| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,2,1,1,2,1,1,0,1,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1200'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845'] |
| Outer characteristic polynomial of the knot is: t^7+47t^5+37t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1200'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 608*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 + 384*K1**2*K2**5 - 1984*K1**2*K2**4 - 896*K1**2*K2**3*K4 + 3104*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6592*K1**2*K2**2 - 704*K1**2*K2*K4 + 5464*K1**2*K2 - 160*K1**2*K3**2 - 160*K1**2*K4**2 - 3656*K1**2 + 2304*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 + 32*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4312*K1*K2*K3 + 912*K1*K3*K4 + 144*K1*K4*K5 - 288*K2**6 + 576*K2**4*K4 - 2136*K2**4 - 832*K2**2*K3**2 - 464*K2**2*K4**2 + 1368*K2**2*K4 - 1230*K2**2 + 192*K2*K3*K5 + 72*K2*K4*K6 - 1012*K3**2 - 550*K4**2 - 52*K5**2 - 2*K6**2 + 2652 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1200'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10133', 'vk6.10186', 'vk6.10329', 'vk6.10420', 'vk6.17664', 'vk6.17713', 'vk6.24231', 'vk6.24280', 'vk6.29916', 'vk6.29959', 'vk6.30022', 'vk6.30075', 'vk6.36495', 'vk6.36591', 'vk6.43592', 'vk6.43704', 'vk6.51621', 'vk6.51658', 'vk6.51707', 'vk6.51726', 'vk6.55694', 'vk6.55753', 'vk6.60264', 'vk6.60328', 'vk6.63338', 'vk6.63367', 'vk6.63389', 'vk6.63405', 'vk6.65404', 'vk6.65445', 'vk6.68544', 'vk6.68577'] |
| 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 | O1O2O3O4U2U3U5O6O5U1U4U6 |
| R3 orbit | {'O1O2O3O4U2U3U5O6O5U1U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U4O6O5U6U2U3 |
| Gauss code of K* | O1O2O3U4U3O5O6O4U5U1U2U6 |
| Gauss code of -K* | O1O2O3U4U1O4O5O6U2U5U6U3 |
| 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 -2 0 2 1 1],[ 2 0 -1 1 3 2 1],[ 2 1 0 1 2 2 1],[ 0 -1 -1 0 1 0 1],[-2 -3 -2 -1 0 -2 0],[-1 -2 -2 0 2 0 1],[-1 -1 -1 -1 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 0 -2 -2],[ 0 1 1 0 0 -1 -1],[ 2 2 1 2 1 0 1],[ 2 3 1 2 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,0,2,1,2,3,1,1,1,1,0,2,2,1,1,-1] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,1,1,2,2,1,1,2,1,1,0,1,-1,-1,1] |
| Phi of -K | [-2,-2,0,1,1,2,-1,1,1,2,2,1,1,2,1,1,0,1,-1,-1,1] |
| Phi of K* | [-2,-1,-1,0,2,2,-1,1,1,1,2,1,1,1,1,0,2,2,1,1,-1] |
| Phi of -K* | [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,2,2,1,0,1,-1,0,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w |
| Inner characteristic polynomial | t^6+33t^4+12t^2 |
| Outer characteristic polynomial | t^7+47t^5+37t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | -192*K1**4*K2**2 + 224*K1**4*K2 - 608*K1**4 + 128*K1**3*K2**3*K3 + 224*K1**3*K2*K3 + 384*K1**2*K2**5 - 1984*K1**2*K2**4 - 896*K1**2*K2**3*K4 + 3104*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 6592*K1**2*K2**2 - 704*K1**2*K2*K4 + 5464*K1**2*K2 - 160*K1**2*K3**2 - 160*K1**2*K4**2 - 3656*K1**2 + 2304*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 1024*K1*K2**2*K3 + 32*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4312*K1*K2*K3 + 912*K1*K3*K4 + 144*K1*K4*K5 - 288*K2**6 + 576*K2**4*K4 - 2136*K2**4 - 832*K2**2*K3**2 - 464*K2**2*K4**2 + 1368*K2**2*K4 - 1230*K2**2 + 192*K2*K3*K5 + 72*K2*K4*K6 - 1012*K3**2 - 550*K4**2 - 52*K5**2 - 2*K6**2 + 2652 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |