| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,1,1,1,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1544'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+32t^5+96t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1544'] |
| 2-strand cable arrow polynomial of the knot is: -2576*K1**4 + 1248*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1312*K1**3*K3 + 352*K1**2*K2**2*K4 - 2496*K1**2*K2**2 - 1728*K1**2*K2*K4 + 6288*K1**2*K2 - 1456*K1**2*K3**2 - 320*K1**2*K4**2 - 4292*K1**2 + 96*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6456*K1*K2*K3 + 2696*K1*K3*K4 + 464*K1*K4*K5 + 24*K1*K5*K6 - 72*K2**4 - 80*K2**2*K3**2 - 112*K2**2*K4**2 + 1152*K2**2*K4 - 3764*K2**2 + 480*K2*K3*K5 + 112*K2*K4*K6 - 2368*K3**2 - 1150*K4**2 - 260*K5**2 - 28*K6**2 + 3860 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1544'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11490', 'vk6.11798', 'vk6.12814', 'vk6.13147', 'vk6.17075', 'vk6.17318', 'vk6.20894', 'vk6.21062', 'vk6.22305', 'vk6.22488', 'vk6.23797', 'vk6.28369', 'vk6.31255', 'vk6.31612', 'vk6.32828', 'vk6.35591', 'vk6.36050', 'vk6.40011', 'vk6.40307', 'vk6.42069', 'vk6.43285', 'vk6.46549', 'vk6.46768', 'vk6.48023', 'vk6.52245', 'vk6.53402', 'vk6.57706', 'vk6.57724', 'vk6.58891', 'vk6.59954', 'vk6.64420', 'vk6.69749'] |
| 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 | O1O2O3U4O5U1U6O4O6U5U3U2 |
| R3 orbit | {'O1O2O3U4O5U1U6O4O6U5U3U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U1U4O5O6U5U3O4U6 |
| Gauss code of K* | O1O2O3U4U3U2O5U1O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U3O5U2U1U6 |
| 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 -2 1 1 -1 0 1],[ 2 0 2 1 2 0 3],[-1 -2 0 0 -1 -1 0],[-1 -1 0 0 -1 -1 0],[ 1 -2 1 1 0 1 1],[ 0 0 1 1 -1 0 0],[-1 -3 0 0 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 -3],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[ 0 0 1 1 0 -1 0],[ 1 1 1 1 1 0 -2],[ 2 3 1 2 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,1,3,0,1,1,1,1,1,2,1,0,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,1,1,1,0,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,1,1,1,0,0,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,1,1,0,0,1,2,1,1,0,0,2,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,0,1,2,3,1,1,1,1,1,1,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 7z^2+28z+29 |
| Enhanced Jones-Krushkal polynomial | 7w^3z^2+28w^2z+29w |
| Inner characteristic polynomial | t^6+24t^4+61t^2+1 |
| Outer characteristic polynomial | t^7+32t^5+96t^3+7t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -2576*K1**4 + 1248*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1312*K1**3*K3 + 352*K1**2*K2**2*K4 - 2496*K1**2*K2**2 - 1728*K1**2*K2*K4 + 6288*K1**2*K2 - 1456*K1**2*K3**2 - 320*K1**2*K4**2 - 4292*K1**2 + 96*K1*K2**2*K3*K4 - 512*K1*K2**2*K3 - 384*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6456*K1*K2*K3 + 2696*K1*K3*K4 + 464*K1*K4*K5 + 24*K1*K5*K6 - 72*K2**4 - 80*K2**2*K3**2 - 112*K2**2*K4**2 + 1152*K2**2*K4 - 3764*K2**2 + 480*K2*K3*K5 + 112*K2*K4*K6 - 2368*K3**2 - 1150*K4**2 - 260*K5**2 - 28*K6**2 + 3860 |
| 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}, {3, 4}, {1, 2}]] |
| If K is slice | False |