| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,1,1,1,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.460', '7.24816'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+30t^5+31t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.460'] |
| 2-strand cable arrow polynomial of the knot is: -512*K1**6 - 1024*K1**4*K2**2 + 1792*K1**4*K2 - 1728*K1**4 + 576*K1**3*K2*K3 - 256*K1**3*K3 - 576*K1**2*K2**4 + 1728*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3936*K1**2*K2**2 - 352*K1**2*K2*K4 + 3288*K1**2*K2 - 32*K1**2*K3**2 - 612*K1**2 + 544*K1*K2**3*K3 - 800*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2144*K1*K2*K3 + 112*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 952*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 656*K2**2*K4 - 510*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 236*K3**2 - 74*K4**2 - 16*K5**2 - 2*K6**2 + 880 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.460'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.403', 'vk6.436', 'vk6.440', 'vk6.843', 'vk6.846', 'vk6.882', 'vk6.887', 'vk6.1120', 'vk6.1589', 'vk6.1649', 'vk6.1762', 'vk6.2034', 'vk6.2064', 'vk6.2066', 'vk6.2142', 'vk6.2239', 'vk6.2700', 'vk6.2732', 'vk6.2844', 'vk6.3140', 'vk6.12051', 'vk6.13044', 'vk6.13540', 'vk6.13551', 'vk6.13729', 'vk6.13742', 'vk6.19468', 'vk6.19470', 'vk6.19761', 'vk6.19765', 'vk6.25790', 'vk6.25816', 'vk6.26635', 'vk6.28454', 'vk6.37922', 'vk6.39321', 'vk6.40219', 'vk6.41501', 'vk6.44916', 'vk6.46854'] |
| 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 | O1O2O3O4O5U6U3O6U4U5U1U2 |
| R3 orbit | {'O1O2O3O4O5U4U6U3O6U5U1U2', 'O1O2O3O4O5U6U3O6U4U5U1U2'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U4U5U1U2O6U3U6 |
| Gauss code of K* | O1O2O3O4U3U4U5U1U2O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U3U4U6U1U2 |
| 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 1 -1 0 2 -1],[ 1 0 1 -1 0 2 0],[-1 -1 0 -1 0 2 -2],[ 1 1 1 0 0 1 1],[ 0 0 0 0 0 1 0],[-2 -2 -2 -1 -1 0 -2],[ 1 0 2 -1 0 2 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 -2 -1 -1 -2 -2],[-1 2 0 0 -1 -1 -2],[ 0 1 0 0 0 0 0],[ 1 1 1 0 0 1 1],[ 1 2 1 0 -1 0 0],[ 1 2 2 0 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,2,0,0,0,-1,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,1,1,1,0,-1,-1] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,-1,1,1,2,0,1,0,1,1,1,1,1,1,-1] |
| Phi of K* | [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,1,1,1,0,-1,-1] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,1,0,1,1,0,2,2,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+22t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+30t^5+31t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -512*K1**6 - 1024*K1**4*K2**2 + 1792*K1**4*K2 - 1728*K1**4 + 576*K1**3*K2*K3 - 256*K1**3*K3 - 576*K1**2*K2**4 + 1728*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3936*K1**2*K2**2 - 352*K1**2*K2*K4 + 3288*K1**2*K2 - 32*K1**2*K3**2 - 612*K1**2 + 544*K1*K2**3*K3 - 800*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2144*K1*K2*K3 + 112*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 952*K2**4 - 176*K2**2*K3**2 - 48*K2**2*K4**2 + 656*K2**2*K4 - 510*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 236*K3**2 - 74*K4**2 - 16*K5**2 - 2*K6**2 + 880 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |