| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,1,3,5,1,0,1,2,0,1,2,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.600', '7.17789'] |
| 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+76t^5+51t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.600'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 624*K1**4 + 96*K1**2*K2**3 - 752*K1**2*K2**2 + 1304*K1**2*K2 - 272*K1**2*K3**2 - 48*K1**2*K4**2 - 964*K1**2 + 96*K1*K2**3*K3 + 1384*K1*K2*K3 + 424*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 208*K2**4 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 176*K2**2*K4 - 774*K2**2 + 104*K2*K3*K5 + 16*K2*K4*K6 - 596*K3**2 - 212*K4**2 - 40*K5**2 - 10*K6**2 + 1026 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.600'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11538', 'vk6.11871', 'vk6.12888', 'vk6.13197', 'vk6.20343', 'vk6.21684', 'vk6.27647', 'vk6.29191', 'vk6.31313', 'vk6.31710', 'vk6.32471', 'vk6.32888', 'vk6.39073', 'vk6.41329', 'vk6.45829', 'vk6.47498', 'vk6.52313', 'vk6.52575', 'vk6.53157', 'vk6.53459', 'vk6.57214', 'vk6.58435', 'vk6.61828', 'vk6.62959', 'vk6.63814', 'vk6.63948', 'vk6.64260', 'vk6.64458', 'vk6.66823', 'vk6.67691', 'vk6.69463', 'vk6.70185'] |
| 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 | O1O2O3O4U2O5U3O6U1U6U5U4 |
| R3 orbit | {'O1O2O3O4U2O5U3O6U1U6U5U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U5U6U4O6U2O5U3 |
| Gauss code of K* | O1O2O3O4U1U5U6U4O5U3O6U2 |
| Gauss code of -K* | O1O2O3O4U3O5U2O6U1U5U6U4 |
| 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 -3 -2 -1 3 2 1],[ 3 0 -1 1 5 3 1],[ 2 1 0 1 2 1 0],[ 1 -1 -1 0 2 1 0],[-3 -5 -2 -2 0 0 0],[-2 -3 -1 -1 0 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 0 0 -2 -2 -5],[-2 0 0 0 -1 -1 -3],[-1 0 0 0 0 0 -1],[ 1 2 1 0 0 -1 -1],[ 2 2 1 0 1 0 1],[ 3 5 3 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,0,0,2,2,5,0,1,1,3,0,0,1,1,1,-1] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,1,3,5,1,0,1,2,0,1,2,0,0,0] |
| Phi of -K | [-3,-2,-1,1,2,3,2,1,3,2,1,0,3,3,3,2,2,2,1,2,1] |
| Phi of K* | [-3,-2,-1,1,2,3,1,2,2,3,1,1,2,3,2,2,3,3,0,1,2] |
| Phi of -K* | [-3,-2,-1,1,2,3,-1,1,1,3,5,1,0,1,2,0,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 9z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+11w^2z+19w |
| Inner characteristic polynomial | t^6+48t^4+15t^2 |
| Outer characteristic polynomial | t^7+76t^5+51t^3 |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -64*K1**6 + 64*K1**4*K2 - 624*K1**4 + 96*K1**2*K2**3 - 752*K1**2*K2**2 + 1304*K1**2*K2 - 272*K1**2*K3**2 - 48*K1**2*K4**2 - 964*K1**2 + 96*K1*K2**3*K3 + 1384*K1*K2*K3 + 424*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 208*K2**4 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 176*K2**2*K4 - 774*K2**2 + 104*K2*K3*K5 + 16*K2*K4*K6 - 596*K3**2 - 212*K4**2 - 40*K5**2 - 10*K6**2 + 1026 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}]] |
| If K is slice | True |