| Min(phi) over symmetries of the knot is: [-3,-3,0,2,2,2,0,0,1,3,3,1,2,3,4,1,0,2,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.517'] |
| Arrow polynomial of the knot is: -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.133', '6.517', '6.545', '6.1198', '6.1251', '6.1906'] |
| Outer characteristic polynomial of the knot is: t^7+87t^5+220t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.517'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**2*K2**4 + 1632*K1**2*K2**3 - 3808*K1**2*K2**2 - 224*K1**2*K2*K4 + 3512*K1**2*K2 - 2840*K1**2 + 992*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4128*K1*K2*K3 + 328*K1*K3*K4 + 184*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1216*K2**6 - 128*K2**4*K3**2 - 192*K2**4*K4**2 + 1632*K2**4*K4 - 3328*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 896*K2**2*K3**2 - 672*K2**2*K4**2 + 2904*K2**2*K4 - 192*K2**2*K5**2 - 8*K2**2*K6**2 - 1032*K2**2 + 808*K2*K3*K5 + 112*K2*K4*K6 - 1152*K3**2 - 514*K4**2 - 200*K5**2 - 8*K6**2 + 2352 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.517'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73190', 'vk6.73206', 'vk6.73628', 'vk6.74307', 'vk6.74406', 'vk6.74949', 'vk6.75017', 'vk6.75104', 'vk6.75123', 'vk6.75560', 'vk6.75588', 'vk6.76519', 'vk6.76586', 'vk6.76930', 'vk6.78046', 'vk6.78066', 'vk6.78526', 'vk6.78554', 'vk6.79359', 'vk6.79781', 'vk6.79859', 'vk6.79966', 'vk6.80819', 'vk6.80886', 'vk6.83685', 'vk6.84712', 'vk6.84819', 'vk6.85275', 'vk6.85639', 'vk6.87702', 'vk6.88374', 'vk6.89497'] |
| 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 | O1O2O3O4O5U2U1U5O6U3U4U6 |
| R3 orbit | {'O1O2O3O4O5U2U1U5O6U3U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U2U3O6U1U5U4 |
| Gauss code of K* | O1O2O3U4O5O6O4U2U1U5U6U3 |
| Gauss code of -K* | O1O2O3U1O4O5O6U4U2U3U6U5 |
| 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 2 2 2],[ 3 0 0 3 4 2 2],[ 3 0 0 2 3 1 2],[ 0 -3 -2 0 1 0 2],[-2 -4 -3 -1 0 0 1],[-2 -2 -1 0 0 0 0],[-2 -2 -2 -2 -1 0 0]] |
| Primitive based matrix | [[ 0 2 2 2 0 -3 -3],[-2 0 1 0 -1 -3 -4],[-2 -1 0 0 -2 -2 -2],[-2 0 0 0 0 -1 -2],[ 0 1 2 0 0 -2 -3],[ 3 3 2 1 2 0 0],[ 3 4 2 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-2,0,3,3,-1,0,1,3,4,0,2,2,2,0,1,2,2,3,0] |
| Phi over symmetry | [-3,-3,0,2,2,2,0,0,1,3,3,1,2,3,4,1,0,2,-1,0,0] |
| Phi of -K | [-3,-3,0,2,2,2,0,0,1,3,3,1,2,3,4,1,0,2,-1,0,0] |
| Phi of K* | [-2,-2,-2,0,3,3,-1,0,0,3,3,0,1,1,2,2,3,4,0,1,0] |
| Phi of -K* | [-3,-3,0,2,2,2,0,2,1,2,3,3,2,2,4,0,2,1,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 2t^3-3t^2 |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w |
| Inner characteristic polynomial | t^6+57t^4+61t^2+1 |
| Outer characteristic polynomial | t^7+87t^5+220t^3+12t |
| Flat arrow polynomial | -8*K1**4 + 8*K1**2*K2 + 4*K1**2 - 2*K1*K3 - K2 |
| 2-strand cable arrow polynomial | -256*K1**2*K2**4 + 1632*K1**2*K2**3 - 3808*K1**2*K2**2 - 224*K1**2*K2*K4 + 3512*K1**2*K2 - 2840*K1**2 + 992*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4128*K1*K2*K3 + 328*K1*K3*K4 + 184*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1216*K2**6 - 128*K2**4*K3**2 - 192*K2**4*K4**2 + 1632*K2**4*K4 - 3328*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 896*K2**2*K3**2 - 672*K2**2*K4**2 + 2904*K2**2*K4 - 192*K2**2*K5**2 - 8*K2**2*K6**2 - 1032*K2**2 + 808*K2*K3*K5 + 112*K2*K4*K6 - 1152*K3**2 - 514*K4**2 - 200*K5**2 - 8*K6**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}], [{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}], [{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}]] |
| If K is slice | False |