| Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,1,2,3,3,1,0,0,1,0,1,1,0,0,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.714'] |
| 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+40t^5+89t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.714'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 352*K1**4*K2 - 2144*K1**4 + 320*K1**3*K2*K3 - 480*K1**3*K3 - 3040*K1**2*K2**2 - 128*K1**2*K2*K4 + 6896*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4676*K1**2 + 128*K1*K2**3*K3 - 608*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4800*K1*K2*K3 + 840*K1*K3*K4 + 144*K1*K4*K5 - 224*K2**4 - 160*K2**2*K3**2 - 8*K2**2*K4**2 + 608*K2**2*K4 - 3670*K2**2 + 208*K2*K3*K5 + 8*K2*K4*K6 - 1716*K3**2 - 452*K4**2 - 96*K5**2 - 2*K6**2 + 3738 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.714'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20025', 'vk6.20074', 'vk6.21295', 'vk6.21356', 'vk6.27072', 'vk6.27135', 'vk6.28775', 'vk6.28824', 'vk6.38473', 'vk6.38532', 'vk6.40660', 'vk6.40729', 'vk6.45353', 'vk6.45428', 'vk6.47120', 'vk6.47170', 'vk6.56824', 'vk6.56879', 'vk6.57956', 'vk6.58017', 'vk6.61338', 'vk6.61405', 'vk6.62512', 'vk6.62562', 'vk6.66544', 'vk6.66579', 'vk6.67331', 'vk6.67370', 'vk6.69186', 'vk6.69227', 'vk6.69935', 'vk6.69968'] |
| 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 | O1O2O3O4U5O6U4U2O5U3U1U6 |
| R3 orbit | {'O1O2O3O4U5O6U4U2O5U3U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U4U2O6U3U1O5U6 |
| Gauss code of K* | O1O2O3U2U4U1U5O6U3O5O4U6 |
| Gauss code of -K* | O1O2O3U4O5O6U1O4U6U3U5U2 |
| 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 -1 -1 0 0 -1 3],[ 1 0 0 1 1 -1 3],[ 1 0 0 0 0 0 2],[ 0 -1 0 0 1 -1 1],[ 0 -1 0 -1 0 0 0],[ 1 1 0 1 0 0 3],[-3 -3 -2 -1 0 -3 0]] |
| Primitive based matrix | [[ 0 3 0 0 -1 -1 -1],[-3 0 0 -1 -2 -3 -3],[ 0 0 0 -1 0 0 -1],[ 0 1 1 0 0 -1 -1],[ 1 2 0 0 0 0 0],[ 1 3 0 1 0 0 1],[ 1 3 1 1 0 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,0,0,1,1,1,0,1,2,3,3,1,0,0,1,0,1,1,0,0,-1] |
| Phi over symmetry | [-3,0,0,1,1,1,0,1,2,3,3,1,0,0,1,0,1,1,0,0,-1] |
| Phi of -K | [-1,-1,-1,0,0,3,-1,0,0,1,1,0,0,0,1,1,1,2,-1,2,3] |
| Phi of K* | [-3,0,0,1,1,1,2,3,1,1,2,1,0,0,1,0,1,1,-1,0,0] |
| Phi of -K* | [-1,-1,-1,0,0,3,-1,0,1,1,3,0,0,1,3,0,0,2,-1,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+3t |
| Normalized Jones-Krushkal polynomial | z^2+18z+33 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+18w^2z+33w |
| Inner characteristic polynomial | t^6+28t^4+43t^2 |
| Outer characteristic polynomial | t^7+40t^5+89t^3+4t |
| Flat arrow polynomial | -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5 |
| 2-strand cable arrow polynomial | -64*K1**6 + 352*K1**4*K2 - 2144*K1**4 + 320*K1**3*K2*K3 - 480*K1**3*K3 - 3040*K1**2*K2**2 - 128*K1**2*K2*K4 + 6896*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 4676*K1**2 + 128*K1*K2**3*K3 - 608*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4800*K1*K2*K3 + 840*K1*K3*K4 + 144*K1*K4*K5 - 224*K2**4 - 160*K2**2*K3**2 - 8*K2**2*K4**2 + 608*K2**2*K4 - 3670*K2**2 + 208*K2*K3*K5 + 8*K2*K4*K6 - 1716*K3**2 - 452*K4**2 - 96*K5**2 - 2*K6**2 + 3738 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]] |
| If K is slice | False |