| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,2,3,3,2,2,0,1,1,0,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1824'] |
| 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+22t^5+72t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1824'] |
| 2-strand cable arrow polynomial of the knot is: 1344*K1**4*K2 - 4432*K1**4 + 416*K1**3*K2*K3 - 1632*K1**3*K3 - 3872*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 9400*K1**2*K2 - 976*K1**2*K3**2 - 96*K1**2*K3*K5 - 5516*K1**2 - 480*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7080*K1*K2*K3 + 1712*K1*K3*K4 + 136*K1*K4*K5 - 104*K2**4 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 760*K2**2*K4 - 4756*K2**2 + 288*K2*K3*K5 + 32*K2*K4*K6 - 2464*K3**2 - 758*K4**2 - 124*K5**2 - 12*K6**2 + 4852 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1824'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4867', 'vk6.5212', 'vk6.6449', 'vk6.6870', 'vk6.8414', 'vk6.8835', 'vk6.9766', 'vk6.10059', 'vk6.11666', 'vk6.12019', 'vk6.13012', 'vk6.20498', 'vk6.20771', 'vk6.21863', 'vk6.27908', 'vk6.29406', 'vk6.29740', 'vk6.32663', 'vk6.33006', 'vk6.39335', 'vk6.39811', 'vk6.46375', 'vk6.47605', 'vk6.47952', 'vk6.48825', 'vk6.49096', 'vk6.51348', 'vk6.51561', 'vk6.53273', 'vk6.57367', 'vk6.64342', 'vk6.66920'] |
| 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 | O1O2O3U1U3O4U2O5O6U4U6U5 |
| R3 orbit | {'O1O2O3U1U3O4U2O5O6U4U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U6O5O4U2O6U1U3 |
| Gauss code of K* | O1O2O3U4U5U6O4O6U1O5U3U2 |
| Gauss code of -K* | O1O2O3U2U1O4U3O5O6U5U4U6 |
| 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 0 1 -1 1 1],[ 2 0 2 1 1 0 0],[ 0 -2 0 0 1 1 1],[-1 -1 0 0 0 0 0],[ 1 -1 -1 0 0 2 1],[-1 0 -1 0 -2 0 0],[-1 0 -1 0 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 0],[-1 0 0 0 -1 -2 0],[ 0 0 1 1 0 1 -2],[ 1 0 1 2 -1 0 -1],[ 2 1 0 0 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,0,0,1,0,1,1,0,1,2,0,-1,2,1] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,2,3,3,2,2,0,1,1,0,0,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,0,0,2,3,3,2,2,0,1,1,0,0,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,0,3,0,0,1,3,1,2,2,2,0,0] |
| Phi of -K* | [-2,-1,0,1,1,1,1,2,0,0,1,-1,1,2,0,1,1,0,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 6z^2+27z+31 |
| Enhanced Jones-Krushkal polynomial | 6w^3z^2+27w^2z+31w |
| Inner characteristic polynomial | t^6+14t^4+19t^2+1 |
| Outer characteristic polynomial | t^7+22t^5+72t^3+5t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | 1344*K1**4*K2 - 4432*K1**4 + 416*K1**3*K2*K3 - 1632*K1**3*K3 - 3872*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 9400*K1**2*K2 - 976*K1**2*K3**2 - 96*K1**2*K3*K5 - 5516*K1**2 - 480*K1*K2**2*K3 - 96*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7080*K1*K2*K3 + 1712*K1*K3*K4 + 136*K1*K4*K5 - 104*K2**4 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 760*K2**2*K4 - 4756*K2**2 + 288*K2*K3*K5 + 32*K2*K4*K6 - 2464*K3**2 - 758*K4**2 - 124*K5**2 - 12*K6**2 + 4852 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]] |
| If K is slice | False |