| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,2,1,3,4,0,0,2,2,0,0,1,-1,0,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.769'] |
| Arrow polynomial of the knot is: -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.425', '6.655', '6.755', '6.769', '6.792', '6.1240', '6.1494', '6.1522', '6.1534', '6.1587', '6.1707', '6.1746', '6.1747', '6.1786', '6.1814', '6.1828', '6.1835', '6.1854', '6.1870'] |
| Outer characteristic polynomial of the knot is: t^7+68t^5+118t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.769'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 1088*K1**4*K2 - 4160*K1**4 + 128*K1**3*K2*K3 - 96*K1**3*K3 + 864*K1**2*K2**3 - 7088*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 8720*K1**2*K2 - 1248*K1**2*K3**2 - 128*K1**2*K3*K5 - 48*K1**2*K4**2 - 3544*K1**2 - 1056*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 7440*K1*K2*K3 + 1936*K1*K3*K4 + 232*K1*K4*K5 + 8*K1*K5*K6 - 1592*K2**4 - 512*K2**2*K3**2 - 16*K2**2*K4**2 + 1752*K2**2*K4 - 3452*K2**2 + 640*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 2196*K3**2 - 902*K4**2 - 220*K5**2 - 12*K6**2 + 4204 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.769'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11484', 'vk6.11788', 'vk6.12805', 'vk6.13141', 'vk6.17033', 'vk6.17275', 'vk6.20867', 'vk6.20942', 'vk6.22276', 'vk6.22352', 'vk6.23758', 'vk6.28337', 'vk6.31242', 'vk6.31591', 'vk6.32815', 'vk6.35544', 'vk6.35993', 'vk6.39969', 'vk6.40102', 'vk6.42042', 'vk6.42952', 'vk6.43246', 'vk6.46506', 'vk6.46622', 'vk6.52237', 'vk6.53073', 'vk6.53391', 'vk6.55453', 'vk6.58858', 'vk6.59935', 'vk6.64406', 'vk6.69728'] |
| 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 | O1O2O3O4U3O5U1U6U2O6U5U4 |
| R3 orbit | {'O1O2O3O4U3O5U1U6U2O6U5U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5O6U3U6U4O5U2 |
| Gauss code of K* | O1O2O3U2O4O5U1U3U6U5O6U4 |
| Gauss code of -K* | O1O2O3U4O5U6U5U1U3O6O4U2 |
| 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 -1 3 2 -1],[ 3 0 1 0 4 2 2],[ 0 -1 0 0 2 0 0],[ 1 0 0 0 1 0 1],[-3 -4 -2 -1 0 0 -3],[-2 -2 0 0 0 0 -2],[ 1 -2 0 -1 3 2 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -1 -3],[-3 0 0 -2 -1 -3 -4],[-2 0 0 0 0 -2 -2],[ 0 2 0 0 0 0 -1],[ 1 1 0 0 0 1 0],[ 1 3 2 0 -1 0 -2],[ 3 4 2 1 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,1,3,0,2,1,3,4,0,0,2,2,0,0,1,-1,0,2] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,2,1,3,4,0,0,2,2,0,0,1,-1,0,2] |
| Phi of -K | [-3,-1,-1,0,2,3,0,2,2,3,2,1,1,1,1,1,3,3,2,1,1] |
| Phi of K* | [-3,-2,0,1,1,3,1,1,1,3,2,2,1,3,3,1,1,2,-1,0,2] |
| Phi of -K* | [-3,-1,-1,0,2,3,0,2,1,2,4,1,0,0,1,0,2,3,0,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+44t^4+63t^2+4 |
| Outer characteristic polynomial | t^7+68t^5+118t^3+9t |
| Flat arrow polynomial | -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6 |
| 2-strand cable arrow polynomial | -192*K1**6 - 128*K1**4*K2**2 + 1088*K1**4*K2 - 4160*K1**4 + 128*K1**3*K2*K3 - 96*K1**3*K3 + 864*K1**2*K2**3 - 7088*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 8720*K1**2*K2 - 1248*K1**2*K3**2 - 128*K1**2*K3*K5 - 48*K1**2*K4**2 - 3544*K1**2 - 1056*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 7440*K1*K2*K3 + 1936*K1*K3*K4 + 232*K1*K4*K5 + 8*K1*K5*K6 - 1592*K2**4 - 512*K2**2*K3**2 - 16*K2**2*K4**2 + 1752*K2**2*K4 - 3452*K2**2 + 640*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 2196*K3**2 - 902*K4**2 - 220*K5**2 - 12*K6**2 + 4204 |
| 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}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |