| Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,1,3,4,0,0,1,2,0,2,2,0,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.807'] |
| Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932'] |
| Outer characteristic polynomial of the knot is: t^7+98t^5+138t^3+5t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.807'] |
| 2-strand cable arrow polynomial of the knot is: -752*K1**4 + 768*K1**3*K2*K3 + 32*K1**3*K3*K4 - 736*K1**3*K3 - 2240*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4520*K1**2*K2 - 1136*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 3984*K1**2 + 96*K1*K2**3*K3 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 128*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4984*K1*K2*K3 + 1064*K1*K3*K4 + 80*K1*K4*K5 - 80*K2**4 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 552*K2**2*K4 - 3100*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 1732*K3**2 - 424*K4**2 - 52*K5**2 - 4*K6**2 + 3054 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.807'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11024', 'vk6.11104', 'vk6.12190', 'vk6.12299', 'vk6.18192', 'vk6.18527', 'vk6.24646', 'vk6.25075', 'vk6.30593', 'vk6.30690', 'vk6.31859', 'vk6.31907', 'vk6.36780', 'vk6.37228', 'vk6.44021', 'vk6.44362', 'vk6.51823', 'vk6.51892', 'vk6.52691', 'vk6.52787', 'vk6.55998', 'vk6.56271', 'vk6.60533', 'vk6.60874', 'vk6.63503', 'vk6.63549', 'vk6.63981', 'vk6.64027', 'vk6.65657', 'vk6.65939', 'vk6.68703', 'vk6.68912'] |
| 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 | O1O2O3O4U1O5U2U4U6U5O6U3 |
| R3 orbit | {'O1O2O3O4U1O5U2U4U6U5O6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2O5U6U5U1U3O6U4 |
| Gauss code of K* | O1O2O3O4U3O5U6U1U5U2O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5U3U6U4U5O6U2 |
| 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 -2 2 1 3 -1],[ 3 0 1 3 2 2 3],[ 2 -1 0 3 1 2 1],[-2 -3 -3 0 -1 2 -3],[-1 -2 -1 1 0 1 -2],[-3 -2 -2 -2 -1 0 -3],[ 1 -3 -1 3 2 3 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -2 -3],[-3 0 -2 -1 -3 -2 -2],[-2 2 0 -1 -3 -3 -3],[-1 1 1 0 -2 -1 -2],[ 1 3 3 2 0 -1 -3],[ 2 2 3 1 1 0 -1],[ 3 2 3 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,2,3,2,1,3,2,2,1,3,3,3,2,1,2,1,3,1] |
| Phi over symmetry | [-3,-2,-1,1,2,3,-1,1,1,3,4,0,0,1,2,0,2,2,0,-1,0] |
| Phi of -K | [-3,-2,-1,1,2,3,0,-1,2,2,4,0,2,1,3,0,0,1,0,1,-1] |
| Phi of K* | [-3,-2,-1,1,2,3,-1,1,1,3,4,0,0,1,2,0,2,2,0,-1,0] |
| Phi of -K* | [-3,-2,-1,1,2,3,1,3,2,3,2,1,1,3,2,2,3,3,1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | 0 |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+21w^2z+27w |
| Inner characteristic polynomial | t^6+70t^4+86t^2+1 |
| Outer characteristic polynomial | t^7+98t^5+138t^3+5t |
| Flat arrow polynomial | -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -752*K1**4 + 768*K1**3*K2*K3 + 32*K1**3*K3*K4 - 736*K1**3*K3 - 2240*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4520*K1**2*K2 - 1136*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 3984*K1**2 + 96*K1*K2**3*K3 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 - 128*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4984*K1*K2*K3 + 1064*K1*K3*K4 + 80*K1*K4*K5 - 80*K2**4 - 288*K2**2*K3**2 - 16*K2**2*K4**2 + 552*K2**2*K4 - 3100*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 16*K3**2*K6 - 1732*K3**2 - 424*K4**2 - 52*K5**2 - 4*K6**2 + 3054 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {2, 5}, {1, 4}, {3}]] |
| If K is slice | False |