| Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,2,1,3,4,1,1,1,1,0,1,2,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.426'] |
| Arrow polynomial of the knot is: 4*K1**3 - 6*K1*K2 + 2*K3 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.426', '6.861', '6.1388'] |
| Outer characteristic polynomial of the knot is: t^7+64t^5+126t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.426'] |
| 2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 2528*K1**4 + 1088*K1**3*K2*K3 - 928*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3584*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 4800*K1**2*K2 - 2816*K1**2*K3**2 - 96*K1**2*K3*K5 - 32*K1**2*K4**2 - 2808*K1**2 + 384*K1*K2**3*K3 - 928*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6784*K1*K2*K3 + 2728*K1*K3*K4 + 128*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 464*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 88*K2**2*K4**2 + 736*K2**2*K4 - 2604*K2**2 + 456*K2*K3*K5 + 120*K2*K4*K6 + 32*K3**2*K6 - 2288*K3**2 - 732*K4**2 - 112*K5**2 - 52*K6**2 + 3050 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.426'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20140', 'vk6.20154', 'vk6.20162', 'vk6.20164', 'vk6.21432', 'vk6.21450', 'vk6.27252', 'vk6.27278', 'vk6.27286', 'vk6.27292', 'vk6.28914', 'vk6.28940', 'vk6.28948', 'vk6.38677', 'vk6.38695', 'vk6.38711', 'vk6.38721', 'vk6.40891', 'vk6.40901', 'vk6.47265', 'vk6.47282', 'vk6.47292', 'vk6.56973', 'vk6.56979', 'vk6.56990', 'vk6.57006', 'vk6.58127', 'vk6.62676', 'vk6.62684', 'vk6.67464', 'vk6.70038', 'vk6.70044'] |
| 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 | O1O2O3O4O5U6U1O6U3U5U4U2 |
| R3 orbit | {'O1O2O3O4O5U6U1O6U3U5U4U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U4U2U1U3O6U5U6 |
| Gauss code of K* | O1O2O3O4U5U4U1U3U2O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U3U2U4U1U6 |
| 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 2 -1],[ 3 0 3 0 2 1 3],[-1 -3 0 -2 1 1 -1],[ 1 0 2 0 2 1 1],[-2 -2 -1 -2 0 0 -2],[-2 -1 -1 -1 0 0 -2],[ 1 -3 1 -1 2 2 0]] |
| Primitive based matrix | [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -2 -1],[-2 0 0 -1 -2 -2 -2],[-1 1 1 0 -2 -1 -3],[ 1 1 2 2 0 1 0],[ 1 2 2 1 -1 0 -3],[ 3 1 2 3 0 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-2,-1,1,1,3,0,1,1,2,1,1,2,2,2,2,1,3,-1,0,3] |
| Phi over symmetry | [-3,-1,-1,1,2,2,-1,2,1,3,4,1,1,1,1,0,1,2,0,0,0] |
| Phi of -K | [-3,-1,-1,1,2,2,-1,2,1,3,4,1,1,1,1,0,1,2,0,0,0] |
| Phi of K* | [-2,-2,-1,1,1,3,0,0,1,1,3,0,1,2,4,1,0,1,-1,-1,2] |
| Phi of -K* | [-3,-1,-1,1,2,2,0,3,3,1,2,1,2,1,2,1,2,2,1,1,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-2t^2+t |
| Normalized Jones-Krushkal polynomial | 8z^2+29z+27 |
| Enhanced Jones-Krushkal polynomial | 8w^3z^2+29w^2z+27w |
| Inner characteristic polynomial | t^6+44t^4+54t^2 |
| Outer characteristic polynomial | t^7+64t^5+126t^3+7t |
| Flat arrow polynomial | 4*K1**3 - 6*K1*K2 + 2*K3 + 1 |
| 2-strand cable arrow polynomial | 768*K1**4*K2 - 2528*K1**4 + 1088*K1**3*K2*K3 - 928*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3584*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 512*K1**2*K2*K4 + 4800*K1**2*K2 - 2816*K1**2*K3**2 - 96*K1**2*K3*K5 - 32*K1**2*K4**2 - 2808*K1**2 + 384*K1*K2**3*K3 - 928*K1*K2**2*K3 - 128*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6784*K1*K2*K3 + 2728*K1*K3*K4 + 128*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 464*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 88*K2**2*K4**2 + 736*K2**2*K4 - 2604*K2**2 + 456*K2*K3*K5 + 120*K2*K4*K6 + 32*K3**2*K6 - 2288*K3**2 - 732*K4**2 - 112*K5**2 - 52*K6**2 + 3050 |
| 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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {3, 4}, {2}, {1}]] |
| If K is slice | False |