| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,2,2,0,0,1,1,0,1,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1576'] |
| 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+22t^5+27t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1576'] |
| 2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 192*K1**4*K2**2 + 96*K1**4*K2 - 208*K1**4 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 192*K1**3*K3 - 1536*K1**2*K2**4 + 3136*K1**2*K2**3 - 7104*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 5552*K1**2*K2 - 112*K1**2*K3**2 - 16*K1**2*K4**2 - 4048*K1**2 + 1760*K1*K2**3*K3 - 640*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5216*K1*K2*K3 + 456*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2552*K2**4 - 608*K2**2*K3**2 - 48*K2**2*K4**2 + 1344*K2**2*K4 - 1238*K2**2 + 176*K2*K3*K5 + 16*K2*K4*K6 - 1236*K3**2 - 330*K4**2 - 28*K5**2 - 2*K6**2 + 2776 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1576'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71572', 'vk6.71685', 'vk6.72101', 'vk6.72309', 'vk6.74031', 'vk6.74594', 'vk6.76072', 'vk6.76786', 'vk6.77192', 'vk6.77291', 'vk6.77493', 'vk6.77651', 'vk6.79014', 'vk6.79592', 'vk6.80553', 'vk6.81005', 'vk6.81103', 'vk6.81143', 'vk6.81163', 'vk6.81211', 'vk6.81313', 'vk6.81458', 'vk6.82260', 'vk6.83502', 'vk6.83830', 'vk6.83980', 'vk6.85393', 'vk6.86327', 'vk6.87105', 'vk6.88015', 'vk6.88338', 'vk6.88968'] |
| 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 | O1O2O3U4O5U2U3O6O4U1U5U6 |
| R3 orbit | {'O1O2O3U4O5U2U3O6O4U1U5U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5U3O6O4U1U2O5U6 |
| Gauss code of K* | O1O2O3U1U4U5O6U2O4O5U3U6 |
| Gauss code of -K* | O1O2O3U4U1O5O6U2O4U5U6U3 |
| 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 1 0 1 1],[ 2 0 0 2 0 2 1],[ 1 0 0 1 0 1 0],[-1 -2 -1 0 -1 0 0],[ 0 0 0 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 0 0 0 -1 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -1 -2],[-1 -1 0 0 0 0 -1],[-1 0 0 0 -1 -1 -2],[ 0 1 0 1 0 0 0],[ 1 1 0 1 0 0 0],[ 2 2 1 2 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,-1,0,1,1,2,0,0,0,1,1,1,2,0,0,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,0,1,2,2,0,0,1,1,0,1,1,-1,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,2,1,1,2,1,1,1,2,0,0,1,0,-1,0] |
| Phi of K* | [-1,-1,-1,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,0,1,2,2,0,0,1,1,0,1,1,-1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-6w^3z+23w^2z+19w |
| Inner characteristic polynomial | t^6+14t^4+10t^2 |
| Outer characteristic polynomial | t^7+22t^5+27t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 4*K1*K2 - K1 + 3*K2 + K3 + 4 |
| 2-strand cable arrow polynomial | 128*K1**4*K2**3 - 192*K1**4*K2**2 + 96*K1**4*K2 - 208*K1**4 - 128*K1**3*K2**2*K3 + 256*K1**3*K2*K3 - 192*K1**3*K3 - 1536*K1**2*K2**4 + 3136*K1**2*K2**3 - 7104*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 5552*K1**2*K2 - 112*K1**2*K3**2 - 16*K1**2*K4**2 - 4048*K1**2 + 1760*K1*K2**3*K3 - 640*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5216*K1*K2*K3 + 456*K1*K3*K4 + 40*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2552*K2**4 - 608*K2**2*K3**2 - 48*K2**2*K4**2 + 1344*K2**2*K4 - 1238*K2**2 + 176*K2*K3*K5 + 16*K2*K4*K6 - 1236*K3**2 - 330*K4**2 - 28*K5**2 - 2*K6**2 + 2776 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |