| Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,1,2,2,3,2,1,2,3,1,1,2,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.699'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874'] |
| Outer characteristic polynomial of the knot is: t^7+62t^5+150t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.699'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 192*K1**4*K2 - 1424*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 544*K1**3*K2*K3 - 640*K1**3*K3 - 896*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2976*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 7968*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 8936*K1**2*K2 - 592*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 6260*K1**2 - 128*K1*K2**4*K3 + 2080*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 - 128*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 7528*K1*K2*K3 + 1112*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 2424*K2**4 - 1168*K2**2*K3**2 - 112*K2**2*K4**2 + 1504*K2**2*K4 - 3110*K2**2 + 496*K2*K3*K5 + 8*K2*K4*K6 - 2168*K3**2 - 574*K4**2 - 108*K5**2 - 2*K6**2 + 4580 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.699'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81557', 'vk6.81633', 'vk6.81639', 'vk6.81832', 'vk6.81837', 'vk6.82052', 'vk6.82215', 'vk6.82225', 'vk6.82327', 'vk6.82337', 'vk6.82542', 'vk6.82547', 'vk6.82999', 'vk6.83117', 'vk6.83124', 'vk6.83565', 'vk6.83572', 'vk6.83924', 'vk6.84071', 'vk6.84085', 'vk6.84532', 'vk6.84895', 'vk6.84898', 'vk6.85891', 'vk6.85902', 'vk6.86400', 'vk6.86413', 'vk6.86455', 'vk6.86462', 'vk6.88831', 'vk6.89757', 'vk6.89877'] |
| 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 | O1O2O3O4U5O6U1U2O5U3U6U4 |
| R3 orbit | {'O1O2O3O4U5O6U1U2O5U3U6U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U2O6U3U4O5U6 |
| Gauss code of K* | O1O2O3U4U5U1U3O6U2O4O5U6 |
| Gauss code of -K* | O1O2O3U4O5O6U2O4U1U3U5U6 |
| 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 -1 0 3 -1 2],[ 3 0 1 1 3 2 2],[ 1 -1 0 0 2 1 1],[ 0 -1 0 0 2 0 0],[-3 -3 -2 -2 0 -2 -1],[ 1 -2 -1 0 2 0 2],[-2 -2 -1 0 1 -2 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -1 -3],[-3 0 -1 -2 -2 -2 -3],[-2 1 0 0 -1 -2 -2],[ 0 2 0 0 0 0 -1],[ 1 2 1 0 0 1 -1],[ 1 2 2 0 -1 0 -2],[ 3 3 2 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,1,3,1,2,2,2,3,0,1,2,2,0,0,1,-1,1,2] |
| Phi over symmetry | [-3,-2,0,1,1,3,0,1,2,2,3,2,1,2,3,1,1,2,-1,0,1] |
| Phi of -K | [-3,-1,-1,0,2,3,0,1,2,3,3,1,1,1,2,1,2,2,2,1,0] |
| Phi of K* | [-3,-2,0,1,1,3,0,1,2,2,3,2,1,2,3,1,1,2,-1,0,1] |
| Phi of -K* | [-3,-1,-1,0,2,3,1,2,1,2,3,1,0,1,2,0,2,2,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+20z+29 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2-4w^3z+24w^2z+29w |
| Inner characteristic polynomial | t^6+38t^4+73t^2+1 |
| Outer characteristic polynomial | t^7+62t^5+150t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 192*K1**4*K2 - 1424*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 544*K1**3*K2*K3 - 640*K1**3*K3 - 896*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2976*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 7968*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 8936*K1**2*K2 - 592*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 6260*K1**2 - 128*K1*K2**4*K3 + 2080*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 - 128*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 7528*K1*K2*K3 + 1112*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 2424*K2**4 - 1168*K2**2*K3**2 - 112*K2**2*K4**2 + 1504*K2**2*K4 - 3110*K2**2 + 496*K2*K3*K5 + 8*K2*K4*K6 - 2168*K3**2 - 574*K4**2 - 108*K5**2 - 2*K6**2 + 4580 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |