| Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,0,2,3,1,3,1,1,0,2,1,0,2,1,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.619'] |
| 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+60t^5+137t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.619'] |
| 2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 576*K1**4 + 896*K1**2*K2**3 - 3888*K1**2*K2**2 - 576*K1**2*K2*K4 + 5016*K1**2*K2 - 4032*K1**2 + 352*K1*K2**3*K3 - 416*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 4376*K1*K2*K3 + 472*K1*K3*K4 + 8*K1*K4*K5 - 64*K2**6 + 320*K2**4*K4 - 1904*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 352*K2**2*K4**2 + 1848*K2**2*K4 - 2508*K2**2 + 400*K2*K3*K5 + 144*K2*K4*K6 - 1244*K3**2 - 564*K4**2 - 36*K5**2 - 12*K6**2 + 3122 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.619'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71597', 'vk6.71600', 'vk6.71721', 'vk6.71725', 'vk6.72142', 'vk6.72145', 'vk6.72336', 'vk6.74048', 'vk6.74050', 'vk6.74617', 'vk6.76804', 'vk6.77212', 'vk6.77223', 'vk6.77524', 'vk6.77535', 'vk6.77669', 'vk6.79047', 'vk6.79056', 'vk6.79614', 'vk6.79624', 'vk6.80567', 'vk6.80580', 'vk6.81019', 'vk6.81032', 'vk6.81343', 'vk6.81350', 'vk6.81392', 'vk6.85410', 'vk6.85427', 'vk6.85491', 'vk6.87976', 'vk6.89322'] |
| 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 | O1O2O3O4U5O6U1O5U3U6U2U4 |
| R3 orbit | {'O1O2O3O4U5O6U1O5U3U6U2U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U3U5U2O6U4O5U6 |
| Gauss code of K* | O1O2O3O4U5U3U1U4O6U2O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6U3O5U1U4U2U6 |
| 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 -3 0 -1 3 0 1],[ 3 0 2 0 3 3 1],[ 0 -2 0 -1 2 1 0],[ 1 0 1 0 2 1 0],[-3 -3 -2 -2 0 -2 -1],[ 0 -3 -1 -1 2 0 1],[-1 -1 0 0 1 -1 0]] |
| Primitive based matrix | [[ 0 3 1 0 0 -1 -3],[-3 0 -1 -2 -2 -2 -3],[-1 1 0 0 -1 0 -1],[ 0 2 0 0 1 -1 -2],[ 0 2 1 -1 0 -1 -3],[ 1 2 0 1 1 0 0],[ 3 3 1 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,0,1,3,1,2,2,2,3,0,1,0,1,-1,1,2,1,3,0] |
| Phi over symmetry | [-3,-1,0,0,1,3,0,2,3,1,3,1,1,0,2,1,0,2,1,2,1] |
| Phi of -K | [-3,-1,0,0,1,3,2,0,1,3,3,0,0,2,2,1,0,1,1,1,1] |
| Phi of K* | [-3,-1,0,0,1,3,1,1,1,2,3,0,1,2,3,-1,0,0,0,1,2] |
| Phi of -K* | [-3,-1,0,0,1,3,0,2,3,1,3,1,1,0,2,1,0,2,1,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-4w^3z+25w^2z+27w |
| Inner characteristic polynomial | t^6+40t^4+67t^2+4 |
| Outer characteristic polynomial | t^7+60t^5+137t^3+16t |
| Flat arrow polynomial | 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3 |
| 2-strand cable arrow polynomial | 96*K1**4*K2 - 576*K1**4 + 896*K1**2*K2**3 - 3888*K1**2*K2**2 - 576*K1**2*K2*K4 + 5016*K1**2*K2 - 4032*K1**2 + 352*K1*K2**3*K3 - 416*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 4376*K1*K2*K3 + 472*K1*K3*K4 + 8*K1*K4*K5 - 64*K2**6 + 320*K2**4*K4 - 1904*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 352*K2**2*K4**2 + 1848*K2**2*K4 - 2508*K2**2 + 400*K2*K3*K5 + 144*K2*K4*K6 - 1244*K3**2 - 564*K4**2 - 36*K5**2 - 12*K6**2 + 3122 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {3, 4}, {1, 2}]] |
| If K is slice | False |