| Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,1,1,2,4,3,1,1,3,2,0,-1,0,1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.447'] |
| Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.80', '6.163', '6.227', '6.447'] |
| Outer characteristic polynomial of the knot is: t^7+100t^5+307t^3+15t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.447'] |
| 2-strand cable arrow polynomial of the knot is: -752*K1**4 - 160*K1**3*K3 - 1152*K1**2*K2**4 + 2240*K1**2*K2**3 - 6384*K1**2*K2**2 - 192*K1**2*K2*K4 + 5456*K1**2*K2 - 112*K1**2*K3**2 - 3396*K1**2 + 3296*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 544*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5224*K1*K2*K3 + 312*K1*K3*K4 - 288*K2**6 - 640*K2**4*K3**2 - 288*K2**4*K4**2 + 960*K2**4*K4 - 3632*K2**4 + 384*K2**3*K3*K5 + 160*K2**3*K4*K6 - 128*K2**3*K6 - 1568*K2**2*K3**2 - 488*K2**2*K4**2 + 2104*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 868*K2**2 + 544*K2*K3*K5 + 144*K2*K4*K6 + 8*K3**2*K6 - 1220*K3**2 - 322*K4**2 - 32*K5**2 - 12*K6**2 + 2664 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.447'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11414', 'vk6.11699', 'vk6.12716', 'vk6.13071', 'vk6.20264', 'vk6.21579', 'vk6.27524', 'vk6.29102', 'vk6.31151', 'vk6.31482', 'vk6.32305', 'vk6.32744', 'vk6.38927', 'vk6.41145', 'vk6.45682', 'vk6.47403', 'vk6.52169', 'vk6.52402', 'vk6.52990', 'vk6.53315', 'vk6.57093', 'vk6.58258', 'vk6.61665', 'vk6.62831', 'vk6.63743', 'vk6.63845', 'vk6.64165', 'vk6.64359', 'vk6.66732', 'vk6.67603', 'vk6.69381', 'vk6.70116'] |
| 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 | O1O2O3O4O5U6U3O6U1U2U4U5 |
| R3 orbit | {'O1O2O3O4O5U6U3O6U1U2U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U1U2U4U5O6U3U6 |
| Gauss code of K* | O1O2O3O4U1U2U5U3U4O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U1U2U6U3U4 |
| 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 -1 -1 2 4 -1],[ 3 0 1 1 3 4 2],[ 1 -1 0 1 2 3 0],[ 1 -1 -1 0 0 1 1],[-2 -3 -2 0 0 1 -2],[-4 -4 -3 -1 -1 0 -4],[ 1 -2 0 -1 2 4 0]] |
| Primitive based matrix | [[ 0 4 2 -1 -1 -1 -3],[-4 0 -1 -1 -3 -4 -4],[-2 1 0 0 -2 -2 -3],[ 1 1 0 0 -1 1 -1],[ 1 3 2 1 0 0 -1],[ 1 4 2 -1 0 0 -2],[ 3 4 3 1 1 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-2,1,1,1,3,1,1,3,4,4,0,2,2,3,1,-1,1,0,1,2] |
| Phi over symmetry | [-4,-2,1,1,1,3,1,1,2,4,3,1,1,3,2,0,-1,0,1,1,1] |
| Phi of -K | [-3,-1,-1,-1,2,4,0,1,1,2,3,0,1,1,1,-1,1,2,3,4,1] |
| Phi of K* | [-4,-2,1,1,1,3,1,1,2,4,3,1,1,3,2,0,-1,0,1,1,1] |
| Phi of -K* | [-3,-1,-1,-1,2,4,1,1,2,3,4,-1,1,0,1,0,2,3,2,4,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+t^3-t^2+3t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+8w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+68t^4+170t^2+4 |
| Outer characteristic polynomial | t^7+100t^5+307t^3+15t |
| Flat arrow polynomial | 4*K1**3 + 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -752*K1**4 - 160*K1**3*K3 - 1152*K1**2*K2**4 + 2240*K1**2*K2**3 - 6384*K1**2*K2**2 - 192*K1**2*K2*K4 + 5456*K1**2*K2 - 112*K1**2*K3**2 - 3396*K1**2 + 3296*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 - 544*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5224*K1*K2*K3 + 312*K1*K3*K4 - 288*K2**6 - 640*K2**4*K3**2 - 288*K2**4*K4**2 + 960*K2**4*K4 - 3632*K2**4 + 384*K2**3*K3*K5 + 160*K2**3*K4*K6 - 128*K2**3*K6 - 1568*K2**2*K3**2 - 488*K2**2*K4**2 + 2104*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 868*K2**2 + 544*K2*K3*K5 + 144*K2*K4*K6 + 8*K3**2*K6 - 1220*K3**2 - 322*K4**2 - 32*K5**2 - 12*K6**2 + 2664 |
| 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}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |