| Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,1,1,1,1,2,-1,0,0,0,0,0,0,1,1,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1766'] |
| Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.374', '6.446', '6.527', '6.1218', '6.1237', '6.1276', '6.1498', '6.1523', '6.1595', '6.1703', '6.1751', '6.1766', '6.1849', '6.1926'] |
| Outer characteristic polynomial of the knot is: t^7+17t^5+26t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1766', '6.1806'] |
| 2-strand cable arrow polynomial of the knot is: -320*K1**6 - 192*K1**4*K2**2 + 2528*K1**4*K2 - 5552*K1**4 + 416*K1**3*K2*K3 - 1856*K1**3*K3 + 1280*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 7280*K1**2*K2**2 - 672*K1**2*K2*K4 + 12880*K1**2*K2 - 624*K1**2*K3**2 - 128*K1**2*K4**2 - 6416*K1**2 + 256*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8632*K1*K2*K3 + 1320*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1352*K2**4 - 96*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1920*K2**2*K4 - 5450*K2**2 + 360*K2*K3*K5 + 72*K2*K4*K6 - 2312*K3**2 - 686*K4**2 - 80*K5**2 - 6*K6**2 + 5532 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1766'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16553', 'vk6.16646', 'vk6.18142', 'vk6.18478', 'vk6.22952', 'vk6.23073', 'vk6.24597', 'vk6.25010', 'vk6.34953', 'vk6.35074', 'vk6.36740', 'vk6.37159', 'vk6.42522', 'vk6.42633', 'vk6.44008', 'vk6.44320', 'vk6.54800', 'vk6.54884', 'vk6.55956', 'vk6.56256', 'vk6.59228', 'vk6.59307', 'vk6.60490', 'vk6.60856', 'vk6.64782', 'vk6.64847', 'vk6.65621', 'vk6.65928', 'vk6.68080', 'vk6.68145', 'vk6.68692', 'vk6.68903'] |
| 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 | O1O2O3U4U1O5O4U6U2O6U5U3 |
| R3 orbit | {'O1O2O3U4U1O5O4U6U2O6U5U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U1U4O5U2U5O6O4U3U6 |
| Gauss code of K* | O1O2U1O3O4U5U2U4O6O5U3U6 |
| Gauss code of -K* | O1O2U3O4O3U5U2O6O5U1U4U6 |
| 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 0 2 0 0 -1],[ 1 0 0 1 1 0 0],[ 0 0 0 1 0 -1 0],[-2 -1 -1 0 -1 -1 -2],[ 0 -1 0 1 0 0 -1],[ 0 0 1 1 0 0 0],[ 1 0 0 2 1 0 0]] |
| Primitive based matrix | [[ 0 2 0 0 0 -1 -1],[-2 0 -1 -1 -1 -1 -2],[ 0 1 0 1 0 0 0],[ 0 1 -1 0 0 0 0],[ 0 1 0 0 0 -1 -1],[ 1 1 0 0 1 0 0],[ 1 2 0 0 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,0,0,0,1,1,1,1,1,1,2,-1,0,0,0,0,0,0,1,1,0] |
| Phi over symmetry | [-2,0,0,0,1,1,1,1,1,1,2,-1,0,0,0,0,0,0,1,1,0] |
| Phi of -K | [-1,-1,0,0,0,2,0,0,1,1,1,0,1,1,2,0,0,1,-1,1,1] |
| Phi of K* | [-2,0,0,0,1,1,1,1,1,1,2,-1,0,1,1,0,1,1,0,0,0] |
| Phi of -K* | [-1,-1,0,0,0,2,0,0,0,1,1,0,0,1,2,-1,0,1,0,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^2+2t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^6+11t^4+13t^2+1 |
| Outer characteristic polynomial | t^7+17t^5+26t^3+4t |
| Flat arrow polynomial | 4*K1**3 - 10*K1**2 - 8*K1*K2 + K1 + 5*K2 + 3*K3 + 6 |
| 2-strand cable arrow polynomial | -320*K1**6 - 192*K1**4*K2**2 + 2528*K1**4*K2 - 5552*K1**4 + 416*K1**3*K2*K3 - 1856*K1**3*K3 + 1280*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 7280*K1**2*K2**2 - 672*K1**2*K2*K4 + 12880*K1**2*K2 - 624*K1**2*K3**2 - 128*K1**2*K4**2 - 6416*K1**2 + 256*K1*K2**3*K3 - 1824*K1*K2**2*K3 - 224*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8632*K1*K2*K3 + 1320*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 1352*K2**4 - 96*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1920*K2**2*K4 - 5450*K2**2 + 360*K2*K3*K5 + 72*K2*K4*K6 - 2312*K3**2 - 686*K4**2 - 80*K5**2 - 6*K6**2 + 5532 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}]] |
| If K is slice | False |