| Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,2,2,3,3,1,1,0,0,0,0,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.730'] |
| Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384'] |
| Outer characteristic polynomial of the knot is: t^7+34t^5+57t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.730'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 448*K1**4*K2 - 1568*K1**4 + 384*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1328*K1**2*K2**2 + 2128*K1**2*K2 - 480*K1**2*K3**2 - 112*K1**2*K4**2 - 788*K1**2 + 1752*K1*K2*K3 + 560*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 192*K2**2*K4 - 1182*K2**2 + 72*K2*K3*K5 + 8*K2*K4*K6 - 748*K3**2 - 272*K4**2 - 40*K5**2 - 2*K6**2 + 1326 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.730'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4071', 'vk6.4104', 'vk6.4244', 'vk6.4324', 'vk6.5313', 'vk6.5346', 'vk6.5515', 'vk6.5529', 'vk6.5636', 'vk6.5650', 'vk6.7472', 'vk6.7708', 'vk6.8944', 'vk6.8977', 'vk6.9106', 'vk6.9186', 'vk6.14533', 'vk6.15286', 'vk6.15415', 'vk6.15755', 'vk6.16172', 'vk6.26276', 'vk6.26721', 'vk6.29847', 'vk6.29880', 'vk6.33920', 'vk6.34208', 'vk6.38212', 'vk6.38238', 'vk6.44989', 'vk6.45011', 'vk6.48566', 'vk6.49176', 'vk6.49273', 'vk6.49287', 'vk6.50254', 'vk6.51591', 'vk6.53979', 'vk6.54484', 'vk6.63310'] |
| 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 | O1O2O3O4U1O5U4U5U3O6U2U6 |
| R3 orbit | {'O1O2O3O4U1O5U4U5U3O6U2U6', 'O1O2O3O4U1U3O5U4U5O6U2U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U5U3O5U2U6U1O6U4 |
| Gauss code of K* | O1O2O3U4O5O4U6U5U3U1O6U2 |
| Gauss code of -K* | O1O2O3U2O4U3U1U5U4O6O5U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 0 1 0 1 1],[ 3 0 3 2 1 1 1],[ 0 -3 0 0 -1 1 1],[-1 -2 0 0 -1 1 0],[ 0 -1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 -1 0 -1],[ 0 0 1 1 0 -1 -3],[ 0 1 1 0 1 0 -1],[ 3 2 1 1 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,0,3,-1,0,0,1,2,0,1,1,1,1,0,1,1,3,1] |
| Phi over symmetry | [-3,0,0,1,1,1,0,2,2,3,3,1,1,0,0,0,0,1,-1,0,0] |
| Phi of -K | [-3,0,0,1,1,1,0,2,2,3,3,1,1,0,0,0,0,1,-1,0,0] |
| Phi of K* | [-1,-1,-1,0,0,3,-1,0,0,0,3,0,0,1,2,1,0,3,1,2,0] |
| Phi of -K* | [-3,0,0,1,1,1,1,3,1,1,2,1,0,1,1,1,1,0,0,0,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 12z+25 |
| Enhanced Jones-Krushkal polynomial | 12w^2z+25w |
| Inner characteristic polynomial | t^6+22t^4+17t^2 |
| Outer characteristic polynomial | t^7+34t^5+57t^3 |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -192*K1**6 - 192*K1**4*K2**2 + 448*K1**4*K2 - 1568*K1**4 + 384*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1328*K1**2*K2**2 + 2128*K1**2*K2 - 480*K1**2*K3**2 - 112*K1**2*K4**2 - 788*K1**2 + 1752*K1*K2*K3 + 560*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 192*K2**2*K4 - 1182*K2**2 + 72*K2*K3*K5 + 8*K2*K4*K6 - 748*K3**2 - 272*K4**2 - 40*K5**2 - 2*K6**2 + 1326 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {4, 5}, {3}, {1, 2}]] |
| If K is slice | False |