| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,3,3,3,0,2,2,2,-1,0,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.456'] |
| Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369'] |
| Outer characteristic polynomial of the knot is: t^7+64t^5+125t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.456'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**6 - 256*K1**4*K2**2 + 512*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 816*K1**2*K2**2 + 976*K1**2*K2 - 192*K1**2*K3**2 - 288*K1**2 + 880*K1*K2*K3 + 192*K1*K3*K4 + 8*K1*K4*K5 - 96*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 224*K2**2*K4 - 662*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 412*K3**2 - 164*K4**2 - 76*K5**2 - 18*K6**2 + 706 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.456'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11042', 'vk6.11122', 'vk6.11565', 'vk6.11905', 'vk6.12208', 'vk6.12317', 'vk6.13220', 'vk6.19237', 'vk6.19316', 'vk6.19532', 'vk6.19611', 'vk6.22393', 'vk6.22712', 'vk6.22814', 'vk6.26045', 'vk6.26084', 'vk6.26508', 'vk6.28421', 'vk6.30619', 'vk6.30716', 'vk6.31347', 'vk6.31357', 'vk6.31759', 'vk6.31927', 'vk6.32523', 'vk6.32924', 'vk6.34761', 'vk6.38118', 'vk6.40135', 'vk6.40156', 'vk6.42376', 'vk6.44640', 'vk6.44746', 'vk6.46665', 'vk6.52339', 'vk6.52601', 'vk6.52817', 'vk6.56628', 'vk6.64718', 'vk6.66282'] |
| 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 | O1O2O3O4O5U6U3O6U2U5U1U4 |
| R3 orbit | {'O1O2O3O4U5U2O5U1U6U3O6U4', 'O1O2O3O4O5U6U3O6U2U5U1U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U2U5U1U4O6U3U6 |
| Gauss code of K* | O1O2O3O4U3U1U5U4U2O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U3U1U6U4U2 |
| 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 -1 -2 -1 3 2 -1],[ 1 0 -1 0 3 2 0],[ 2 1 0 1 3 2 1],[ 1 0 -1 0 1 0 1],[-3 -3 -3 -1 0 0 -3],[-2 -2 -2 0 0 0 -2],[ 1 0 -1 -1 3 2 0]] |
| Primitive based matrix | [[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -1 -3 -3 -3],[-2 0 0 0 -2 -2 -2],[ 1 1 0 0 1 0 -1],[ 1 3 2 -1 0 0 -1],[ 1 3 2 0 0 0 -1],[ 2 3 2 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,1,1,1,2,0,1,3,3,3,0,2,2,2,-1,0,1,0,1,1] |
| Phi over symmetry | [-3,-2,1,1,1,2,0,1,3,3,3,0,2,2,2,-1,0,1,0,1,1] |
| Phi of -K | [-2,-1,-1,-1,2,3,0,0,0,2,2,-1,0,3,3,0,1,1,1,1,1] |
| Phi of K* | [-3,-2,1,1,1,2,1,1,1,3,2,1,1,3,2,0,-1,0,0,0,0] |
| Phi of -K* | [-2,-1,-1,-1,2,3,1,1,1,2,3,-1,0,2,3,0,0,1,2,3,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+3t |
| Normalized Jones-Krushkal polynomial | 9z+19 |
| Enhanced Jones-Krushkal polynomial | 9w^2z+19w |
| Inner characteristic polynomial | t^6+44t^4+75t^2 |
| Outer characteristic polynomial | t^7+64t^5+125t^3 |
| Flat arrow polynomial | -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -256*K1**6 - 256*K1**4*K2**2 + 512*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 816*K1**2*K2**2 + 976*K1**2*K2 - 192*K1**2*K3**2 - 288*K1**2 + 880*K1*K2*K3 + 192*K1*K3*K4 + 8*K1*K4*K5 - 96*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 224*K2**2*K4 - 662*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 412*K3**2 - 164*K4**2 - 76*K5**2 - 18*K6**2 + 706 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {2, 3}]] |
| If K is slice | False |