| Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,1,4,3,3,0,1,1,1,2,2,2,1,2,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.101'] |
| Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K2**2 + 3*K2 + 2*K3 + 6 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.101', '6.505'] |
| Outer characteristic polynomial of the knot is: t^7+90t^5+60t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.101'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 448*K1**4*K2 - 1744*K1**4 + 192*K1**3*K2*K3 - 416*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2496*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 384*K1**2*K2*K4 + 4056*K1**2*K2 - 720*K1**2*K3**2 - 96*K1**2*K3*K5 - 144*K1**2*K4**2 - 2136*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3784*K1*K2*K3 - 32*K1*K3**2*K5 + 1120*K1*K3*K4 + 136*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 496*K2**2*K3**2 + 32*K2**2*K4**3 - 184*K2**2*K4**2 + 704*K2**2*K4 - 1860*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 416*K2*K3*K5 + 64*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 80*K3**2*K6 - 1112*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 358*K4**2 - 88*K5**2 - 20*K6**2 + 2052 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.101'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11473', 'vk6.11777', 'vk6.12792', 'vk6.13128', 'vk6.16348', 'vk6.16390', 'vk6.19194', 'vk6.19195', 'vk6.19488', 'vk6.19489', 'vk6.22370', 'vk6.22765', 'vk6.22773', 'vk6.25991', 'vk6.25992', 'vk6.26378', 'vk6.28342', 'vk6.31228', 'vk6.31577', 'vk6.34627', 'vk6.34633', 'vk6.34710', 'vk6.34718', 'vk6.35548', 'vk6.35997', 'vk6.38088', 'vk6.39966', 'vk6.40126', 'vk6.42345', 'vk6.42351', 'vk6.42955', 'vk6.43250', 'vk6.44572', 'vk6.44573', 'vk6.46640', 'vk6.52226', 'vk6.56540', 'vk6.59141', 'vk6.59149', 'vk6.66260'] |
| 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 | O1O2O3O4O5O6U2U6U3U5U1U4 |
| R3 orbit | {'O1O2O3O4O5O6U2U6U3U5U1U4', 'O1O2O3O4O5U1U5U2U6U3O6U4'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3O4O5O6U3U6U2U4U1U5 |
| Gauss code of K* | O1O2O3O4O5O6U5U1U3U6U4U2 |
| Gauss code of -K* | O1O2O3O4O5O6U5U3U1U4U6U2 |
| 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 -4 -1 3 2 1],[ 1 0 -3 0 3 2 1],[ 4 3 0 2 4 3 1],[ 1 0 -2 0 2 1 0],[-3 -3 -4 -2 0 0 0],[-2 -2 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 2 1 -1 -1 -4],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 -1 -2 -3],[-1 0 0 0 0 -1 -1],[ 1 2 1 0 0 0 -2],[ 1 3 2 1 0 0 -3],[ 4 4 3 1 2 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,1,1,4,0,0,2,3,4,0,1,2,3,0,1,1,0,2,3] |
| Phi over symmetry | [-4,-1,-1,1,2,3,0,1,4,3,3,0,1,1,1,2,2,2,1,2,1] |
| Phi of -K | [-4,-1,-1,1,2,3,0,1,4,3,3,0,1,1,1,2,2,2,1,2,1] |
| Phi of K* | [-3,-2,-1,1,1,4,1,2,1,2,3,1,1,2,3,1,2,4,0,0,1] |
| Phi of -K* | [-4,-1,-1,1,2,3,2,3,1,3,4,0,0,1,2,1,2,3,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^3-t^2+t |
| Normalized Jones-Krushkal polynomial | 2z^2+15z+23 |
| Enhanced Jones-Krushkal polynomial | 2w^3z^2+15w^2z+23w |
| Inner characteristic polynomial | t^6+58t^4+23t^2+1 |
| Outer characteristic polynomial | t^7+90t^5+60t^3+4t |
| Flat arrow polynomial | 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K2**2 + 3*K2 + 2*K3 + 6 |
| 2-strand cable arrow polynomial | -64*K1**6 + 448*K1**4*K2 - 1744*K1**4 + 192*K1**3*K2*K3 - 416*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2496*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 384*K1**2*K2*K4 + 4056*K1**2*K2 - 720*K1**2*K3**2 - 96*K1**2*K3*K5 - 144*K1**2*K4**2 - 2136*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3784*K1*K2*K3 - 32*K1*K3**2*K5 + 1120*K1*K3*K4 + 136*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 496*K2**2*K3**2 + 32*K2**2*K4**3 - 184*K2**2*K4**2 + 704*K2**2*K4 - 1860*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 416*K2*K3*K5 + 64*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 80*K3**2*K6 - 1112*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 358*K4**2 - 88*K5**2 - 20*K6**2 + 2052 |
| 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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]] |
| If K is slice | False |