| Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,2,2,4,2,1,1,2,1,-1,-1,1,1,1,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.652'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166'] |
| Outer characteristic polynomial of the knot is: t^7+64t^5+91t^3+8t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.652'] |
| 2-strand cable arrow polynomial of the knot is: -416*K1**4 - 128*K1**3*K3 + 1184*K1**2*K2**3 - 3632*K1**2*K2**2 - 352*K1**2*K2*K4 + 5544*K1**2*K2 - 256*K1**2*K3**2 - 4840*K1**2 + 256*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4736*K1*K2*K3 + 1136*K1*K3*K4 + 8*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 448*K2**4*K4 - 1776*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 248*K2**2*K4**2 + 1824*K2**2*K4 - 2908*K2**2 - 64*K2*K3**2*K4 + 384*K2*K3*K5 + 144*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 1716*K3**2 - 756*K4**2 - 92*K5**2 - 60*K6**2 + 3650 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.652'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73762', 'vk6.73763', 'vk6.73803', 'vk6.73805', 'vk6.73900', 'vk6.73902', 'vk6.73939', 'vk6.73940', 'vk6.75747', 'vk6.75749', 'vk6.75904', 'vk6.75905', 'vk6.78701', 'vk6.78702', 'vk6.78759', 'vk6.78761', 'vk6.78900', 'vk6.78902', 'vk6.80324', 'vk6.80326', 'vk6.80360', 'vk6.80361', 'vk6.80448', 'vk6.80449', 'vk6.81719', 'vk6.81721', 'vk6.82493', 'vk6.82496', 'vk6.84459', 'vk6.84463', 'vk6.88344', 'vk6.88352'] |
| 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 | O1O2O3O4U2O5U1U4O6U5U3U6 |
| R3 orbit | {'O1O2O3O4U2O5U1U4O6U5U3U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U5U2U6O5U1U4O6U3 |
| Gauss code of K* | O1O2O3U4U5U2U6O5U1O4O6U3 |
| Gauss code of -K* | O1O2O3U1O4O5U3O6U4U2U6U5 |
| 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 -2 1 1 1 2],[ 3 0 0 4 2 2 2],[ 2 0 0 2 1 1 1],[-1 -4 -2 0 -1 1 2],[-1 -2 -1 1 0 1 1],[-1 -2 -1 -1 -1 0 1],[-2 -2 -1 -2 -1 -1 0]] |
| Primitive based matrix | [[ 0 2 1 1 1 -2 -3],[-2 0 -1 -1 -2 -1 -2],[-1 1 0 1 1 -1 -2],[-1 1 -1 0 -1 -1 -2],[-1 2 -1 1 0 -2 -4],[ 2 1 1 1 2 0 0],[ 3 2 2 2 4 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,-1,2,3,1,1,2,1,2,-1,-1,1,2,1,1,2,2,4,0] |
| Phi over symmetry | [-3,-2,1,1,1,2,0,2,2,4,2,1,1,2,1,-1,-1,1,1,1,2] |
| Phi of -K | [-3,-2,1,1,1,2,1,0,2,2,3,1,2,2,3,-1,1,-1,1,0,0] |
| Phi of K* | [-2,-1,-1,-1,2,3,-1,0,0,3,3,-1,1,1,0,1,2,2,2,2,1] |
| Phi of -K* | [-3,-2,1,1,1,2,0,2,2,4,2,1,1,2,1,-1,-1,1,1,1,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^3-3t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-2w^3z+23w^2z+27w |
| Inner characteristic polynomial | t^6+44t^4+15t^2 |
| Outer characteristic polynomial | t^7+64t^5+91t^3+8t |
| Flat arrow polynomial | 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3 |
| 2-strand cable arrow polynomial | -416*K1**4 - 128*K1**3*K3 + 1184*K1**2*K2**3 - 3632*K1**2*K2**2 - 352*K1**2*K2*K4 + 5544*K1**2*K2 - 256*K1**2*K3**2 - 4840*K1**2 + 256*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4736*K1*K2*K3 + 1136*K1*K3*K4 + 8*K1*K4*K5 + 8*K1*K5*K6 - 288*K2**6 + 448*K2**4*K4 - 1776*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 248*K2**2*K4**2 + 1824*K2**2*K4 - 2908*K2**2 - 64*K2*K3**2*K4 + 384*K2*K3*K5 + 144*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 1716*K3**2 - 756*K4**2 - 92*K5**2 - 60*K6**2 + 3650 |
| 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}, {4, 5}, {3}, {2}, {1}], [{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 |