| Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,3,2,2,4,2,1,2,2,0,2,1,1,0,-2] |
| Flat knots (up to 7 crossings) with same phi are :['6.165'] |
| Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.165'] |
| Outer characteristic polynomial of the knot is: t^7+83t^5+140t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.165'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 192*K1**4*K2 - 304*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 + 256*K1**2*K2**5 - 1920*K1**2*K2**4 + 4224*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7984*K1**2*K2**2 - 320*K1**2*K2*K4 + 5704*K1**2*K2 - 80*K1**2*K3**2 - 3496*K1**2 + 128*K1*K2**5*K3 - 512*K1*K2**4*K3 + 3680*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2528*K1*K2**2*K3 - 352*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5896*K1*K2*K3 + 264*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 1760*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 1600*K2**4*K4 - 4576*K2**4 + 160*K2**3*K3*K5 - 160*K2**3*K6 - 1616*K2**2*K3**2 - 192*K2**2*K4**2 + 3000*K2**2*K4 - 166*K2**2 + 432*K2*K3*K5 + 16*K2*K4*K6 - 1196*K3**2 - 264*K4**2 - 20*K5**2 - 2*K6**2 + 2534 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.165'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70466', 'vk6.70481', 'vk6.70522', 'vk6.70595', 'vk6.70640', 'vk6.70663', 'vk6.70749', 'vk6.70837', 'vk6.70919', 'vk6.70948', 'vk6.70999', 'vk6.71100', 'vk6.71156', 'vk6.71171', 'vk6.71232', 'vk6.71293', 'vk6.71311', 'vk6.71326', 'vk6.73555', 'vk6.74354', 'vk6.74998', 'vk6.75316', 'vk6.76569', 'vk6.76648', 'vk6.76980', 'vk6.78294', 'vk6.79390', 'vk6.79942', 'vk6.81505', 'vk6.86859', 'vk6.88080', 'vk6.89222'] |
| 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 | O1O2O3O4O5U1O6U4U5U6U2U3 |
| R3 orbit | {'O1O2O3O4O5U1O6U4U5U6U2U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U3U4U6U1U2O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U4U5U1U2O6U3 |
| Gauss code of -K* | O1O2O3O4O5U3O6U4U5U1U2U6 |
| 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 2 -1 1 2],[ 4 0 3 4 1 2 2],[ 0 -3 0 1 -2 0 2],[-2 -4 -1 0 -2 0 2],[ 1 -1 2 2 0 1 2],[-1 -2 0 0 -1 0 1],[-2 -2 -2 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 2 2 1 0 -1 -4],[-2 0 2 0 -1 -2 -4],[-2 -2 0 -1 -2 -2 -2],[-1 0 1 0 0 -1 -2],[ 0 1 2 0 0 -2 -3],[ 1 2 2 1 2 0 -1],[ 4 4 2 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,0,1,4,-2,0,1,2,4,1,2,2,2,0,1,2,2,3,1] |
| Phi over symmetry | [-4,-1,0,1,2,2,1,3,2,2,4,2,1,2,2,0,2,1,1,0,-2] |
| Phi of -K | [-4,-1,0,1,2,2,2,1,3,2,4,-1,1,1,1,1,1,0,1,0,-2] |
| Phi of K* | [-2,-2,-1,0,1,4,-2,0,0,1,4,1,1,1,2,1,1,3,-1,1,2] |
| Phi of -K* | [-4,-1,0,1,2,2,1,3,2,2,4,2,1,2,2,0,2,1,1,0,-2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-2t^2 |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w |
| Inner characteristic polynomial | t^6+57t^4+26t^2+1 |
| Outer characteristic polynomial | t^7+83t^5+140t^3+12t |
| Flat arrow polynomial | -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3 |
| 2-strand cable arrow polynomial | -128*K1**4*K2**2 + 192*K1**4*K2 - 304*K1**4 + 64*K1**3*K2*K3 - 192*K1**3*K3 + 256*K1**2*K2**5 - 1920*K1**2*K2**4 + 4224*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7984*K1**2*K2**2 - 320*K1**2*K2*K4 + 5704*K1**2*K2 - 80*K1**2*K3**2 - 3496*K1**2 + 128*K1*K2**5*K3 - 512*K1*K2**4*K3 + 3680*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2528*K1*K2**2*K3 - 352*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5896*K1*K2*K3 + 264*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 1760*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 1600*K2**4*K4 - 4576*K2**4 + 160*K2**3*K3*K5 - 160*K2**3*K6 - 1616*K2**2*K3**2 - 192*K2**2*K4**2 + 3000*K2**2*K4 - 166*K2**2 + 432*K2*K3*K5 + 16*K2*K4*K6 - 1196*K3**2 - 264*K4**2 - 20*K5**2 - 2*K6**2 + 2534 |
| 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}, {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}, {1, 3}, {2}]] |
| If K is slice | False |