| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,3,1,1,1,2,1,0,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1497'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928'] |
| Outer characteristic polynomial of the knot is: t^7+31t^5+49t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1497'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 256*K1**4*K2 - 880*K1**4 + 352*K1**3*K2*K3 - 224*K1**3*K3 + 1088*K1**2*K2**3 - 3472*K1**2*K2**2 - 1024*K1**2*K2*K4 + 5040*K1**2*K2 - 304*K1**2*K3**2 - 4132*K1**2 + 96*K1*K2**3*K3 - 672*K1*K2**2*K3 + 4264*K1*K2*K3 + 1264*K1*K3*K4 + 8*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 256*K2**4*K4 - 1056*K2**4 - 288*K2**2*K3**2 - 528*K2**2*K4**2 + 1640*K2**2*K4 - 2998*K2**2 + 344*K2*K3*K5 + 304*K2*K4*K6 + 24*K3**2*K6 - 1500*K3**2 - 880*K4**2 - 120*K5**2 - 66*K6**2 + 3358 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1497'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11290', 'vk6.11368', 'vk6.12555', 'vk6.12666', 'vk6.18351', 'vk6.18689', 'vk6.24796', 'vk6.25253', 'vk6.30968', 'vk6.31093', 'vk6.32150', 'vk6.32269', 'vk6.36982', 'vk6.37435', 'vk6.44167', 'vk6.44487', 'vk6.52062', 'vk6.52141', 'vk6.52901', 'vk6.52962', 'vk6.56125', 'vk6.56349', 'vk6.60645', 'vk6.60986', 'vk6.63671', 'vk6.63716', 'vk6.64103', 'vk6.64148', 'vk6.65781', 'vk6.66040', 'vk6.68785', 'vk6.68994'] |
| 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 | O1O2O3U1O4U5U3O5O6U4U2U6 |
| R3 orbit | {'O1O2O3U1O4U5U3O5O6U4U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U5O4O6U1U6O5U3 |
| Gauss code of K* | O1O2O3U4U2U5O4U1O6O5U6U3 |
| Gauss code of -K* | O1O2O3U1U4O5O4U3O6U5U2U6 |
| 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 0 1 0 -1 2],[ 2 0 2 1 1 1 1],[ 0 -2 0 0 1 -1 2],[-1 -1 0 0 0 -1 1],[ 0 -1 -1 0 0 0 1],[ 1 -1 1 1 0 0 2],[-2 -1 -2 -1 -1 -2 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -2 -1],[-1 1 0 0 0 -1 -1],[ 0 1 0 0 -1 0 -1],[ 0 2 0 1 0 -1 -2],[ 1 2 1 0 1 0 -1],[ 2 1 1 1 2 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,1,1,2,2,1,0,0,1,1,1,0,1,1,2,1] |
| Phi over symmetry | [-2,-1,0,0,1,2,0,0,1,1,3,1,1,1,2,1,0,0,1,1,0] |
| Phi of -K | [-2,-1,0,0,1,2,0,0,1,2,3,0,1,1,1,-1,1,0,1,1,0] |
| Phi of K* | [-2,-1,0,0,1,2,0,0,1,1,3,1,1,1,2,1,0,0,1,1,0] |
| Phi of -K* | [-2,-1,0,0,1,2,1,1,2,1,1,0,1,1,2,-1,0,1,0,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+23z+31 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-2w^3z+25w^2z+31w |
| Inner characteristic polynomial | t^6+21t^4+23t^2 |
| Outer characteristic polynomial | t^7+31t^5+49t^3+9t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 256*K1**4*K2 - 880*K1**4 + 352*K1**3*K2*K3 - 224*K1**3*K3 + 1088*K1**2*K2**3 - 3472*K1**2*K2**2 - 1024*K1**2*K2*K4 + 5040*K1**2*K2 - 304*K1**2*K3**2 - 4132*K1**2 + 96*K1*K2**3*K3 - 672*K1*K2**2*K3 + 4264*K1*K2*K3 + 1264*K1*K3*K4 + 8*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 256*K2**4*K4 - 1056*K2**4 - 288*K2**2*K3**2 - 528*K2**2*K4**2 + 1640*K2**2*K4 - 2998*K2**2 + 344*K2*K3*K5 + 304*K2*K4*K6 + 24*K3**2*K6 - 1500*K3**2 - 880*K4**2 - 120*K5**2 - 66*K6**2 + 3358 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {3, 5}, {2, 4}]] |
| If K is slice | False |