| Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,1,3,1,3,0,1,0,0,1,0,1,1,2,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1287'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+53t^5+87t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1287'] |
| 2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 128*K1**4 - 576*K1**2*K2**4 + 928*K1**2*K2**3 - 6448*K1**2*K2**2 - 32*K1**2*K2*K4 + 5624*K1**2*K2 - 3496*K1**2 + 576*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 5144*K1*K2*K3 + 80*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1048*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1896*K2**2 + 32*K2*K3*K5 - 1048*K3**2 - 94*K4**2 + 2356 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1287'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4737', 'vk6.5063', 'vk6.6271', 'vk6.6714', 'vk6.8234', 'vk6.8681', 'vk6.9623', 'vk6.9943', 'vk6.20646', 'vk6.22077', 'vk6.28136', 'vk6.29565', 'vk6.39570', 'vk6.41801', 'vk6.46189', 'vk6.47807', 'vk6.48777', 'vk6.48989', 'vk6.49587', 'vk6.49792', 'vk6.50787', 'vk6.51002', 'vk6.51275', 'vk6.51473', 'vk6.57562', 'vk6.58732', 'vk6.62240', 'vk6.63186', 'vk6.67040', 'vk6.67913', 'vk6.69669', 'vk6.70350'] |
| 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 | O1O2O3U1O4O5U6U4O6U3U2U5 |
| R3 orbit | {'O1O2O3U1O4O5U6U4O6U3U2U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U2U1O5U6U5O4O6U3 |
| Gauss code of K* | O1O2O3U4U2U1O4U5U3O6O5U6 |
| Gauss code of -K* | O1O2O3U4O5O4U1U5O6U3U2U6 |
| 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 -2 0 0 0 3 -1],[ 2 0 2 1 0 2 2],[ 0 -2 0 0 1 3 -1],[ 0 -1 0 0 1 2 -1],[ 0 0 -1 -1 0 0 0],[-3 -2 -3 -2 0 0 -3],[ 1 -2 1 1 0 3 0]] |
| Primitive based matrix | [[ 0 3 0 0 0 -1 -2],[-3 0 0 -2 -3 -3 -2],[ 0 0 0 -1 -1 0 0],[ 0 2 1 0 0 -1 -1],[ 0 3 1 0 0 -1 -2],[ 1 3 0 1 1 0 -2],[ 2 2 0 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,0,0,0,1,2,0,2,3,3,2,1,1,0,0,0,1,1,1,2,2] |
| Phi over symmetry | [-3,0,0,0,1,2,0,1,3,1,3,0,1,0,0,1,0,1,1,2,-1] |
| Phi of -K | [-2,-1,0,0,0,3,-1,0,1,2,3,0,0,1,1,0,-1,0,-1,1,3] |
| Phi of K* | [-3,0,0,0,1,2,0,1,3,1,3,0,1,0,0,1,0,1,1,2,-1] |
| Phi of -K* | [-2,-1,0,0,0,3,2,0,1,2,2,0,1,1,3,-1,-1,0,0,2,3] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2-8w^3z+26w^2z+17w |
| Inner characteristic polynomial | t^6+39t^4+52t^2+1 |
| Outer characteristic polynomial | t^7+53t^5+87t^3+12t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | 32*K1**4*K2 - 128*K1**4 - 576*K1**2*K2**4 + 928*K1**2*K2**3 - 6448*K1**2*K2**2 - 32*K1**2*K2*K4 + 5624*K1**2*K2 - 3496*K1**2 + 576*K1*K2**3*K3 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 + 5144*K1*K2*K3 + 80*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 1048*K2**4 - 144*K2**2*K3**2 - 8*K2**2*K4**2 + 688*K2**2*K4 - 1896*K2**2 + 32*K2*K3*K5 - 1048*K3**2 - 94*K4**2 + 2356 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]] |
| If K is slice | False |