K3*K4 + 2648*K1*K2*K3 + 2984*K1*K3*K4 + 392*K1*K4*K5 - 72*K2**2*K4**2 + 432*K2**2*K4 - 1426*K2**2 - 160*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 176*K2*K3*K5 + 104*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 96*K3**2*K4**2 + 88*K3**2*K6 - 2076*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1344*K4**2 - 172*K5**2 - 46*K6**2 - 16*K7**2 - 2*K8**2 + 2352 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {3, 5}, {4}, {2}, {1}], [{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}, {1, 3}, {2}]] |
| If K is slice | False |