| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,2,-1,0,0,1,0,1,1,0,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1926'] |
| 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+19t^5+35t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1926'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 1312*K1**4*K2 - 3568*K1**4 + 736*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1632*K1**3*K3 + 448*K1**2*K2**3 - 5536*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 11608*K1**2*K2 - 1456*K1**2*K3**2 - 128*K1**2*K3*K5 - 112*K1**2*K4**2 - 8600*K1**2 + 256*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 96*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 10232*K1*K2*K3 - 128*K1*K2*K4*K5 + 2424*K1*K3*K4 + 328*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 808*K2**4 - 32*K2**3*K6 - 496*K2**2*K3**2 - 256*K2**2*K4**2 + 1704*K2**2*K4 - 6666*K2**2 + 608*K2*K3*K5 + 264*K2*K4*K6 - 3648*K3**2 - 1182*K4**2 - 248*K5**2 - 86*K6**2 + 6996 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1926'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16576', 'vk6.16667', 'vk6.18136', 'vk6.18472', 'vk6.22979', 'vk6.23098', 'vk6.24595', 'vk6.25008', 'vk6.34976', 'vk6.35095', 'vk6.36726', 'vk6.37145', 'vk6.42549', 'vk6.42658', 'vk6.43998', 'vk6.44310', 'vk6.54807', 'vk6.54889', 'vk6.55938', 'vk6.56234', 'vk6.59239', 'vk6.59315', 'vk6.60476', 'vk6.60838', 'vk6.64789', 'vk6.64852', 'vk6.65591', 'vk6.65898', 'vk6.68091', 'vk6.68154', 'vk6.68666', 'vk6.68877'] |
| 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 | O1O2O3U1U4U2O5O4O6U3U6U5 |
| R3 orbit | {'O1O2O3U1U4U2O5O4O6U3U6U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U5O6O4O5U3U1U6 |
| Gauss code of K* | O1O2O3U4U5U1O4O6O5U3U6U2 |
| Gauss code of -K* | O1O2O3U2U4U1O5O4O6U3U5U6 |
| 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 1 0 0 0 1],[ 2 0 1 2 1 1 1],[-1 -1 0 0 -1 0 1],[ 0 -2 0 0 0 1 1],[ 0 -1 1 0 0 0 1],[ 0 -1 0 -1 0 0 0],[-1 -1 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 0 0 0 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 0 -1 -1 -1],[ 0 0 0 0 -1 0 -1],[ 0 0 1 1 0 0 -2],[ 0 1 1 0 0 0 -1],[ 2 1 1 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,0,0,0,2,-1,0,0,1,1,0,1,1,1,1,0,1,0,2,1] |
| Phi over symmetry | [-2,0,0,0,1,1,0,1,1,2,2,-1,0,0,1,0,1,1,0,0,1] |
| Phi of -K | [-2,0,0,0,1,1,0,1,1,2,2,-1,0,0,1,0,1,1,0,0,1] |
| Phi of K* | [-1,-1,0,0,0,2,-1,0,0,1,2,0,1,1,2,0,0,1,1,0,1] |
| Phi of -K* | [-2,0,0,0,1,1,1,1,2,1,1,0,-1,0,0,0,1,1,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+13t^4+16t^2 |
| Outer characteristic polynomial | t^7+19t^5+35t^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 | -192*K1**6 - 192*K1**4*K2**2 + 1312*K1**4*K2 - 3568*K1**4 + 736*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1632*K1**3*K3 + 448*K1**2*K2**3 - 5536*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 11608*K1**2*K2 - 1456*K1**2*K3**2 - 128*K1**2*K3*K5 - 112*K1**2*K4**2 - 8600*K1**2 + 256*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 96*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 10232*K1*K2*K3 - 128*K1*K2*K4*K5 + 2424*K1*K3*K4 + 328*K1*K4*K5 + 64*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 808*K2**4 - 32*K2**3*K6 - 496*K2**2*K3**2 - 256*K2**2*K4**2 + 1704*K2**2*K4 - 6666*K2**2 + 608*K2*K3*K5 + 264*K2*K4*K6 - 3648*K3**2 - 1182*K4**2 - 248*K5**2 - 86*K6**2 + 6996 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}]] |
| If K is slice | False |