| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,3,2,3,1,2,2,2,0,1,1,2,2,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.900'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.217', '6.219', '6.304', '6.349', '6.390', '6.400', '6.416', '6.515', '6.518', '6.530', '6.582', '6.616', '6.629', '6.641', '6.645', '6.702', '6.710', '6.715', '6.729', '6.733', '6.734', '6.802', '6.840', '6.845', '6.854', '6.860', '6.900', '6.905', '6.921', '6.924', '6.979', '6.980', '6.996', '6.1044', '6.1067', '6.1086', '6.1100', '6.1139', '6.1145', '6.1149', '6.1167', '6.1169', '6.1183', '6.1314'] |
| Outer characteristic polynomial of the knot is: t^7+71t^5+60t^3+10t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.900'] |
| 2-strand cable arrow polynomial of the knot is: -1040*K1**4 + 1792*K1**3*K2*K3 - 256*K1**3*K3 - 1024*K1**2*K2**2*K3**2 - 5184*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1472*K1**2*K2*K4 + 4672*K1**2*K2 - 1200*K1**2*K3**2 - 2916*K1**2 + 1216*K1*K2**3*K3 + 1312*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6272*K1*K2*K3 - 224*K1*K2*K4*K5 + 1784*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 328*K2**4 - 1104*K2**2*K3**2 - 488*K2**2*K4**2 + 968*K2**2*K4 - 2310*K2**2 - 96*K2*K3**2*K4 + 328*K2*K3*K5 + 200*K2*K4*K6 - 1816*K3**2 - 650*K4**2 - 20*K5**2 - 18*K6**2 + 2624 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.900'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16336', 'vk6.16377', 'vk6.18064', 'vk6.18404', 'vk6.22671', 'vk6.22750', 'vk6.24507', 'vk6.24932', 'vk6.34617', 'vk6.34696', 'vk6.36648', 'vk6.37074', 'vk6.42310', 'vk6.42339', 'vk6.43930', 'vk6.44251', 'vk6.54607', 'vk6.54644', 'vk6.55884', 'vk6.56174', 'vk6.59096', 'vk6.59132', 'vk6.60404', 'vk6.60765', 'vk6.64642', 'vk6.64688', 'vk6.65518', 'vk6.65836', 'vk6.67997', 'vk6.68021', 'vk6.68604', 'vk6.68823'] |
| 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 | O1O2O3O4U2U5O6O5U1U3U4U6 |
| R3 orbit | {'O1O2O3O4U2U5O6O5U1U3U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U1U2U4O6O5U6U3 |
| Gauss code of K* | O1O2O3O4U1U5U2U3O5O6U4U6 |
| Gauss code of -K* | O1O2O3O4U5U1O5O6U2U3U6U4 |
| 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 0 2 1 2],[ 3 0 0 2 3 3 2],[ 2 0 0 1 2 2 2],[ 0 -2 -1 0 1 0 1],[-2 -3 -2 -1 0 -2 0],[-1 -3 -2 0 2 0 2],[-2 -2 -2 -1 0 -2 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -2 -3],[-2 0 0 -2 -1 -2 -2],[-2 0 0 -2 -1 -2 -3],[-1 2 2 0 0 -2 -3],[ 0 1 1 0 0 -1 -2],[ 2 2 2 2 1 0 0],[ 3 2 3 3 2 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,2,3,0,2,1,2,2,2,1,2,3,0,2,3,1,2,0] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,2,3,2,3,1,2,2,2,0,1,1,2,2,0] |
| Phi of -K | [-3,-2,0,1,2,2,1,1,1,2,3,1,1,2,2,1,1,1,-1,-1,0] |
| Phi of K* | [-2,-2,-1,0,2,3,0,-1,1,2,2,-1,1,2,3,1,1,1,1,1,1] |
| Phi of -K* | [-3,-2,0,1,2,2,0,2,3,2,3,1,2,2,2,0,1,1,2,2,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-t^2-t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+49t^4+21t^2 |
| Outer characteristic polynomial | t^7+71t^5+60t^3+10t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 + K1 + K2 + K3 + 2 |
| 2-strand cable arrow polynomial | -1040*K1**4 + 1792*K1**3*K2*K3 - 256*K1**3*K3 - 1024*K1**2*K2**2*K3**2 - 5184*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1472*K1**2*K2*K4 + 4672*K1**2*K2 - 1200*K1**2*K3**2 - 2916*K1**2 + 1216*K1*K2**3*K3 + 1312*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6272*K1*K2*K3 - 224*K1*K2*K4*K5 + 1784*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 328*K2**4 - 1104*K2**2*K3**2 - 488*K2**2*K4**2 + 968*K2**2*K4 - 2310*K2**2 - 96*K2*K3**2*K4 + 328*K2*K3*K5 + 200*K2*K4*K6 - 1816*K3**2 - 650*K4**2 - 20*K5**2 - 18*K6**2 + 2624 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |