| Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,1,2,1,3,4,0,1,1,2,1,2,3,-1,-1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.417'] |
| 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+96t^5+81t^3 |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.417'] |
| 2-strand cable arrow polynomial of the knot is: 24*K1**2*K2 - 928*K1**2*K3**2 - 48*K1**2*K6**2 - 656*K1**2 + 1312*K1*K2*K3 + 856*K1*K3*K4 + 32*K1*K4*K5 + 64*K1*K5*K6 + 40*K1*K6*K7 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 88*K2**2*K4 - 8*K2**2*K6**2 - 546*K2**2 + 152*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 632*K3**2 - 262*K4**2 - 84*K5**2 - 38*K6**2 - 12*K7**2 - 2*K8**2 + 742 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.417'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16985', 'vk6.17226', 'vk6.19921', 'vk6.20217', 'vk6.21144', 'vk6.21509', 'vk6.23389', 'vk6.26840', 'vk6.26856', 'vk6.27413', 'vk6.28615', 'vk6.29026', 'vk6.35446', 'vk6.38276', 'vk6.38292', 'vk6.38827', 'vk6.40409', 'vk6.41020', 'vk6.42882', 'vk6.45151', 'vk6.45159', 'vk6.45594', 'vk6.46998', 'vk6.47356', 'vk6.55148', 'vk6.56704', 'vk6.56711', 'vk6.57794', 'vk6.58165', 'vk6.59524', 'vk6.61123', 'vk6.61561', 'vk6.62367', 'vk6.62736', 'vk6.64964', 'vk6.66406', 'vk6.67161', 'vk6.68256', 'vk6.69057', 'vk6.69842'] |
| 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 | O1O2O3O4O5U6U1O6U2U4U3U5 |
| R3 orbit | {'O1O2O3O4O5U6U1O6U2U4U3U5', 'O1O2O3O4O5U2U6U1O6U4U3U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U1U3U2U4O6U5U6 |
| Gauss code of K* | O1O2O3O4U5U1U3U2U4O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U1U3U2U4U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 2 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 -2 1 1 4 -1],[ 3 0 0 2 1 3 3],[ 2 0 0 2 1 3 2],[-1 -2 -2 0 0 2 -1],[-1 -1 -1 0 0 1 -1],[-4 -3 -3 -2 -1 0 -4],[ 1 -3 -2 1 1 4 0]] |
| Primitive based matrix | [[ 0 4 1 1 -1 -2 -3],[-4 0 -1 -2 -4 -3 -3],[-1 1 0 0 -1 -1 -1],[-1 2 0 0 -1 -2 -2],[ 1 4 1 1 0 -2 -3],[ 2 3 1 2 2 0 0],[ 3 3 1 2 3 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-1,-1,1,2,3,1,2,4,3,3,0,1,1,1,1,2,2,2,3,0] |
| Phi over symmetry | [-4,-1,-1,1,2,3,1,2,1,3,4,0,1,1,2,1,2,3,-1,-1,1] |
| Phi of -K | [-3,-2,-1,1,1,4,1,-1,2,3,4,-1,1,2,3,1,1,1,0,1,2] |
| Phi of K* | [-4,-1,-1,1,2,3,1,2,1,3,4,0,1,1,2,1,2,3,-1,-1,1] |
| Phi of -K* | [-3,-2,-1,1,1,4,0,3,1,2,3,2,1,2,3,1,1,4,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+t^3+t^2-t |
| Normalized Jones-Krushkal polynomial | 7z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^3z+11w^2z+15w |
| Inner characteristic polynomial | t^6+64t^4+24t^2 |
| Outer characteristic polynomial | t^7+96t^5+81t^3 |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | 24*K1**2*K2 - 928*K1**2*K3**2 - 48*K1**2*K6**2 - 656*K1**2 + 1312*K1*K2*K3 + 856*K1*K3*K4 + 32*K1*K4*K5 + 64*K1*K5*K6 + 40*K1*K6*K7 - 128*K2**2*K3**2 - 8*K2**2*K4**2 + 88*K2**2*K4 - 8*K2**2*K6**2 - 546*K2**2 + 152*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 632*K3**2 - 262*K4**2 - 84*K5**2 - 38*K6**2 - 12*K7**2 - 2*K8**2 + 742 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}]] |
| If K is slice | False |