| Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,0,1,2,2,0,1,1,2,1,2,2,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.906'] |
| Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070'] |
| Outer characteristic polynomial of the knot is: t^7+61t^5+92t^3+14t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.906'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 288*K1**4*K2 - 1952*K1**4 + 128*K1**3*K2**3*K3 + 1088*K1**3*K2*K3 - 672*K1**3*K3 - 448*K1**2*K2**4 + 512*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7488*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 11000*K1**2*K2 - 896*K1**2*K3**2 - 64*K1**2*K4**2 - 8280*K1**2 + 1984*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 864*K1*K2**2*K5 + 224*K1*K2*K3**3 - 384*K1*K2*K3*K4 + 10520*K1*K2*K3 + 1480*K1*K3*K4 + 256*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1824*K2**4 - 32*K2**3*K6 - 1344*K2**2*K3**2 - 64*K2**2*K4**2 + 2320*K2**2*K4 - 6082*K2**2 + 1104*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 - 3260*K3**2 - 832*K4**2 - 244*K5**2 - 6*K6**2 + 6438 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.906'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16508', 'vk6.16599', 'vk6.18084', 'vk6.18422', 'vk6.22939', 'vk6.23034', 'vk6.24535', 'vk6.24954', 'vk6.34914', 'vk6.35023', 'vk6.36674', 'vk6.37098', 'vk6.42485', 'vk6.42596', 'vk6.43954', 'vk6.44271', 'vk6.54751', 'vk6.54846', 'vk6.55900', 'vk6.56186', 'vk6.59215', 'vk6.59278', 'vk6.60430', 'vk6.60785', 'vk6.64765', 'vk6.64828', 'vk6.65542', 'vk6.65854', 'vk6.68067', 'vk6.68130', 'vk6.68624', 'vk6.68839'] |
| 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 | O1O2O3O4U2U5O6O5U3U1U4U6 |
| R3 orbit | {'O1O2O3O4U2U5O6O5U3U1U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U4U2O6O5U6U3 |
| Gauss code of K* | O1O2O3O4U2U5U1U3O5O6U4U6 |
| Gauss code of -K* | O1O2O3O4U5U1O5O6U2U4U6U3 |
| 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 -2 -2 -1 2 1 2],[ 2 0 -1 1 3 2 2],[ 2 1 0 1 2 2 2],[ 1 -1 -1 0 1 1 1],[-2 -3 -2 -1 0 -2 0],[-1 -2 -2 -1 2 0 2],[-2 -2 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -2 -2],[-2 0 0 -2 -1 -2 -2],[-2 0 0 -2 -1 -2 -3],[-1 2 2 0 -1 -2 -2],[ 1 1 1 1 0 -1 -1],[ 2 2 2 2 1 0 1],[ 2 2 3 2 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,2,2,0,2,1,2,2,2,1,2,3,1,2,2,1,1,-1] |
| Phi over symmetry | [-2,-2,-1,1,2,2,-1,0,1,2,2,0,1,1,2,1,2,2,-1,-1,0] |
| Phi of -K | [-2,-2,-1,1,2,2,-1,0,1,2,2,0,1,1,2,1,2,2,-1,-1,0] |
| Phi of K* | [-2,-2,-1,1,2,2,0,-1,2,1,2,-1,2,2,2,1,1,1,0,0,-1] |
| Phi of -K* | [-2,-2,-1,1,2,2,-1,1,2,2,3,1,2,2,2,1,1,1,2,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+43t^4+36t^2+4 |
| Outer characteristic polynomial | t^7+61t^5+92t^3+14t |
| Flat arrow polynomial | 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 288*K1**4*K2 - 1952*K1**4 + 128*K1**3*K2**3*K3 + 1088*K1**3*K2*K3 - 672*K1**3*K3 - 448*K1**2*K2**4 + 512*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7488*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 704*K1**2*K2*K4 + 11000*K1**2*K2 - 896*K1**2*K3**2 - 64*K1**2*K4**2 - 8280*K1**2 + 1984*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 864*K1*K2**2*K5 + 224*K1*K2*K3**3 - 384*K1*K2*K3*K4 + 10520*K1*K2*K3 + 1480*K1*K3*K4 + 256*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1824*K2**4 - 32*K2**3*K6 - 1344*K2**2*K3**2 - 64*K2**2*K4**2 + 2320*K2**2*K4 - 6082*K2**2 + 1104*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 - 3260*K3**2 - 832*K4**2 - 244*K5**2 - 6*K6**2 + 6438 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | True |