| Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,1,0,3,4,3,0,2,3,2,1,1,1,0,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.76'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511'] |
| Outer characteristic polynomial of the knot is: t^7+109t^5+65t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.76'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 224*K1**4*K2 - 272*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 + 160*K1**2*K2**3 - 1600*K1**2*K2**2 - 288*K1**2*K2*K4 + 1912*K1**2*K2 - 144*K1**2*K3**2 - 1464*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 1968*K1*K2*K3 - 32*K1*K2*K4*K5 + 416*K1*K3*K4 + 168*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**4 - 224*K2**2*K3**2 - 328*K2**2*K4**2 + 712*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 1306*K2**2 + 384*K2*K3*K5 + 96*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 668*K3**2 - 394*K4**2 - 160*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 1298 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.76'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11421', 'vk6.11712', 'vk6.12727', 'vk6.13076', 'vk6.18074', 'vk6.18412', 'vk6.22035', 'vk6.22657', 'vk6.22742', 'vk6.24521', 'vk6.28091', 'vk6.29537', 'vk6.31162', 'vk6.31499', 'vk6.34603', 'vk6.34681', 'vk6.36658', 'vk6.39500', 'vk6.41714', 'vk6.42300', 'vk6.42331', 'vk6.43936', 'vk6.46100', 'vk6.47752', 'vk6.52175', 'vk6.52415', 'vk6.54601', 'vk6.54639', 'vk6.55894', 'vk6.58663', 'vk6.59083', 'vk6.59124', 'vk6.60418', 'vk6.63125', 'vk6.63746', 'vk6.63850', 'vk6.64673', 'vk6.65528', 'vk6.68013', 'vk6.68610'] |
| 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 | O1O2O3O4O5O6U2U3U6U5U1U4 |
| R3 orbit | {'O1O2O3O4O5O6U2U3U6U5U1U4', 'O1O2O3O4O5U1U2U5U6U3O6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U3U6U2U1U4U5 |
| Gauss code of K* | O1O2O3O4O5O6U5U1U2U6U4U3 |
| Gauss code of -K* | O1O2O3O4O5O6U4U3U1U5U6U2 |
| 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 -1 -4 -2 3 2 2],[ 1 0 -3 -1 3 2 2],[ 4 3 0 1 4 3 2],[ 2 1 -1 0 3 2 1],[-3 -3 -4 -3 0 0 0],[-2 -2 -3 -2 0 0 0],[-2 -2 -2 -1 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 2 -1 -2 -4],[-3 0 0 0 -3 -3 -4],[-2 0 0 0 -2 -1 -2],[-2 0 0 0 -2 -2 -3],[ 1 3 2 2 0 -1 -3],[ 2 3 1 2 1 0 -1],[ 4 4 2 3 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-2,1,2,4,0,0,3,3,4,0,2,1,2,2,2,3,1,3,1] |
| Phi over symmetry | [-4,-2,-1,2,2,3,1,0,3,4,3,0,2,3,2,1,1,1,0,1,1] |
| Phi of -K | [-4,-2,-1,2,2,3,1,0,3,4,3,0,2,3,2,1,1,1,0,1,1] |
| Phi of K* | [-3,-2,-2,1,2,4,1,1,1,2,3,0,1,2,3,1,3,4,0,0,1] |
| Phi of -K* | [-4,-2,-1,2,2,3,1,3,2,3,4,1,1,2,3,2,2,3,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2+18w^2z+17w |
| Inner characteristic polynomial | t^6+71t^4+34t^2+1 |
| Outer characteristic polynomial | t^7+109t^5+65t^3+4t |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 224*K1**4*K2 - 272*K1**4 + 384*K1**3*K2*K3 - 224*K1**3*K3 + 160*K1**2*K2**3 - 1600*K1**2*K2**2 - 288*K1**2*K2*K4 + 1912*K1**2*K2 - 144*K1**2*K3**2 - 1464*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 1968*K1*K2*K3 - 32*K1*K2*K4*K5 + 416*K1*K3*K4 + 168*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**4 - 224*K2**2*K3**2 - 328*K2**2*K4**2 + 712*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 1306*K2**2 + 384*K2*K3*K5 + 96*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 668*K3**2 - 394*K4**2 - 160*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 1298 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {2, 4}, {1, 3}]] |
| If K is slice | False |