| Min(phi) over symmetries of the knot is: [-4,0,1,1,1,1,1,1,3,4,4,0,1,0,1,0,-1,0,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.459'] |
| Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506'] |
| Outer characteristic polynomial of the knot is: t^7+66t^5+105t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.459'] |
| 2-strand cable arrow polynomial of the knot is: -976*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 672*K1**3*K3 + 96*K1**2*K2**2*K4 - 704*K1**2*K2**2 - 544*K1**2*K2*K4 + 2632*K1**2*K2 - 496*K1**2*K3**2 - 160*K1**2*K4**2 - 1980*K1**2 - 64*K1*K2**2*K3 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 2336*K1*K2*K3 + 1152*K1*K3*K4 + 328*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4 - 48*K2**2*K4**2 + 496*K2**2*K4 - 1596*K2**2 + 232*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 + 8*K3**2*K6 - 1008*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 668*K4**2 - 200*K5**2 - 60*K6**2 - 12*K7**2 - 2*K8**2 + 1788 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.459'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11494', 'vk6.11805', 'vk6.12822', 'vk6.13151', 'vk6.13394', 'vk6.13487', 'vk6.13678', 'vk6.13776', 'vk6.14203', 'vk6.14450', 'vk6.15675', 'vk6.16127', 'vk6.16767', 'vk6.16782', 'vk6.16883', 'vk6.19040', 'vk6.19313', 'vk6.19606', 'vk6.22481', 'vk6.23195', 'vk6.23264', 'vk6.23789', 'vk6.26499', 'vk6.28386', 'vk6.33149', 'vk6.33211', 'vk6.33305', 'vk6.35182', 'vk6.36030', 'vk6.40028', 'vk6.40292', 'vk6.42673', 'vk6.42688', 'vk6.44737', 'vk6.46759', 'vk6.48016', 'vk6.52257', 'vk6.53414', 'vk6.53570', 'vk6.53700'] |
| 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 | O1O2O3O4O5U6U3O6U4U2U1U5 |
| R3 orbit | {'O1O2O3O4O5U6U3O6U4U2U1U5', 'O1O2O3O4O5U4U6U3O6U2U1U5'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5U1U5U4U2O6U3U6 |
| Gauss code of K* | O1O2O3O4U3U2U5U1U4O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U1U4U6U3U2 |
| 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 -1 0 4 -1],[ 1 0 0 0 1 4 0],[ 1 0 0 0 1 3 0],[ 1 0 0 0 0 1 1],[ 0 -1 -1 0 0 1 0],[-4 -4 -3 -1 -1 0 -4],[ 1 0 0 -1 0 4 0]] |
| Primitive based matrix | [[ 0 4 0 -1 -1 -1 -1],[-4 0 -1 -1 -3 -4 -4],[ 0 1 0 0 -1 0 -1],[ 1 1 0 0 0 1 0],[ 1 3 1 0 0 0 0],[ 1 4 0 -1 0 0 0],[ 1 4 1 0 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,0,1,1,1,1,1,1,3,4,4,0,1,0,1,0,-1,0,0,0,0] |
| Phi over symmetry | [-4,0,1,1,1,1,1,1,3,4,4,0,1,0,1,0,-1,0,0,0,0] |
| Phi of -K | [-1,-1,-1,-1,0,4,-1,0,0,1,4,0,0,1,1,0,0,1,0,2,3] |
| Phi of K* | [-4,0,1,1,1,1,3,1,1,2,4,0,1,0,1,0,0,0,0,-1,0] |
| Phi of -K* | [-1,-1,-1,-1,0,4,-1,0,0,0,4,0,0,0,1,0,1,3,1,4,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+4t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2+17w^2z+19w |
| Inner characteristic polynomial | t^6+46t^4+63t^2+1 |
| Outer characteristic polynomial | t^7+66t^5+105t^3+4t |
| Flat arrow polynomial | -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -976*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 672*K1**3*K3 + 96*K1**2*K2**2*K4 - 704*K1**2*K2**2 - 544*K1**2*K2*K4 + 2632*K1**2*K2 - 496*K1**2*K3**2 - 160*K1**2*K4**2 - 1980*K1**2 - 64*K1*K2**2*K3 - 64*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 2336*K1*K2*K3 + 1152*K1*K3*K4 + 328*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4 - 48*K2**2*K4**2 + 496*K2**2*K4 - 1596*K2**2 + 232*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 + 8*K3**2*K6 - 1008*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 668*K4**2 - 200*K5**2 - 60*K6**2 - 12*K7**2 - 2*K8**2 + 1788 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |