| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,1,3,3,3,0,2,1,1,1,2,2,2,3,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.419'] |
| 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+84t^5+141t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.419'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 512*K1**4*K2 - 976*K1**4 + 320*K1**3*K2*K3 - 96*K1**3*K3 - 320*K1**2*K2**4 + 512*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 2592*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 3312*K1**2*K2 - 112*K1**2*K3**2 - 32*K1**2*K4**2 - 1880*K1**2 + 704*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 608*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2392*K1*K2*K3 + 336*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 528*K2**4 - 416*K2**2*K3**2 - 48*K2**2*K4**2 + 480*K2**2*K4 - 1286*K2**2 + 136*K2*K3*K5 + 8*K2*K4*K6 - 644*K3**2 - 180*K4**2 - 28*K5**2 - 2*K6**2 + 1554 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.419'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17009', 'vk6.17250', 'vk6.19983', 'vk6.20241', 'vk6.21156', 'vk6.21545', 'vk6.23417', 'vk6.26872', 'vk6.27006', 'vk6.27462', 'vk6.28636', 'vk6.29060', 'vk6.35490', 'vk6.38305', 'vk6.38416', 'vk6.38875', 'vk6.40432', 'vk6.41075', 'vk6.42919', 'vk6.45171', 'vk6.45302', 'vk6.45642', 'vk6.47009', 'vk6.47379', 'vk6.55194', 'vk6.56719', 'vk6.56797', 'vk6.57810', 'vk6.58211', 'vk6.59577', 'vk6.61301', 'vk6.62384', 'vk6.64993', 'vk6.66415', 'vk6.66509', 'vk6.67181', 'vk6.67549', 'vk6.68277', 'vk6.69155', 'vk6.69853'] |
| 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 | O1O2O3O4O5U6U1O6U2U5U3U4 |
| R3 orbit | {'O1O2O3O4O5U2U6U1O6U5U3U4', 'O1O2O3O4O5U6U1O6U2U5U3U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U2U3U1U4O6U5U6 |
| Gauss code of K* | O1O2O3O4U5U1U3U4U2O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U3U1U2U4U6 |
| 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 -3 -2 1 3 2 -1],[ 3 0 0 2 3 1 3],[ 2 0 0 2 3 1 2],[-1 -2 -2 0 1 0 -1],[-3 -3 -3 -1 0 0 -3],[-2 -1 -1 0 0 0 -2],[ 1 -3 -2 1 3 2 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 0 -1 -3 -3 -3],[-2 0 0 0 -2 -1 -1],[-1 1 0 0 -1 -2 -2],[ 1 3 2 1 0 -2 -3],[ 2 3 1 2 2 0 0],[ 3 3 1 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,0,1,3,3,3,0,2,1,1,1,2,2,2,3,0] |
| Phi over symmetry | [-3,-2,-1,1,2,3,0,1,3,3,3,0,2,1,1,1,2,2,2,3,0] |
| Phi of -K | [-3,-2,-1,1,2,3,1,-1,2,4,3,-1,1,3,2,1,1,1,1,1,1] |
| Phi of K* | [-3,-2,-1,1,2,3,1,1,1,2,3,1,1,3,4,1,1,2,-1,-1,1] |
| Phi of -K* | [-3,-2,-1,1,2,3,0,3,2,1,3,2,2,1,3,1,2,3,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+56t^4+57t^2+1 |
| Outer characteristic polynomial | t^7+84t^5+141t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -128*K1**6 - 256*K1**4*K2**2 + 512*K1**4*K2 - 976*K1**4 + 320*K1**3*K2*K3 - 96*K1**3*K3 - 320*K1**2*K2**4 + 512*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 2592*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 3312*K1**2*K2 - 112*K1**2*K3**2 - 32*K1**2*K4**2 - 1880*K1**2 + 704*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 608*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2392*K1*K2*K3 + 336*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 528*K2**4 - 416*K2**2*K3**2 - 48*K2**2*K4**2 + 480*K2**2*K4 - 1286*K2**2 + 136*K2*K3*K5 + 8*K2*K4*K6 - 644*K3**2 - 180*K4**2 - 28*K5**2 - 2*K6**2 + 1554 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | False |