| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,2,0,1,0,-1,1,0,1,0,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1761'] |
| 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+29t^5+76t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1761'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 704*K1**4*K2 - 4112*K1**4 + 448*K1**3*K2*K3 - 1216*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5808*K1**2*K2**2 - 576*K1**2*K2*K4 + 10816*K1**2*K2 - 624*K1**2*K3**2 - 48*K1**2*K4**2 - 5728*K1**2 + 256*K1*K2**3*K3 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7136*K1*K2*K3 + 736*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 984*K2**4 - 160*K2**2*K3**2 - 48*K2**2*K4**2 + 944*K2**2*K4 - 4310*K2**2 + 160*K2*K3*K5 + 16*K2*K4*K6 - 1912*K3**2 - 358*K4**2 - 40*K5**2 - 2*K6**2 + 4708 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1761'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16938', 'vk6.17181', 'vk6.20546', 'vk6.21945', 'vk6.23338', 'vk6.23633', 'vk6.28004', 'vk6.29469', 'vk6.35382', 'vk6.35803', 'vk6.39412', 'vk6.41603', 'vk6.42859', 'vk6.43138', 'vk6.45992', 'vk6.47666', 'vk6.55089', 'vk6.55342', 'vk6.57426', 'vk6.58595', 'vk6.59491', 'vk6.59783', 'vk6.62097', 'vk6.63073', 'vk6.64938', 'vk6.65146', 'vk6.66966', 'vk6.67825', 'vk6.68231', 'vk6.68374', 'vk6.69581', 'vk6.70276'] |
| 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 | O1O2O3U4U5O4O6U3U1O5U2U6 |
| R3 orbit | {'O1O2O3U4U5O4O6U3U1O5U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2O5U3U1O4O6U5U6 |
| Gauss code of K* | O1O2U3O4O5U2U4U1O6O3U6U5 |
| Gauss code of -K* | O1O2U3O4O5U1U6O3O6U5U2U4 |
| 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 0 -1 0 2],[ 1 0 0 0 1 1 2],[ 0 0 0 1 -1 0 1],[ 0 0 -1 0 0 1 0],[ 1 -1 1 0 0 0 3],[ 0 -1 0 -1 0 0 2],[-2 -2 -1 0 -3 -2 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 0 -1 -2 -2 -3],[ 0 0 0 -1 1 0 0],[ 0 1 1 0 0 0 -1],[ 0 2 -1 0 0 -1 0],[ 1 2 0 0 1 0 1],[ 1 3 0 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,0,1,2,2,3,1,-1,0,0,0,0,1,1,0,-1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,2,0,1,0,-1,1,0,1,0,1,1,1,-1] |
| Phi of -K | [-1,-1,0,0,0,2,-1,0,1,1,1,1,0,1,0,0,1,0,-1,1,2] |
| Phi of K* | [-2,0,0,0,1,1,0,1,2,0,1,0,-1,1,0,1,0,1,1,1,-1] |
| Phi of -K* | [-1,-1,0,0,0,2,-1,0,0,1,3,0,1,0,2,1,-1,0,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 2z^2+21z+35 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+21w^2z+35w |
| Inner characteristic polynomial | t^6+23t^4+55t^2+4 |
| Outer characteristic polynomial | t^7+29t^5+76t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6 |
| 2-strand cable arrow polynomial | -64*K1**6 + 704*K1**4*K2 - 4112*K1**4 + 448*K1**3*K2*K3 - 1216*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5808*K1**2*K2**2 - 576*K1**2*K2*K4 + 10816*K1**2*K2 - 624*K1**2*K3**2 - 48*K1**2*K4**2 - 5728*K1**2 + 256*K1*K2**3*K3 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 7136*K1*K2*K3 + 736*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 984*K2**4 - 160*K2**2*K3**2 - 48*K2**2*K4**2 + 944*K2**2*K4 - 4310*K2**2 + 160*K2*K3*K5 + 16*K2*K4*K6 - 1912*K3**2 - 358*K4**2 - 40*K5**2 - 2*K6**2 + 4708 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {3, 5}, {2, 4}, {1}]] |
| If K is slice | False |