| Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,4,2,3,3,2,1,2,2,1,1,3,0,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.267'] |
| Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449'] |
| Outer characteristic polynomial of the knot is: t^7+96t^5+78t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.267'] |
| 2-strand cable arrow polynomial of the knot is: -16*K1**4 - 480*K1**3*K3 - 384*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 3024*K1**2*K2 - 528*K1**2*K3**2 - 4120*K1**2 + 160*K1*K2**3*K3 - 896*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 4360*K1*K2*K3 + 1272*K1*K3*K4 + 40*K1*K4*K5 + 32*K1*K5*K6 - 200*K2**4 - 32*K2**3*K6 - 640*K2**2*K3**2 - 8*K2**2*K4**2 + 832*K2**2*K4 - 8*K2**2*K6**2 - 3298*K2**2 - 64*K2*K3**2*K4 + 736*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 160*K3**2*K6 - 2264*K3**2 + 8*K3*K4*K7 - 692*K4**2 - 240*K5**2 - 126*K6**2 - 8*K7**2 - 2*K8**2 + 3380 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.267'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73741', 'vk6.73858', 'vk6.74205', 'vk6.74830', 'vk6.75654', 'vk6.75858', 'vk6.76387', 'vk6.76883', 'vk6.78665', 'vk6.78854', 'vk6.79239', 'vk6.79720', 'vk6.80285', 'vk6.80413', 'vk6.80729', 'vk6.81079', 'vk6.81622', 'vk6.81813', 'vk6.81995', 'vk6.82318', 'vk6.82368', 'vk6.82726', 'vk6.83214', 'vk6.84234', 'vk6.84315', 'vk6.84417', 'vk6.84500', 'vk6.85648', 'vk6.86538', 'vk6.87558', 'vk6.88262', 'vk6.89410'] |
| 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 | O1O2O3O4O5U1U3O6U5U4U2U6 |
| R3 orbit | {'O1O2O3O4O5U1U3O6U5U4U2U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U4U2U1O6U3U5 |
| Gauss code of K* | O1O2O3O4U5U3U6U2U1O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U4U3U5U2U6 |
| 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 -4 0 -1 1 1 3],[ 4 0 4 1 3 2 3],[ 0 -4 0 -2 1 1 3],[ 1 -1 2 0 2 1 2],[-1 -3 -1 -2 0 0 2],[-1 -2 -1 -1 0 0 1],[-3 -3 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 1 1 0 -1 -4],[-3 0 -1 -2 -3 -2 -3],[-1 1 0 0 -1 -1 -2],[-1 2 0 0 -1 -2 -3],[ 0 3 1 1 0 -2 -4],[ 1 2 1 2 2 0 -1],[ 4 3 2 3 4 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,-1,0,1,4,1,2,3,2,3,0,1,1,2,1,2,3,2,4,1] |
| Phi over symmetry | [-4,-1,0,1,1,3,1,4,2,3,3,2,1,2,2,1,1,3,0,1,2] |
| Phi of -K | [-4,-1,0,1,1,3,2,0,2,3,4,-1,0,1,2,0,0,0,0,0,1] |
| Phi of K* | [-3,-1,-1,0,1,4,0,1,0,2,4,0,0,0,2,0,1,3,-1,0,2] |
| Phi of -K* | [-4,-1,0,1,1,3,1,4,2,3,3,2,1,2,2,1,1,3,0,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w |
| Inner characteristic polynomial | t^6+68t^4+18t^2+1 |
| Outer characteristic polynomial | t^7+96t^5+78t^3+8t |
| Flat arrow polynomial | -2*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 + 2*K2 + K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -16*K1**4 - 480*K1**3*K3 - 384*K1**2*K2**2 + 32*K1**2*K2*K3**2 + 3024*K1**2*K2 - 528*K1**2*K3**2 - 4120*K1**2 + 160*K1*K2**3*K3 - 896*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 4360*K1*K2*K3 + 1272*K1*K3*K4 + 40*K1*K4*K5 + 32*K1*K5*K6 - 200*K2**4 - 32*K2**3*K6 - 640*K2**2*K3**2 - 8*K2**2*K4**2 + 832*K2**2*K4 - 8*K2**2*K6**2 - 3298*K2**2 - 64*K2*K3**2*K4 + 736*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 160*K3**2*K6 - 2264*K3**2 + 8*K3*K4*K7 - 692*K4**2 - 240*K5**2 - 126*K6**2 - 8*K7**2 - 2*K8**2 + 3380 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |