| Min(phi) over symmetries of the knot is: [-5,-1,1,1,2,2,1,4,5,2,3,2,2,1,2,0,1,1,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.42'] |
| Arrow polynomial of the knot is: K1 - 2*K2*K3 + K5 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.13', '6.30', '6.33', '6.42', '6.56'] |
| Outer characteristic polynomial of the knot is: t^7+108t^5+122t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.42'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**4 + 128*K1**3*K2*K3 - 32*K1**3*K3 - 704*K1**2*K2**2 - 128*K1**2*K2*K4 + 2160*K1**2*K2 - 1440*K1**2*K3**2 - 192*K1**2*K3*K5 - 64*K1**2*K4**2 - 3552*K1**2 - 320*K1*K2**2*K3 - 32*K1*K2*K3*K6 + 3920*K1*K2*K3 - 32*K1*K2*K5*K6 + 128*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 3280*K1*K3*K4 + 296*K1*K4*K5 + 88*K1*K5*K6 + 48*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 128*K2**4 - 192*K2**2*K3**2 + 448*K2**2*K4 - 2474*K2**2 - 256*K2*K3**2*K4 + 488*K2*K3*K5 + 216*K2*K4*K6 + 40*K2*K5*K7 - 320*K3**4 - 480*K3**2*K4**2 + 392*K3**2*K6 - 2584*K3**2 + 360*K3*K4*K7 - 8*K4**2*K6**2 - 1404*K4**2 - 320*K5**2 - 212*K6**2 - 104*K7**2 + 3554 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.42'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20043', 'vk6.20107', 'vk6.21327', 'vk6.21385', 'vk6.27104', 'vk6.27184', 'vk6.28797', 'vk6.28867', 'vk6.38487', 'vk6.38593', 'vk6.40688', 'vk6.40781', 'vk6.45381', 'vk6.45473', 'vk6.47136', 'vk6.47209', 'vk6.56844', 'vk6.56924', 'vk6.57984', 'vk6.58056', 'vk6.61366', 'vk6.61466', 'vk6.62534', 'vk6.62613', 'vk6.66558', 'vk6.66628', 'vk6.67350', 'vk6.67412', 'vk6.69205', 'vk6.69268', 'vk6.69951', 'vk6.70005'] |
| 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 | O1O2O3O4O5O6U1U4U6U5U3U2 |
| R3 orbit | {'O1O2O3O4O5O6U1U4U6U5U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5O6U5U4U2U1U3U6 |
| Gauss code of K* | O1O2O3O4O5O6U1U6U5U2U4U3 |
| Gauss code of -K* | O1O2O3O4O5O6U4U3U5U2U1U6 |
| 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 -5 1 1 -1 2 2],[ 5 0 5 4 1 3 2],[-1 -5 0 0 -2 1 1],[-1 -4 0 0 -2 1 1],[ 1 -1 2 2 0 2 1],[-2 -3 -1 -1 -2 0 0],[-2 -2 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 1 -1 -5],[-2 0 0 -1 -1 -1 -2],[-2 0 0 -1 -1 -2 -3],[-1 1 1 0 0 -2 -4],[-1 1 1 0 0 -2 -5],[ 1 1 2 2 2 0 -1],[ 5 2 3 4 5 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,-1,1,5,0,1,1,1,2,1,1,2,3,0,2,4,2,5,1] |
| Phi over symmetry | [-5,-1,1,1,2,2,1,4,5,2,3,2,2,1,2,0,1,1,1,1,0] |
| Phi of -K | [-5,-1,1,1,2,2,3,1,2,4,5,0,0,1,2,0,0,0,0,0,0] |
| Phi of K* | [-2,-2,-1,-1,1,5,0,0,0,1,4,0,0,2,5,0,0,1,0,2,3] |
| Phi of -K* | [-5,-1,1,1,2,2,1,4,5,2,3,2,2,1,2,0,1,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^5-2t^2-t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+72t^4+22t^2+1 |
| Outer characteristic polynomial | t^7+108t^5+122t^3+8t |
| Flat arrow polynomial | K1 - 2*K2*K3 + K5 + 1 |
| 2-strand cable arrow polynomial | -512*K1**4 + 128*K1**3*K2*K3 - 32*K1**3*K3 - 704*K1**2*K2**2 - 128*K1**2*K2*K4 + 2160*K1**2*K2 - 1440*K1**2*K3**2 - 192*K1**2*K3*K5 - 64*K1**2*K4**2 - 3552*K1**2 - 320*K1*K2**2*K3 - 32*K1*K2*K3*K6 + 3920*K1*K2*K3 - 32*K1*K2*K5*K6 + 128*K1*K3**3*K4 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 3280*K1*K3*K4 + 296*K1*K4*K5 + 88*K1*K5*K6 + 48*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 128*K2**4 - 192*K2**2*K3**2 + 448*K2**2*K4 - 2474*K2**2 - 256*K2*K3**2*K4 + 488*K2*K3*K5 + 216*K2*K4*K6 + 40*K2*K5*K7 - 320*K3**4 - 480*K3**2*K4**2 + 392*K3**2*K6 - 2584*K3**2 + 360*K3*K4*K7 - 8*K4**2*K6**2 - 1404*K4**2 - 320*K5**2 - 212*K6**2 - 104*K7**2 + 3554 |
| 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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |