| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,3,2,4,1,2,1,2,1,1,2,1,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.231', '7.17815'] |
| 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+80t^5+20t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.231'] |
| 2-strand cable arrow polynomial of the knot is: -96*K1**4 + 64*K1**2*K2**3 - 352*K1**2*K2**2 + 632*K1**2*K2 - 32*K1**2*K3**2 - 16*K1**2*K4**2 - 576*K1**2 + 32*K1*K2**3*K3 + 368*K1*K2*K3 + 152*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 112*K2**4 - 32*K2**2*K3**2 - 16*K2**2*K4**2 + 112*K2**2*K4 - 342*K2**2 + 48*K2*K3*K5 + 16*K2*K4*K6 - 188*K3**2 - 112*K4**2 - 28*K5**2 - 10*K6**2 + 462 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.231'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11230', 'vk6.11309', 'vk6.12491', 'vk6.12602', 'vk6.18234', 'vk6.18570', 'vk6.24702', 'vk6.25116', 'vk6.30900', 'vk6.31023', 'vk6.32084', 'vk6.32203', 'vk6.36822', 'vk6.37284', 'vk6.44062', 'vk6.44402', 'vk6.51976', 'vk6.52071', 'vk6.52857', 'vk6.52904', 'vk6.56027', 'vk6.56302', 'vk6.60572', 'vk6.60911', 'vk6.63632', 'vk6.63677', 'vk6.64060', 'vk6.64105', 'vk6.65686', 'vk6.65977', 'vk6.68731', 'vk6.68940', 'vk6.71608', 'vk6.71611', 'vk6.71733', 'vk6.72152', 'vk6.72156', 'vk6.72329', 'vk6.77206', 'vk6.77227', 'vk6.77231', 'vk6.77539', 'vk6.77542', 'vk6.78716', 'vk6.78920', 'vk6.78922', 'vk6.79610', 'vk6.81345', 'vk6.81357', 'vk6.85412', 'vk6.85414', 'vk6.87980', 'vk6.87992', 'vk6.88357', 'vk6.89315', 'vk6.89317'] |
| 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 | O1O2O3O4O5U3O6U2U5U6U1U4 |
| R3 orbit | {'O1O2O3O4O5U3U1O6U5U2U6U4', 'O1O2O3O4U2O5U1O6U3U5U6U4', 'O1O2O3O4O5U3U1U4O6U2U5U6', 'O1O2O3O4O5U3O6U2U5U6U1U4'} |
| R3 orbit length | 4 |
| Gauss code of -K | O1O2O3O4O5U2U5U6U1U4O6U3 |
| Gauss code of K* | O1O2O3O4O5U4U1U6U5U2O6U3 |
| Gauss code of -K* | O1O2O3O4O5U3O6U4U1U6U5U2 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -1 -3 -2 3 1 2],[ 1 0 -2 -1 3 1 2],[ 3 2 0 0 4 2 2],[ 2 1 0 0 2 1 1],[-3 -3 -4 -2 0 -1 1],[-1 -1 -2 -1 1 0 1],[-2 -2 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 1 -1 -3 -2 -4],[-2 -1 0 -1 -2 -1 -2],[-1 1 1 0 -1 -1 -2],[ 1 3 2 1 0 -1 -2],[ 2 2 1 1 1 0 0],[ 3 4 2 2 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,-1,1,3,2,4,1,2,1,2,1,1,2,1,2,0] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,3,2,4,1,2,1,2,1,1,2,1,2,0] |
| Phi of -K | [-3,-2,-1,1,2,3,1,0,2,3,2,0,2,3,3,1,1,1,0,1,2] |
| Phi of K* | [-3,-2,-1,1,2,3,2,1,1,3,2,0,1,3,3,1,2,2,0,0,1] |
| Phi of -K* | [-3,-2,-1,1,2,3,0,2,2,2,4,1,1,1,2,1,2,3,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z+11 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+7w^2z+11w |
| Inner characteristic polynomial | t^6+52t^4 |
| Outer characteristic polynomial | t^7+80t^5+20t^3 |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -96*K1**4 + 64*K1**2*K2**3 - 352*K1**2*K2**2 + 632*K1**2*K2 - 32*K1**2*K3**2 - 16*K1**2*K4**2 - 576*K1**2 + 32*K1*K2**3*K3 + 368*K1*K2*K3 + 152*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 112*K2**4 - 32*K2**2*K3**2 - 16*K2**2*K4**2 + 112*K2**2*K4 - 342*K2**2 + 48*K2*K3*K5 + 16*K2*K4*K6 - 188*K3**2 - 112*K4**2 - 28*K5**2 - 10*K6**2 + 462 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]] |
| If K is slice | False |