| Min(phi) over symmetries of the knot is: [-4,-1,2,3,1,4,3,2,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.81'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 6*K1*K2 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.81'] |
| Outer characteristic polynomial of the knot is: t^5+65t^3+14t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.81'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 160*K1**2*K2**3 - 1024*K1**2*K2**2 + 1632*K1**2*K2 - 96*K1**2*K3**2 - 112*K1**2*K4**2 - 1792*K1**2 + 128*K1*K2**3*K3 + 1256*K1*K2*K3 + 584*K1*K3*K4 + 200*K1*K4*K5 + 40*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 328*K2**4 - 192*K2**2*K3**2 - 56*K2**2*K4**2 + 304*K2**2*K4 - 976*K2**2 + 160*K2*K3*K5 + 32*K2*K4*K6 - 48*K3**4 - 32*K3**2*K4**2 + 48*K3**2*K6 - 668*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 434*K4**2 - 140*K5**2 - 48*K6**2 - 24*K7**2 - 2*K8**2 + 1474 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.81'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71368', 'vk6.71427', 'vk6.71890', 'vk6.71949', 'vk6.72442', 'vk6.72592', 'vk6.72709', 'vk6.72801', 'vk6.72864', 'vk6.73020', 'vk6.73368', 'vk6.73530', 'vk6.74267', 'vk6.74388', 'vk6.74448', 'vk6.75061', 'vk6.75536', 'vk6.75828', 'vk6.76440', 'vk6.76633', 'vk6.77025', 'vk6.77744', 'vk6.77795', 'vk6.78252', 'vk6.78502', 'vk6.78631', 'vk6.78824', 'vk6.79311', 'vk6.79426', 'vk6.79842', 'vk6.79896', 'vk6.80261', 'vk6.80772', 'vk6.80872', 'vk6.85149', 'vk6.86512', 'vk6.87209', 'vk6.87351', 'vk6.89256', 'vk6.89429'] |
| 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 | O1O2O3O4O5O6U2U4U6U1U5U3 |
| R3 orbit | {'O1O2O3O4O5O6U2U4U6U1U5U3', 'O1O2O3O4O5U1O6U4U2U6U5U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U4U2U6U1U3U5 |
| Gauss code of K* | O1O2O3O4O5O6U4U1U6U2U5U3 |
| Gauss code of -K* | O1O2O3O4O5O6U4U2U5U1U6U3 |
| 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 -2 -4 2 -1 3 2],[ 2 0 -2 3 0 3 2],[ 4 2 0 4 1 3 2],[-2 -3 -4 0 -2 1 1],[ 1 0 -1 2 0 2 1],[-3 -3 -3 -1 -2 0 0],[-2 -2 -2 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 3 2 -1 -4],[-3 0 -1 -2 -3],[-2 1 0 -2 -4],[ 1 2 2 0 -1],[ 4 3 4 1 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-3,-2,1,4,1,2,3,2,4,1] |
| Phi over symmetry | [-4,-1,2,3,1,4,3,2,2,1] |
| Phi of -K | [-4,-1,2,3,2,2,4,1,2,0] |
| Phi of K* | [-3,-2,1,4,0,2,4,1,2,2] |
| Phi of -K* | [-4,-1,2,3,1,4,3,2,2,1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 9z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^3z+11w^2z+19w |
| Inner characteristic polynomial | t^4+35t^2+1 |
| Outer characteristic polynomial | t^5+65t^3+14t |
| Flat arrow polynomial | 4*K1**3 - 6*K1**2 - 6*K1*K2 - 2*K2**2 + 3*K2 + 2*K3 + K4 + 5 |
| 2-strand cable arrow polynomial | -144*K1**4 + 160*K1**2*K2**3 - 1024*K1**2*K2**2 + 1632*K1**2*K2 - 96*K1**2*K3**2 - 112*K1**2*K4**2 - 1792*K1**2 + 128*K1*K2**3*K3 + 1256*K1*K2*K3 + 584*K1*K3*K4 + 200*K1*K4*K5 + 40*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 + 64*K2**4*K4 - 328*K2**4 - 192*K2**2*K3**2 - 56*K2**2*K4**2 + 304*K2**2*K4 - 976*K2**2 + 160*K2*K3*K5 + 32*K2*K4*K6 - 48*K3**4 - 32*K3**2*K4**2 + 48*K3**2*K6 - 668*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 434*K4**2 - 140*K5**2 - 48*K6**2 - 24*K7**2 - 2*K8**2 + 1474 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]] |
| If K is slice | False |