| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,1,1,1,1,1,2,2,1,2,2,-1,-1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.951'] |
| Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935'] |
| Outer characteristic polynomial of the knot is: t^7+39t^5+79t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.951'] |
| 2-strand cable arrow polynomial of the knot is: -240*K1**4 + 416*K1**3*K2*K3 - 832*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1120*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 3112*K1**2*K2 - 1872*K1**2*K3**2 - 160*K1**2*K3*K5 - 4284*K1**2 + 352*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 32*K1*K2**2*K5 + 256*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6632*K1*K2*K3 - 32*K1*K3**2*K5 + 1704*K1*K3*K4 + 64*K1*K4*K5 - 104*K2**4 - 1232*K2**2*K3**2 - 80*K2**2*K4**2 + 288*K2**2*K4 - 3116*K2**2 - 64*K2*K3**2*K4 + 768*K2*K3*K5 + 96*K2*K4*K6 - 64*K3**4 + 64*K3**2*K6 - 2536*K3**2 - 382*K4**2 - 100*K5**2 - 28*K6**2 + 3324 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.951'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4666', 'vk6.4953', 'vk6.6124', 'vk6.6611', 'vk6.8133', 'vk6.8535', 'vk6.9515', 'vk6.9870', 'vk6.20370', 'vk6.21713', 'vk6.27678', 'vk6.29224', 'vk6.39114', 'vk6.41370', 'vk6.45862', 'vk6.47525', 'vk6.48706', 'vk6.48909', 'vk6.49474', 'vk6.49693', 'vk6.50730', 'vk6.50929', 'vk6.51205', 'vk6.51406', 'vk6.57231', 'vk6.58458', 'vk6.61845', 'vk6.62982', 'vk6.66842', 'vk6.67712', 'vk6.69478', 'vk6.70202'] |
| 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 | O1O2O3O4U5U1O6O5U6U2U4U3 |
| R3 orbit | {'O1O2O3O4U5U1O6O5U6U2U4U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U2U1U3U5O6O5U4U6 |
| Gauss code of K* | O1O2O3O4U5U2U4U3O6O5U1U6 |
| Gauss code of -K* | O1O2O3O4U5U4O6O5U2U1U3U6 |
| 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 -1 2 2 0 -1],[ 2 0 0 2 1 2 -1],[ 1 0 0 2 1 2 -1],[-2 -2 -2 0 0 -1 -1],[-2 -1 -1 0 0 -1 -1],[ 0 -2 -2 1 1 0 -1],[ 1 1 1 1 1 1 0]] |
| Primitive based matrix | [[ 0 2 2 0 -1 -1 -2],[-2 0 0 -1 -1 -1 -1],[-2 0 0 -1 -1 -2 -2],[ 0 1 1 0 -1 -2 -2],[ 1 1 1 1 0 1 1],[ 1 1 2 2 -1 0 0],[ 2 1 2 2 -1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,0,1,1,2,0,1,1,1,1,1,1,2,2,1,2,2,-1,-1,0] |
| Phi over symmetry | [-2,-2,0,1,1,2,0,1,1,1,1,1,1,2,2,1,2,2,-1,-1,0] |
| Phi of -K | [-2,-1,-1,0,2,2,1,2,0,2,3,1,-1,1,2,0,2,2,1,1,0] |
| Phi of K* | [-2,-2,0,1,1,2,0,1,1,2,2,1,2,2,3,-1,0,0,-1,1,2] |
| Phi of -K* | [-2,-1,-1,0,2,2,-1,0,2,1,2,1,1,1,1,2,1,2,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 5z^2+24z+29 |
| Enhanced Jones-Krushkal polynomial | 5w^3z^2-2w^3z+26w^2z+29w |
| Inner characteristic polynomial | t^6+25t^4+14t^2 |
| Outer characteristic polynomial | t^7+39t^5+79t^3+9t |
| Flat arrow polynomial | -2*K1**2 - 4*K1*K2 + 2*K1 + K2 + 2*K3 + 2 |
| 2-strand cable arrow polynomial | -240*K1**4 + 416*K1**3*K2*K3 - 832*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1120*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 3112*K1**2*K2 - 1872*K1**2*K3**2 - 160*K1**2*K3*K5 - 4284*K1**2 + 352*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 32*K1*K2**2*K5 + 256*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6632*K1*K2*K3 - 32*K1*K3**2*K5 + 1704*K1*K3*K4 + 64*K1*K4*K5 - 104*K2**4 - 1232*K2**2*K3**2 - 80*K2**2*K4**2 + 288*K2**2*K4 - 3116*K2**2 - 64*K2*K3**2*K4 + 768*K2*K3*K5 + 96*K2*K4*K6 - 64*K3**4 + 64*K3**2*K6 - 2536*K3**2 - 382*K4**2 - 100*K5**2 - 28*K6**2 + 3324 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |