| Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,1,2,1,-1,1,3,2,0,1,1,-1,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1307'] |
| Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717'] |
| Outer characteristic polynomial of the knot is: t^7+44t^5+200t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1307'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 448*K1**4*K2 - 3104*K1**4 + 192*K1**3*K2*K3 - 192*K1**3*K3 - 3968*K1**2*K2**2 - 256*K1**2*K2*K4 + 7936*K1**2*K2 - 352*K1**2*K3**2 - 5000*K1**2 - 576*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5984*K1*K2*K3 + 1200*K1*K3*K4 + 272*K1*K4*K5 + 64*K1*K5*K6 - 1024*K2**4 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 1856*K2**2*K4 - 4748*K2**2 + 688*K2*K3*K5 + 32*K2*K4*K6 - 2336*K3**2 - 1072*K4**2 - 392*K5**2 - 44*K6**2 + 5014 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1307'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13943', 'vk6.14039', 'vk6.15014', 'vk6.15136', 'vk6.17442', 'vk6.17463', 'vk6.23954', 'vk6.23987', 'vk6.33756', 'vk6.33831', 'vk6.34299', 'vk6.36253', 'vk6.43417', 'vk6.53891', 'vk6.53925', 'vk6.54439', 'vk6.55594', 'vk6.60085', 'vk6.60101', 'vk6.65312'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is -. |
| The reverse -K is |
| The 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 | O1O2O3U4O5O4U6U2O6U1U5U3 |
| R3 orbit | {'O1O2O3U4O5O4U6U2O6U1U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4U3O5U2U5O6O4U6 |
| Gauss code of K* | O1O2O3U1U4U3O5U2U5O6O4U6 |
| Gauss code of -K* | Same |
| Diagrammatic symmetry type | - |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 2 1 1 -1],[ 2 0 1 3 2 1 1],[ 1 -1 0 1 1 -1 1],[-2 -3 -1 0 -1 -1 -2],[-1 -2 -1 1 0 1 -2],[-1 -1 1 1 -1 0 -1],[ 1 -1 -1 2 2 1 0]] |
| Primitive based matrix | [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -1 -2 -3],[-1 1 0 1 -1 -2 -2],[-1 1 -1 0 1 -1 -1],[ 1 1 1 -1 0 1 -1],[ 1 2 2 1 -1 0 -1],[ 2 3 2 1 1 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,1,1,2,1,1,1,2,3,-1,1,2,2,-1,1,1,-1,1,1] |
| Phi over symmetry | [-2,-1,-1,1,1,2,0,0,1,2,1,-1,1,3,2,0,1,1,-1,0,0] |
| Phi of -K | [-2,-1,-1,1,1,2,0,0,1,2,1,-1,1,3,2,0,1,1,-1,0,0] |
| Phi of K* | [-2,-1,-1,1,1,2,0,0,1,2,1,-1,1,3,2,0,1,1,-1,0,0] |
| Phi of -K* | [-2,-1,-1,1,1,2,1,1,1,2,3,-1,1,2,2,-1,1,1,-1,1,1] |
| Symmetry type of based matrix | - |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 2z^2+22z+37 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+22w^2z+37w |
| Inner characteristic polynomial | t^6+32t^4+132t^2+1 |
| Outer characteristic polynomial | t^7+44t^5+200t^3+7t |
| Flat arrow polynomial | -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 448*K1**4*K2 - 3104*K1**4 + 192*K1**3*K2*K3 - 192*K1**3*K3 - 3968*K1**2*K2**2 - 256*K1**2*K2*K4 + 7936*K1**2*K2 - 352*K1**2*K3**2 - 5000*K1**2 - 576*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5984*K1*K2*K3 + 1200*K1*K3*K4 + 272*K1*K4*K5 + 64*K1*K5*K6 - 1024*K2**4 - 224*K2**2*K3**2 - 16*K2**2*K4**2 + 1856*K2**2*K4 - 4748*K2**2 + 688*K2*K3*K5 + 32*K2*K4*K6 - 2336*K3**2 - 1072*K4**2 - 392*K5**2 - 44*K6**2 + 5014 |
| Genus of based matrix | 0 |
| Fillings of based matrix | [[{4, 6}, {2, 5}, {1, 3}]] |
| If K is slice | True |