| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,2,-1,0,1,1,0,0,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1881'] |
| 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+23t^5+58t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1881'] |
| 2-strand cable arrow polynomial of the knot is: -304*K1**4 + 32*K1**3*K2*K3 - 256*K1**3*K3 - 1152*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 2224*K1**2*K2 - 240*K1**2*K3**2 - 2052*K1**2 + 64*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2768*K1*K2*K3 + 552*K1*K3*K4 + 56*K1*K4*K5 - 104*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 416*K2**2*K4 - 1620*K2**2 + 256*K2*K3*K5 + 32*K2*K4*K6 - 1016*K3**2 - 270*K4**2 - 52*K5**2 - 4*K6**2 + 1596 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1881'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4660', 'vk6.4947', 'vk6.6114', 'vk6.6601', 'vk6.8127', 'vk6.8529', 'vk6.9513', 'vk6.9868', 'vk6.20377', 'vk6.21718', 'vk6.27693', 'vk6.29237', 'vk6.39133', 'vk6.41387', 'vk6.45873', 'vk6.47534', 'vk6.48692', 'vk6.48895', 'vk6.49448', 'vk6.49667', 'vk6.50708', 'vk6.50907', 'vk6.51195', 'vk6.51396', 'vk6.57238', 'vk6.58463', 'vk6.61860', 'vk6.62995', 'vk6.66861', 'vk6.67729', 'vk6.69489', 'vk6.70211', 'vk6.82022', 'vk6.82756', 'vk6.85377', 'vk6.86698', 'vk6.86936', 'vk6.87037', 'vk6.87589', 'vk6.89456'] |
| 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 | O1O2O3U4U5O6U2O5O4U6U1U3 |
| R3 orbit | {'O1O2O3U4U1U5O6O4O5U3U2U6', 'O1O2O3U4U5O6U2O5O4U6U1U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U1U3U4O5O6U2O4U6U5 |
| Gauss code of K* | O1O2O3U2U4U3O5O6U1O4U6U5 |
| Gauss code of -K* | O1O2O3U4U5O6U3O5O4U1U6U2 |
| 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 -1 0 2 0 0 -1],[ 1 0 1 2 1 1 -1],[ 0 -1 0 0 1 0 -1],[-2 -2 0 0 -1 -1 -2],[ 0 -1 -1 1 0 0 0],[ 0 -1 0 1 0 0 -1],[ 1 1 1 2 0 1 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -1 -2 -2],[ 0 0 0 1 0 -1 -1],[ 0 1 -1 0 0 0 -1],[ 0 1 0 0 0 -1 -1],[ 1 2 1 0 1 0 1],[ 1 2 1 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,1,1,2,2,-1,0,1,1,0,0,1,1,1,-1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,2,2,-1,0,1,1,0,0,1,1,1,-1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,0,1,1,0,0,0,1,0,-1,2,0,1,1] |
| Phi of K* | [-2,0,0,0,1,1,1,1,2,1,1,0,-1,0,1,0,0,0,0,0,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,1,1,1,2,0,1,1,2,-1,0,1,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+17t^4+35t^2 |
| Outer characteristic polynomial | t^7+23t^5+58t^3+3t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -304*K1**4 + 32*K1**3*K2*K3 - 256*K1**3*K3 - 1152*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 2224*K1**2*K2 - 240*K1**2*K3**2 - 2052*K1**2 + 64*K1*K2**3*K3 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 2768*K1*K2*K3 + 552*K1*K3*K4 + 56*K1*K4*K5 - 104*K2**4 - 240*K2**2*K3**2 - 48*K2**2*K4**2 + 416*K2**2*K4 - 1620*K2**2 + 256*K2*K3*K5 + 32*K2*K4*K6 - 1016*K3**2 - 270*K4**2 - 52*K5**2 - 4*K6**2 + 1596 |
| 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}], [{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}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |