| Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,-1,1,3,2,1,0,0,0,1,1,1,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1046'] |
| Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931'] |
| Outer characteristic polynomial of the knot is: t^7+45t^5+219t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1046'] |
| 2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 832*K1**4*K2 - 2912*K1**4 + 288*K1**3*K2*K3 - 96*K1**3*K3 - 704*K1**2*K2**4 + 2944*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8864*K1**2*K2**2 - 224*K1**2*K2*K4 + 9832*K1**2*K2 - 64*K1**2*K3**2 - 4956*K1**2 + 1184*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 192*K1*K2**2*K5 + 6144*K1*K2*K3 + 152*K1*K3*K4 - 192*K2**6 + 128*K2**4*K4 - 2096*K2**4 - 432*K2**2*K3**2 - 16*K2**2*K4**2 + 1160*K2**2*K4 - 2720*K2**2 + 88*K2*K3*K5 - 1228*K3**2 - 96*K4**2 + 3830 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1046'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4671', 'vk6.4960', 'vk6.6137', 'vk6.6624', 'vk6.8148', 'vk6.8554', 'vk6.9528', 'vk6.9881', 'vk6.20698', 'vk6.22138', 'vk6.28227', 'vk6.29652', 'vk6.39687', 'vk6.41928', 'vk6.46263', 'vk6.47870', 'vk6.48711', 'vk6.48920', 'vk6.49487', 'vk6.49702', 'vk6.50737', 'vk6.50940', 'vk6.51214', 'vk6.51413', 'vk6.57629', 'vk6.58791', 'vk6.62313', 'vk6.63254', 'vk6.67107', 'vk6.67971', 'vk6.69703', 'vk6.70386'] |
| 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 | O1O2O3O4U5U6O5U1O6U4U2U3 |
| R3 orbit | {'O1O2O3O4U5U6O5U1O6U4U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U3U1O5U4O6U5U6 |
| Gauss code of K* | O1O2O3U4U2U3U1O5O6U5O4U6 |
| Gauss code of -K* | O1O2O3U4O5U6O4O6U3U1U2U5 |
| 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 0 2 1 -1 0],[ 2 0 1 2 0 2 3],[ 0 -1 0 1 0 -1 1],[-2 -2 -1 0 0 -3 -1],[-1 0 0 0 0 -2 0],[ 1 -2 1 3 2 0 0],[ 0 -3 -1 1 0 0 0]] |
| Primitive based matrix | [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -3 -2],[-1 0 0 0 0 -2 0],[ 0 1 0 0 1 -1 -1],[ 0 1 0 -1 0 0 -3],[ 1 3 2 1 0 0 -2],[ 2 2 0 1 3 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,0,0,1,2,0,1,1,3,2,0,0,2,0,-1,1,1,0,3,2] |
| Phi over symmetry | [-2,-1,0,0,1,2,-1,-1,1,3,2,1,0,0,0,1,1,1,1,1,1] |
| Phi of -K | [-2,-1,0,0,1,2,-1,-1,1,3,2,1,0,0,0,1,1,1,1,1,1] |
| Phi of K* | [-2,-1,0,0,1,2,1,1,1,0,2,1,1,0,3,-1,1,-1,0,1,-1] |
| Phi of -K* | [-2,-1,0,0,1,2,2,1,3,0,2,1,0,2,3,1,0,1,0,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 5z^2+26z+33 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+26w^2z+33w |
| Inner characteristic polynomial | t^6+35t^4+159t^2 |
| Outer characteristic polynomial | t^7+45t^5+219t^3+7t |
| Flat arrow polynomial | 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | -448*K1**4*K2**2 + 832*K1**4*K2 - 2912*K1**4 + 288*K1**3*K2*K3 - 96*K1**3*K3 - 704*K1**2*K2**4 + 2944*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8864*K1**2*K2**2 - 224*K1**2*K2*K4 + 9832*K1**2*K2 - 64*K1**2*K3**2 - 4956*K1**2 + 1184*K1*K2**3*K3 - 1408*K1*K2**2*K3 - 192*K1*K2**2*K5 + 6144*K1*K2*K3 + 152*K1*K3*K4 - 192*K2**6 + 128*K2**4*K4 - 2096*K2**4 - 432*K2**2*K3**2 - 16*K2**2*K4**2 + 1160*K2**2*K4 - 2720*K2**2 + 88*K2*K3*K5 - 1228*K3**2 - 96*K4**2 + 3830 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |