| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,1,0,0,0,1,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1542'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878'] |
| Outer characteristic polynomial of the knot is: t^7+36t^5+120t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1542'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**6 - 128*K1**4*K2**2 + 1920*K1**4*K2 - 4496*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4208*K1**2*K2**2 - 416*K1**2*K2*K4 + 7496*K1**2*K2 - 720*K1**2*K3**2 - 160*K1**2*K3*K5 - 112*K1**2*K4**2 - 4120*K1**2 + 448*K1*K2**3*K3 - 736*K1*K2**2*K3 - 64*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 5208*K1*K2*K3 + 2216*K1*K3*K4 + 384*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 568*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 912*K2**2*K4 - 3638*K2**2 + 304*K2*K3*K5 + 16*K2*K4*K6 - 2264*K3**2 - 1086*K4**2 - 168*K5**2 - 2*K6**2 + 4380 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1542'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11468', 'vk6.11773', 'vk6.12790', 'vk6.13127', 'vk6.17052', 'vk6.17295', 'vk6.20857', 'vk6.20950', 'vk6.22266', 'vk6.22360', 'vk6.23774', 'vk6.28327', 'vk6.31229', 'vk6.31580', 'vk6.32804', 'vk6.35567', 'vk6.36018', 'vk6.39947', 'vk6.40120', 'vk6.42026', 'vk6.42968', 'vk6.43265', 'vk6.46492', 'vk6.46638', 'vk6.52221', 'vk6.53062', 'vk6.53378', 'vk6.55461', 'vk6.58854', 'vk6.59942', 'vk6.64401', 'vk6.69720'] |
| 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 | O1O2O3U4O5U1U6O4O6U3U5U2 |
| R3 orbit | {'O1O2O3U4O5U1U6O4O6U3U5U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U2U4U1O5O6U5U3O4U6 |
| Gauss code of K* | O1O2O3U4U3U1O5U2O4O6U5U6 |
| Gauss code of -K* | O1O2O3U4U5O4O6U2O5U3U1U6 |
| 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 -1 1 1],[ 2 0 2 0 2 1 3],[-1 -2 0 -1 -1 0 0],[ 0 0 1 0 -1 0 1],[ 1 -2 1 1 0 2 1],[-1 -1 0 0 -2 0 -1],[-1 -3 0 -1 -1 1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -3],[-1 -1 0 0 0 -2 -1],[-1 0 0 0 -1 -1 -2],[ 0 1 0 1 0 -1 0],[ 1 1 2 1 1 0 -2],[ 2 3 1 2 0 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,3,0,0,2,1,1,1,2,1,0,2] |
| Phi over symmetry | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,1,0,0,0,1,0,-1,0] |
| Phi of -K | [-2,-1,0,1,1,1,-1,2,0,1,2,0,1,1,0,0,0,1,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,0,2,0,0,1,0,0,1,1,0,2,-1] |
| Phi of -K* | [-2,-1,0,1,1,1,2,0,1,2,3,1,2,1,1,0,1,1,0,-1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+25z+35 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+25w^2z+35w |
| Inner characteristic polynomial | t^6+28t^4+83t^2+4 |
| Outer characteristic polynomial | t^7+36t^5+120t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 4*K1*K2 - K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -192*K1**6 - 128*K1**4*K2**2 + 1920*K1**4*K2 - 4496*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4208*K1**2*K2**2 - 416*K1**2*K2*K4 + 7496*K1**2*K2 - 720*K1**2*K3**2 - 160*K1**2*K3*K5 - 112*K1**2*K4**2 - 4120*K1**2 + 448*K1*K2**3*K3 - 736*K1*K2**2*K3 - 64*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 5208*K1*K2*K3 + 2216*K1*K3*K4 + 384*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 568*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 912*K2**2*K4 - 3638*K2**2 + 304*K2*K3*K5 + 16*K2*K4*K6 - 2264*K3**2 - 1086*K4**2 - 168*K5**2 - 2*K6**2 + 4380 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}]] |
| If K is slice | False |