| Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,1,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.731'] |
| Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384'] |
| Outer characteristic polynomial of the knot is: t^7+53t^5+27t^3+3t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.731'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 128*K1**4*K2 - 1920*K1**4 + 32*K1**3*K2*K3 - 320*K1**3*K3 + 32*K1**2*K2**2*K4 - 1792*K1**2*K2**2 - 288*K1**2*K2*K4 + 4408*K1**2*K2 - 224*K1**2*K3**2 - 64*K1**2*K4**2 - 2392*K1**2 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3016*K1*K2*K3 + 648*K1*K3*K4 + 72*K1*K4*K5 - 128*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 400*K2**2*K4 - 2134*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 1020*K3**2 - 316*K4**2 - 44*K5**2 - 2*K6**2 + 2178 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.731'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16554', 'vk6.16645', 'vk6.18150', 'vk6.18486', 'vk6.22953', 'vk6.23072', 'vk6.24605', 'vk6.25018', 'vk6.34954', 'vk6.35073', 'vk6.35377', 'vk6.35796', 'vk6.36748', 'vk6.37167', 'vk6.39409', 'vk6.41602', 'vk6.42523', 'vk6.42632', 'vk6.42850', 'vk6.43127', 'vk6.44016', 'vk6.44328', 'vk6.45985', 'vk6.47661', 'vk6.54801', 'vk6.55351', 'vk6.56248', 'vk6.57431', 'vk6.59229', 'vk6.59788', 'vk6.60848', 'vk6.62098', 'vk6.64783', 'vk6.64846', 'vk6.65613', 'vk6.65920', 'vk6.68081', 'vk6.68144', 'vk6.68684', 'vk6.68895'] |
| 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 | O1O2O3O4U1O5U4U6U2O6U5U3 |
| R3 orbit | {'O1O2O3O4U1U3O5U6U2O6U4U5', 'O1O2O3O4U1O5U4U6U2O6U5U3'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4U2U5O6U3U6U1O5U4 |
| Gauss code of K* | O1O2O3U2O4O5U6U3U5U1O6U4 |
| Gauss code of -K* | O1O2O3U4O5U3U6U1U5O6O4U2 |
| 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 0 2 0 2 -1],[ 3 0 2 3 1 2 2],[ 0 -2 0 1 0 1 0],[-2 -3 -1 0 -1 1 -2],[ 0 -1 0 1 0 1 0],[-2 -2 -1 -1 -1 0 -2],[ 1 -2 0 2 0 2 0]] |
| Primitive based matrix | [[ 0 2 2 0 0 -1 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -1 -2 -2],[ 0 1 1 0 0 0 -1],[ 0 1 1 0 0 0 -2],[ 1 2 2 0 0 0 -2],[ 3 3 2 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,0,1,3,-1,1,1,2,3,1,1,2,2,0,0,1,0,2,2] |
| Phi over symmetry | [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,1,1,1,-1] |
| Phi of -K | [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,1,1,1,-1] |
| Phi of K* | [-2,-2,0,0,1,3,-1,1,1,1,3,1,1,1,2,0,1,1,1,2,0] |
| Phi of -K* | [-3,-1,0,0,2,2,2,1,2,2,3,0,0,2,2,0,1,1,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 13z+27 |
| Enhanced Jones-Krushkal polynomial | 13w^2z+27w |
| Inner characteristic polynomial | t^6+35t^4+11t^2 |
| Outer characteristic polynomial | t^7+53t^5+27t^3+3t |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 + 128*K1**4*K2 - 1920*K1**4 + 32*K1**3*K2*K3 - 320*K1**3*K3 + 32*K1**2*K2**2*K4 - 1792*K1**2*K2**2 - 288*K1**2*K2*K4 + 4408*K1**2*K2 - 224*K1**2*K3**2 - 64*K1**2*K4**2 - 2392*K1**2 - 192*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3016*K1*K2*K3 + 648*K1*K3*K4 + 72*K1*K4*K5 - 128*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 400*K2**2*K4 - 2134*K2**2 + 96*K2*K3*K5 + 8*K2*K4*K6 - 1020*K3**2 - 316*K4**2 - 44*K5**2 - 2*K6**2 + 2178 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{4, 6}, {1, 5}, {2, 3}]] |
| If K is slice | False |