| Min(phi) over symmetries of the knot is: [-3,-1,2,2,0,2,4,1,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.383'] |
| Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.383', '6.922', '6.1172', '6.1356', '6.1359'] |
| Outer characteristic polynomial of the knot is: t^5+41t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.383'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 832*K1**4*K2 - 4976*K1**4 + 768*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1120*K1**3*K3 + 320*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7328*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1632*K1**2*K2*K4 + 14224*K1**2*K2 - 1232*K1**2*K3**2 - 96*K1**2*K3*K5 - 640*K1**2*K4**2 - 9332*K1**2 + 320*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 192*K1*K2**2*K5 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 11920*K1*K2*K3 - 160*K1*K2*K4*K5 + 3664*K1*K3*K4 + 824*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 912*K2**4 - 32*K2**3*K6 - 704*K2**2*K3**2 - 392*K2**2*K4**2 + 2552*K2**2*K4 - 7768*K2**2 + 768*K2*K3*K5 + 224*K2*K4*K6 - 64*K3**4 + 56*K3**2*K6 - 4180*K3**2 - 1828*K4**2 - 352*K5**2 - 48*K6**2 + 8138 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.383'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16562', 'vk6.16653', 'vk6.18154', 'vk6.18490', 'vk6.22961', 'vk6.23080', 'vk6.24609', 'vk6.25022', 'vk6.34962', 'vk6.35081', 'vk6.36744', 'vk6.37163', 'vk6.42531', 'vk6.42640', 'vk6.44012', 'vk6.44324', 'vk6.54793', 'vk6.54879', 'vk6.55952', 'vk6.56252', 'vk6.59221', 'vk6.59301', 'vk6.60486', 'vk6.60852', 'vk6.64775', 'vk6.64838', 'vk6.65609', 'vk6.65916', 'vk6.68073', 'vk6.68136', 'vk6.68680', 'vk6.68891'] |
| 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 | O1O2O3O4O5U4U2O6U1U5U6U3 |
| R3 orbit | {'O1O2O3O4O5U4U2O6U1U5U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U3U6U1U5O6U4U2 |
| Gauss code of K* | O1O2O3O4U1U5U4U6U2O6O5U3 |
| Gauss code of -K* | O1O2O3O4U2O5O6U3U6U1U5U4 |
| 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 -3 -2 2 -1 2 2],[ 3 0 0 4 0 3 2],[ 2 0 0 2 0 2 1],[-2 -4 -2 0 -1 0 1],[ 1 0 0 1 0 1 1],[-2 -3 -2 0 -1 0 1],[-2 -2 -1 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 2 -1 -3],[-2 0 1 -1 -4],[-2 -1 0 -1 -2],[ 1 1 1 0 0],[ 3 4 2 0 0]] |
| If based matrix primitive | False |
| Phi of primitive based matrix | [-2,-2,1,3,-1,1,4,1,2,0] |
| Phi over symmetry | [-3,-1,2,2,0,2,4,1,1,-1] |
| Phi of -K | [-3,-1,2,2,2,1,3,2,2,-1] |
| Phi of K* | [-2,-2,1,3,-1,2,3,2,1,2] |
| Phi of -K* | [-3,-1,2,2,0,2,4,1,1,-1] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 2z^2+23z+39 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+23w^2z+39w |
| Inner characteristic polynomial | t^4+23t^2+4 |
| Outer characteristic polynomial | t^5+41t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 12*K1**2 - 10*K1*K2 + 2*K1 + 6*K2 + 4*K3 + 7 |
| 2-strand cable arrow polynomial | -64*K1**6 - 64*K1**4*K2**2 + 832*K1**4*K2 - 4976*K1**4 + 768*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1120*K1**3*K3 + 320*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7328*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 1632*K1**2*K2*K4 + 14224*K1**2*K2 - 1232*K1**2*K3**2 - 96*K1**2*K3*K5 - 640*K1**2*K4**2 - 9332*K1**2 + 320*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1664*K1*K2**2*K3 - 192*K1*K2**2*K5 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 11920*K1*K2*K3 - 160*K1*K2*K4*K5 + 3664*K1*K3*K4 + 824*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 912*K2**4 - 32*K2**3*K6 - 704*K2**2*K3**2 - 392*K2**2*K4**2 + 2552*K2**2*K4 - 7768*K2**2 + 768*K2*K3*K5 + 224*K2*K4*K6 - 64*K3**4 + 56*K3**2*K6 - 4180*K3**2 - 1828*K4**2 - 352*K5**2 - 48*K6**2 + 8138 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}]] |
| If K is slice | False |