| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,2,3,2,1,1,1,1,0,0,0,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.995'] |
| Arrow polynomial of the knot is: -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.458', '6.601', '6.611', '6.995', '6.1026', '6.1037'] |
| Outer characteristic polynomial of the knot is: t^7+50t^5+27t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.995'] |
| 2-strand cable arrow polynomial of the knot is: -176*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 64*K1**2*K2**2*K3**2 - 1920*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 288*K1**2*K2*K4 + 2896*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 2696*K1**2 + 384*K1*K2**3*K3 - 448*K1*K2**2*K3 - 192*K1*K2**2*K5 + 64*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3704*K1*K2*K3 + 664*K1*K3*K4 + 48*K1*K4*K5 - 328*K2**4 - 512*K2**2*K3**2 - 56*K2**2*K4**2 + 608*K2**2*K4 - 1962*K2**2 + 352*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 1180*K3**2 - 266*K4**2 - 36*K5**2 - 6*K6**2 + 1968 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.995'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71644', 'vk6.71655', 'vk6.71658', 'vk6.71666', 'vk6.71821', 'vk6.71831', 'vk6.72245', 'vk6.72257', 'vk6.72260', 'vk6.72269', 'vk6.72373', 'vk6.72380', 'vk6.77260', 'vk6.77273', 'vk6.77356', 'vk6.77370', 'vk6.77372', 'vk6.77376', 'vk6.77608', 'vk6.77618', 'vk6.77700', 'vk6.77712', 'vk6.77715', 'vk6.77719', 'vk6.81407', 'vk6.81414', 'vk6.81441', 'vk6.81695', 'vk6.82450', 'vk6.82455', 'vk6.84424', 'vk6.84436', 'vk6.84439', 'vk6.86956', 'vk6.87165', 'vk6.87169', 'vk6.87774', 'vk6.87998', 'vk6.88119', 'vk6.89636'] |
| 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 | O1O2O3O4U1U5O6U4O5U2U6U3 |
| R3 orbit | {'O1O2O3O4U1U5U3O6O5U2U4U6', 'O1O2O3O4U1U5O6U4O5U2U6U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U5U3O6U1O5U6U4 |
| Gauss code of K* | O1O2O3U4U1U3U5O4O6U2O5U6 |
| Gauss code of -K* | O1O2O3U4O5U2O4O6U5U1U3U6 |
| 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 -1 2 1 0 1],[ 3 0 2 3 1 2 2],[ 1 -2 0 2 1 0 1],[-2 -3 -2 0 0 -2 0],[-1 -1 -1 0 0 -1 0],[ 0 -2 0 2 1 0 1],[-1 -2 -1 0 0 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -2 -2 -3],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[ 0 2 1 1 0 0 -2],[ 1 2 1 1 0 0 -2],[ 3 3 1 2 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,1,3,0,0,2,2,3,0,1,1,1,1,1,2,0,2,2] |
| Phi over symmetry | [-3,-1,0,1,1,2,0,1,2,3,2,1,1,1,1,0,0,0,0,1,1] |
| Phi of -K | [-3,-1,0,1,1,2,0,1,2,3,2,1,1,1,1,0,0,0,0,1,1] |
| Phi of K* | [-2,-1,-1,0,1,3,1,1,0,1,2,0,0,1,2,0,1,3,1,1,0] |
| Phi of -K* | [-3,-1,0,1,1,2,2,2,1,2,3,0,1,1,2,1,1,2,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | 3w^3z^2+16w^2z+21w |
| Inner characteristic polynomial | t^6+34t^4+6t^2 |
| Outer characteristic polynomial | t^7+50t^5+27t^3+3t |
| Flat arrow polynomial | -6*K1**2 - 6*K1*K2 + 3*K1 + 3*K2 + 3*K3 + 4 |
| 2-strand cable arrow polynomial | -176*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 64*K1**2*K2**2*K3**2 - 1920*K1**2*K2**2 + 96*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 288*K1**2*K2*K4 + 2896*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K3*K5 - 16*K1**2*K4**2 - 2696*K1**2 + 384*K1*K2**3*K3 - 448*K1*K2**2*K3 - 192*K1*K2**2*K5 + 64*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3704*K1*K2*K3 + 664*K1*K3*K4 + 48*K1*K4*K5 - 328*K2**4 - 512*K2**2*K3**2 - 56*K2**2*K4**2 + 608*K2**2*K4 - 1962*K2**2 + 352*K2*K3*K5 + 40*K2*K4*K6 - 16*K3**4 + 8*K3**2*K6 - 1180*K3**2 - 266*K4**2 - 36*K5**2 - 6*K6**2 + 1968 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]] |
| If K is slice | False |