| Min(phi) over symmetries of the knot is: [-4,0,0,0,1,3,0,2,3,4,4,1,1,0,1,0,1,1,1,2,2] |
| Flat knots (up to 7 crossings) with same phi are :['6.463'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.95', '6.107', '6.276', '6.292', '6.394', '6.429', '6.463'] |
| Outer characteristic polynomial of the knot is: t^7+85t^5+180t^3+9t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.463'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K3*K4 - 32*K1**2*K2*K4 + 328*K1**2*K2 - 1936*K1**2*K3**2 - 64*K1**2*K3*K5 - 80*K1**2*K4**2 - 2168*K1**2 - 224*K1*K2*K3*K4 + 2536*K1*K2*K3 - 64*K1*K3*K4*K6 + 3104*K1*K3*K4 + 448*K1*K4*K5 + 48*K1*K6*K7 + 24*K1*K7*K8 - 8*K2**2*K4**2 + 248*K2**2*K4 - 1170*K2**2 + 240*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 32*K3**2*K4**2 + 88*K3**2*K6 - 1956*K3**2 + 112*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1344*K4**2 - 260*K5**2 - 70*K6**2 - 72*K7**2 - 18*K8**2 + 2304 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.463'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4663', 'vk6.4952', 'vk6.6121', 'vk6.6610', 'vk6.8134', 'vk6.8538', 'vk6.9516', 'vk6.9873', 'vk6.20364', 'vk6.21707', 'vk6.27668', 'vk6.29214', 'vk6.39108', 'vk6.41364', 'vk6.45860', 'vk6.47523', 'vk6.48703', 'vk6.48908', 'vk6.49471', 'vk6.49692', 'vk6.50731', 'vk6.50932', 'vk6.51206', 'vk6.51409', 'vk6.57233', 'vk6.58460', 'vk6.61851', 'vk6.62988', 'vk6.66852', 'vk6.67722', 'vk6.69484', 'vk6.70208'] |
| 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 | O1O2O3O4O5U6U4O6U1U3U2U5 |
| R3 orbit | {'O1O2O3O4O5U6U4O6U1U3U2U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U1U4U3U5O6U2U6 |
| Gauss code of K* | O1O2O3O4U1U3U2U5U4O6O5U6 |
| Gauss code of -K* | O1O2O3O4U5O6O5U1U6U3U2U4 |
| 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 0 0 0 4 -1],[ 3 0 2 1 1 4 2],[ 0 -2 0 0 1 3 -1],[ 0 -1 0 0 1 2 -1],[ 0 -1 -1 -1 0 0 0],[-4 -4 -3 -2 0 0 -4],[ 1 -2 1 1 0 4 0]] |
| Primitive based matrix | [[ 0 4 0 0 0 -1 -3],[-4 0 0 -2 -3 -4 -4],[ 0 0 0 -1 -1 0 -1],[ 0 2 1 0 0 -1 -1],[ 0 3 1 0 0 -1 -2],[ 1 4 0 1 1 0 -2],[ 3 4 1 1 2 2 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,0,0,0,1,3,0,2,3,4,4,1,1,0,1,0,1,1,1,2,2] |
| Phi over symmetry | [-4,0,0,0,1,3,0,2,3,4,4,1,1,0,1,0,1,1,1,2,2] |
| Phi of -K | [-3,-1,0,0,0,4,0,1,2,2,3,0,0,1,1,0,-1,1,-1,2,4] |
| Phi of K* | [-4,0,0,0,1,3,1,2,4,1,3,0,1,0,1,1,0,2,1,2,0] |
| Phi of -K* | [-3,-1,0,0,0,4,2,1,1,2,4,0,1,1,4,-1,-1,0,0,2,3] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+t^3+t |
| Normalized Jones-Krushkal polynomial | 4z^2+21z+27 |
| Enhanced Jones-Krushkal polynomial | 4w^3z^2-4w^3z+25w^2z+27w |
| Inner characteristic polynomial | t^6+59t^4+96t^2+1 |
| Outer characteristic polynomial | t^7+85t^5+180t^3+9t |
| Flat arrow polynomial | -2*K1*K2 + K1 - 2*K2**2 + K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -64*K1**4 + 32*K1**3*K3*K4 - 32*K1**2*K2*K4 + 328*K1**2*K2 - 1936*K1**2*K3**2 - 64*K1**2*K3*K5 - 80*K1**2*K4**2 - 2168*K1**2 - 224*K1*K2*K3*K4 + 2536*K1*K2*K3 - 64*K1*K3*K4*K6 + 3104*K1*K3*K4 + 448*K1*K4*K5 + 48*K1*K6*K7 + 24*K1*K7*K8 - 8*K2**2*K4**2 + 248*K2**2*K4 - 1170*K2**2 + 240*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 - 32*K3**2*K4**2 + 88*K3**2*K6 - 1956*K3**2 + 112*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 1344*K4**2 - 260*K5**2 - 70*K6**2 - 72*K7**2 - 18*K8**2 + 2304 |
| 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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {3, 5}, {4}, {2}, {1}], [{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 |