| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,2,3,3,-1,0,2,2,0,0,1,0,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.712'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+50t^5+150t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.712'] |
| 2-strand cable arrow polynomial of the knot is: -528*K1**4 - 32*K1**3*K3 + 576*K1**2*K2**3 - 2320*K1**2*K2**2 - 192*K1**2*K2*K4 + 4440*K1**2*K2 - 16*K1**2*K3**2 - 4096*K1**2 + 192*K1*K2**3*K3 - 416*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3616*K1*K2*K3 + 312*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 872*K2**4 - 464*K2**2*K3**2 - 168*K2**2*K4**2 + 1096*K2**2*K4 - 2928*K2**2 + 312*K2*K3*K5 + 48*K2*K4*K6 - 1300*K3**2 - 430*K4**2 - 60*K5**2 + 3124 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.712'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71590', 'vk6.71607', 'vk6.71716', 'vk6.71730', 'vk6.72132', 'vk6.72148', 'vk6.72328', 'vk6.74057', 'vk6.74059', 'vk6.74621', 'vk6.76809', 'vk6.77207', 'vk6.77222', 'vk6.77516', 'vk6.77530', 'vk6.77665', 'vk6.79055', 'vk6.79063', 'vk6.79621', 'vk6.79630', 'vk6.80575', 'vk6.80584', 'vk6.81025', 'vk6.81035', 'vk6.81348', 'vk6.81363', 'vk6.81394', 'vk6.85423', 'vk6.85428', 'vk6.85497', 'vk6.87988', 'vk6.89311'] |
| 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 | O1O2O3O4U5O6U4U2O5U1U3U6 |
| R3 orbit | {'O1O2O3O4U5O6U4U2O5U1U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2U4O6U3U1O5U6 |
| Gauss code of K* | O1O2O3U1U4U2U5O6U3O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U1O4U6U2U5U3 |
| 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 3],[ 2 0 1 2 1 0 3],[ 1 -1 0 0 0 0 2],[-1 -2 0 0 1 -2 1],[ 0 -1 0 -1 0 0 0],[ 1 0 0 2 0 0 3],[-3 -3 -2 -1 0 -3 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 0 -2 -3 -3],[-1 1 0 1 0 -2 -2],[ 0 0 -1 0 0 0 -1],[ 1 2 0 0 0 0 -1],[ 1 3 2 0 0 0 0],[ 2 3 2 1 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,0,2,3,3,-1,0,2,2,0,0,1,0,1,0] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,0,2,3,3,-1,0,2,2,0,0,1,0,1,0] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,1,1,2,0,1,2,2,1,0,1,2,3,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,3,1,2,2,2,0,2,1,1,1,1,0,1,0] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,1,2,3,0,0,2,3,0,0,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 5z^2+24z+29 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2-2w^3z+26w^2z+29w |
| Inner characteristic polynomial | t^6+34t^4+73t^2+1 |
| Outer characteristic polynomial | t^7+50t^5+150t^3+10t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -528*K1**4 - 32*K1**3*K3 + 576*K1**2*K2**3 - 2320*K1**2*K2**2 - 192*K1**2*K2*K4 + 4440*K1**2*K2 - 16*K1**2*K3**2 - 4096*K1**2 + 192*K1*K2**3*K3 - 416*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 3616*K1*K2*K3 + 312*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 160*K2**4*K4 - 872*K2**4 - 464*K2**2*K3**2 - 168*K2**2*K4**2 + 1096*K2**2*K4 - 2928*K2**2 + 312*K2*K3*K5 + 48*K2*K4*K6 - 1300*K3**2 - 430*K4**2 - 60*K5**2 + 3124 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}]] |
| If K is slice | False |