| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,1,0,0,0,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1615', '6.1849'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+20t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1615', '6.1849'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 3616*K1**4*K2 - 6368*K1**4 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 96*K1**3*K3*K4 - 2400*K1**3*K3 + 32*K1**3*K4*K5 + 992*K1**2*K2**3 - 6544*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 12112*K1**2*K2 - 1120*K1**2*K3**2 - 160*K1**2*K3*K5 - 368*K1**2*K4**2 - 32*K1**2*K4*K6 - 32*K1**2*K5**2 - 6648*K1**2 + 256*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8440*K1*K2*K3 - 32*K1*K2*K4*K5 + 2440*K1*K3*K4 + 744*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 808*K2**4 - 32*K2**3*K6 - 240*K2**2*K3**2 - 64*K2**2*K4**2 + 1288*K2**2*K4 - 5210*K2**2 + 440*K2*K3*K5 + 56*K2*K4*K6 - 2752*K3**2 - 1102*K4**2 - 312*K5**2 - 14*K6**2 + 5852 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1849'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4834', 'vk6.5177', 'vk6.6402', 'vk6.6833', 'vk6.8367', 'vk6.8795', 'vk6.9735', 'vk6.10038', 'vk6.11631', 'vk6.11982', 'vk6.12977', 'vk6.20460', 'vk6.20727', 'vk6.21815', 'vk6.27844', 'vk6.29354', 'vk6.31434', 'vk6.32612', 'vk6.39278', 'vk6.39759', 'vk6.41458', 'vk6.46323', 'vk6.47581', 'vk6.47900', 'vk6.49059', 'vk6.49891', 'vk6.51321', 'vk6.51538', 'vk6.53242', 'vk6.57319', 'vk6.62005', 'vk6.64319'] |
| 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 | O1O2O3U2U4O5U1O6O4U6U5U3 |
| R3 orbit | {'O1O2O3U2U4O5U1O6O4U6U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U5O6O5U3O4U6U2 |
| Gauss code of K* | O1O2O3U4U5U3O5O6U2O4U1U6 |
| Gauss code of -K* | O1O2O3U4U3O5U2O4O6U1U6U5 |
| 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 -1 2 1 0 -1],[ 1 0 0 2 1 0 -1],[ 1 0 0 1 1 0 0],[-2 -2 -1 0 0 -1 -1],[-1 -1 -1 0 0 -1 -1],[ 0 0 0 1 1 0 0],[ 1 1 0 1 1 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -1 -2],[-1 0 0 -1 -1 -1 -1],[ 0 1 1 0 0 0 0],[ 1 1 1 0 0 0 1],[ 1 1 1 0 0 0 0],[ 1 2 1 0 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,1,0,0,0,0,-1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,1,0,0,0,0,-1,0] |
| Phi of -K | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,1,1,1,1,2,0,1,1] |
| Phi of K* | [-2,-1,0,1,1,1,1,1,1,2,2,0,1,1,1,1,1,1,-1,0,0] |
| Phi of -K* | [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,1,1,0,1,1,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 3z^2+24z+37 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+24w^2z+37w |
| Inner characteristic polynomial | t^6+12t^4+7t^2 |
| Outer characteristic polynomial | t^7+20t^5+20t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6 |
| 2-strand cable arrow polynomial | -128*K1**6 - 256*K1**4*K2**2 + 3616*K1**4*K2 - 6368*K1**4 - 128*K1**3*K2**2*K3 + 768*K1**3*K2*K3 + 96*K1**3*K3*K4 - 2400*K1**3*K3 + 32*K1**3*K4*K5 + 992*K1**2*K2**3 - 6544*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 12112*K1**2*K2 - 1120*K1**2*K3**2 - 160*K1**2*K3*K5 - 368*K1**2*K4**2 - 32*K1**2*K4*K6 - 32*K1**2*K5**2 - 6648*K1**2 + 256*K1*K2**3*K3 - 1280*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8440*K1*K2*K3 - 32*K1*K2*K4*K5 + 2440*K1*K3*K4 + 744*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 808*K2**4 - 32*K2**3*K6 - 240*K2**2*K3**2 - 64*K2**2*K4**2 + 1288*K2**2*K4 - 5210*K2**2 + 440*K2*K3*K5 + 56*K2*K4*K6 - 2752*K3**2 - 1102*K4**2 - 312*K5**2 - 14*K6**2 + 5852 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}]] |
| If K is slice | False |