| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,0,1,4,4,0,0,2,2,0,1,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.385', '7.17788'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+72t^5+43t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.385'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 448*K1**4*K2 - 672*K1**4 + 128*K1**3*K2*K3 - 448*K1**2*K2**2 + 752*K1**2*K2 - 160*K1**2*K3**2 - 48*K1**2*K4**2 - 236*K1**2 + 520*K1*K2*K3 + 224*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4 - 32*K2**2*K3**2 - 16*K2**2*K4**2 + 80*K2**2*K4 - 412*K2**2 + 56*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 248*K3**2 - 120*K4**2 - 44*K5**2 - 12*K6**2 + 494 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.385'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11271', 'vk6.11351', 'vk6.12536', 'vk6.12649', 'vk6.17054', 'vk6.17296', 'vk6.17629', 'vk6.18916', 'vk6.18992', 'vk6.19343', 'vk6.19636', 'vk6.22259', 'vk6.24088', 'vk6.24182', 'vk6.25510', 'vk6.26115', 'vk6.26533', 'vk6.28320', 'vk6.30945', 'vk6.31070', 'vk6.31241', 'vk6.31592', 'vk6.32125', 'vk6.32246', 'vk6.32816', 'vk6.35561', 'vk6.36012', 'vk6.36426', 'vk6.37653', 'vk6.39940', 'vk6.40100', 'vk6.43524', 'vk6.44776', 'vk6.46487', 'vk6.52021', 'vk6.53392', 'vk6.55462', 'vk6.56657', 'vk6.65394', 'vk6.66111'] |
| 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 | O1O2O3O4O5U4U2O6U1U6U5U3 |
| R3 orbit | {'O1O2O3O4O5U4U2O6U1U6U5U3', 'O1O2O3O4U3O5U2O6U1U6U4U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U3U1U6U5O6U4U2 |
| Gauss code of K* | O1O2O3O4U1U5U4U6U3O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U2U6U1U5U4 |
| 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 -2 2 -1 3 1],[ 3 0 0 4 0 4 1],[ 2 0 0 2 0 2 0],[-2 -4 -2 0 -1 1 0],[ 1 0 0 1 0 1 0],[-3 -4 -2 -1 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 0 -1 -2 -4],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 1 1 1 0 0 0 0],[ 2 2 2 0 0 0 0],[ 3 4 4 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,1,0,1,2,4,0,1,2,4,0,0,1,0,0,0] |
| Phi over symmetry | [-3,-2,-1,1,2,3,0,0,1,4,4,0,0,2,2,0,1,1,0,0,1] |
| Phi of -K | [-3,-2,-1,1,2,3,1,2,3,1,2,1,3,2,3,2,2,3,1,2,0] |
| Phi of K* | [-3,-2,-1,1,2,3,0,2,3,3,2,1,2,2,1,2,3,3,1,2,1] |
| Phi of -K* | [-3,-2,-1,1,2,3,0,0,1,4,4,0,0,2,2,0,1,1,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | 7w^2z+15w |
| Inner characteristic polynomial | t^6+44t^4+11t^2 |
| Outer characteristic polynomial | t^7+72t^5+43t^3 |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -192*K1**6 - 128*K1**4*K2**2 + 448*K1**4*K2 - 672*K1**4 + 128*K1**3*K2*K3 - 448*K1**2*K2**2 + 752*K1**2*K2 - 160*K1**2*K3**2 - 48*K1**2*K4**2 - 236*K1**2 + 520*K1*K2*K3 + 224*K1*K3*K4 + 64*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**4 - 32*K2**2*K3**2 - 16*K2**2*K4**2 + 80*K2**2*K4 - 412*K2**2 + 56*K2*K3*K5 + 16*K2*K4*K6 + 8*K3**2*K6 - 248*K3**2 - 120*K4**2 - 44*K5**2 - 12*K6**2 + 494 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |