| Min(phi) over symmetries of the knot is: [-4,-1,1,2,2,1,4,2,3,2,1,2,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['5.7'] |
| Arrow polynomial of the knot is: 2*K1**2 + 2*K1*K3 - 2*K2 - K4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['5.7', '7.352', '7.708', '7.1617', '7.2251', '7.2542', '7.2567', '7.2640', '7.2644', '7.2800', '7.2949', '7.3302', '7.3362', '7.3493', '7.3509', '7.4303', '7.4629', '7.4697', '7.6335', '7.6420', '7.6432', '7.6551', '7.6861', '7.7036', '7.7119', '7.7184', '7.7262', '7.7309', '7.8137', '7.8190', '7.8335', '7.8340', '7.8350', '7.8355', '7.8507', '7.8519', '7.8762', '7.9411', '7.9426', '7.9481', '7.9546', '7.9559', '7.9683', '7.9839', '7.10077', '7.11566', '7.11632', '7.12328', '7.13123', '7.13129', '7.13176', '7.13221', '7.13236', '7.13256', '7.13440', '7.14552', '7.15117', '7.15149', '7.15213', '7.15219', '7.17074', '7.17255'] |
| Outer characteristic polynomial of the knot is: t^6+67t^4+41t^2 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.7', '7.18621'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 - 128*K1**2*K2**2 + 280*K1**2*K2 - 192*K1**2*K3**2 - 544*K1**2 + 888*K1*K2*K3 + 184*K1*K3*K4 + 8*K1*K4*K5 + 16*K1*K5*K6 - 8*K2**4 - 192*K2**2*K3**2 + 40*K2**2*K4 - 8*K2**2*K6**2 - 524*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 424*K3**2 - 76*K4**2 - 72*K5**2 - 28*K6**2 - 2*K8**2 + 580 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.7'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.607', 'vk5.611', 'vk5.742', 'vk5.746', 'vk5.1090', 'vk5.1098', 'vk5.1256', 'vk5.1262', 'vk5.1608', 'vk5.1614', 'vk5.1724', 'vk5.1730', 'vk5.1840', 'vk5.1847', 'vk5.1907', 'vk5.1911'] |
| 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 | O1O2O3O4O5U1U3U5U4U2 |
| R3 orbit | {'O1O2O3O4O5U1U3U5U4U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U2U1U3U5 |
| Gauss code of K* | O1O2O3O4O5U1U5U2U4U3 |
| Gauss code of -K* | O1O2O3O4O5U3U2U4U1U5 |
| 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 -4 1 -1 2 2],[ 4 0 4 1 3 2],[-1 -4 0 -2 1 1],[ 1 -1 2 0 2 1],[-2 -3 -1 -2 0 0],[-2 -2 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -4],[-2 0 0 -1 -1 -2],[-2 0 0 -1 -2 -3],[-1 1 1 0 -2 -4],[ 1 1 2 2 0 -1],[ 4 2 3 4 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,4,0,1,1,2,1,2,3,2,4,1] |
| Phi over symmetry | [-4,-1,1,2,2,1,4,2,3,2,1,2,1,1,0] |
| Phi of -K | [-4,-1,1,2,2,2,1,3,4,0,1,2,0,0,0] |
| Phi of K* | [-2,-2,-1,1,4,0,0,1,3,0,2,4,0,1,2] |
| Phi of -K* | [-4,-1,1,2,2,1,4,2,3,2,1,2,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-2t^2 |
| Normalized Jones-Krushkal polynomial | -8z-15 |
| Enhanced Jones-Krushkal polynomial | 2w^3z-10w^2z-15w |
| Inner characteristic polynomial | t^5+41t^3+5t |
| Outer characteristic polynomial | t^6+67t^4+41t^2 |
| Flat arrow polynomial | 2*K1**2 + 2*K1*K3 - 2*K2 - K4 |
| 2-strand cable arrow polynomial | -64*K1**4 + 32*K1**3*K2*K3 - 128*K1**2*K2**2 + 280*K1**2*K2 - 192*K1**2*K3**2 - 544*K1**2 + 888*K1*K2*K3 + 184*K1*K3*K4 + 8*K1*K4*K5 + 16*K1*K5*K6 - 8*K2**4 - 192*K2**2*K3**2 + 40*K2**2*K4 - 8*K2**2*K6**2 - 524*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 + 8*K2*K6*K8 - 64*K3**4 + 56*K3**2*K6 - 424*K3**2 - 76*K4**2 - 72*K5**2 - 28*K6**2 - 2*K8**2 + 580 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 5}, {4}, {2, 3}], [{4, 5}, {2, 3}, {1}], [{5}, {1, 4}, {2, 3}], [{5}, {3, 4}, {1, 2}]] |
| If K is slice | False |