| Min(phi) over symmetries of the knot is: [-5,0,0,0,1,4,1,2,3,5,4,0,0,1,1,0,1,2,1,3,3] |
| Flat knots (up to 7 crossings) with same phi are :['6.46'] |
| Arrow polynomial of the knot is: 4*K1*K2**2 - 4*K1*K2 + K1 - 2*K2**2 - 2*K2*K3 + K3 + K4 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.46'] |
| Outer characteristic polynomial of the knot is: t^7+123t^5+78t^3+6t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.46'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 96*K1**3*K3*K4 + 320*K1**2*K2*K4**2 - 416*K1**2*K2*K4 + 984*K1**2*K2 - 208*K1**2*K3**2 - 800*K1**2*K4**2 - 1924*K1**2 - 512*K1*K2*K3*K4 + 992*K1*K2*K3 - 384*K1*K2*K4*K5 + 1840*K1*K3*K4 + 1144*K1*K4*K5 + 120*K1*K5*K6 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 560*K2**2*K4**2 + 1040*K2**2*K4 - 1634*K2**2 + 32*K2*K3*K4**2*K5 - 32*K2*K3*K4*K5 + 424*K2*K3*K5 + 32*K2*K4**3*K6 - 32*K2*K4**2*K6 + 600*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 - 1008*K3**2 + 16*K3*K4*K7 - 56*K4**4 - 32*K4**2*K5**2 - 8*K4**2*K6**2 + 32*K4**2*K8 - 1344*K4**2 + 8*K4*K5*K9 - 504*K5**2 - 174*K6**2 - 4*K7**2 - 2*K8**2 + 2072 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.46'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19954', 'vk6.20099', 'vk6.21205', 'vk6.21378', 'vk6.26939', 'vk6.27168', 'vk6.28689', 'vk6.28853', 'vk6.38355', 'vk6.38578', 'vk6.40501', 'vk6.40768', 'vk6.45218', 'vk6.45465', 'vk6.47039', 'vk6.47203', 'vk6.56746', 'vk6.56916', 'vk6.57849', 'vk6.58051', 'vk6.61189', 'vk6.61451', 'vk6.62425', 'vk6.62604', 'vk6.66450', 'vk6.66620', 'vk6.67221', 'vk6.67408', 'vk6.69096', 'vk6.69264', 'vk6.69877', 'vk6.70003'] |
| 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 | O1O2O3O4O5O6U1U5U4U3U6U2 |
| R3 orbit | {'O1O2O3O4O5O6U1U5U4U3U6U2'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5O6U5U1U4U3U2U6 |
| Gauss code of K* | O1O2O3O4O5O6U1U6U4U3U2U5 |
| Gauss code of -K* | O1O2O3O4O5O6U2U5U4U3U1U6 |
| 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 -5 1 0 0 0 4],[ 5 0 5 3 2 1 4],[-1 -5 0 -1 -1 -1 3],[ 0 -3 1 0 0 0 3],[ 0 -2 1 0 0 0 2],[ 0 -1 1 0 0 0 1],[-4 -4 -3 -3 -2 -1 0]] |
| Primitive based matrix | [[ 0 4 1 0 0 0 -5],[-4 0 -3 -1 -2 -3 -4],[-1 3 0 -1 -1 -1 -5],[ 0 1 1 0 0 0 -1],[ 0 2 1 0 0 0 -2],[ 0 3 1 0 0 0 -3],[ 5 4 5 1 2 3 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-1,0,0,0,5,3,1,2,3,4,1,1,1,5,0,0,1,0,2,3] |
| Phi over symmetry | [-5,0,0,0,1,4,1,2,3,5,4,0,0,1,1,0,1,2,1,3,3] |
| Phi of -K | [-5,0,0,0,1,4,2,3,4,1,5,0,0,0,1,0,0,2,0,3,0] |
| Phi of K* | [-4,-1,0,0,0,5,0,1,2,3,5,0,0,0,1,0,0,2,0,3,4] |
| Phi of -K* | [-5,0,0,0,1,4,1,2,3,5,4,0,0,1,1,0,1,2,1,3,3] |
| Symmetry type of based matrix | c |
| u-polynomial | t^5-t^4-t |
| Normalized Jones-Krushkal polynomial | z^2+6z+9 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w |
| Inner characteristic polynomial | t^6+81t^4+20t^2 |
| Outer characteristic polynomial | t^7+123t^5+78t^3+6t |
| Flat arrow polynomial | 4*K1*K2**2 - 4*K1*K2 + K1 - 2*K2**2 - 2*K2*K3 + K3 + K4 + 2 |
| 2-strand cable arrow polynomial | -144*K1**4 + 96*K1**3*K3*K4 + 320*K1**2*K2*K4**2 - 416*K1**2*K2*K4 + 984*K1**2*K2 - 208*K1**2*K3**2 - 800*K1**2*K4**2 - 1924*K1**2 - 512*K1*K2*K3*K4 + 992*K1*K2*K3 - 384*K1*K2*K4*K5 + 1840*K1*K3*K4 + 1144*K1*K4*K5 + 120*K1*K5*K6 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 560*K2**2*K4**2 + 1040*K2**2*K4 - 1634*K2**2 + 32*K2*K3*K4**2*K5 - 32*K2*K3*K4*K5 + 424*K2*K3*K5 + 32*K2*K4**3*K6 - 32*K2*K4**2*K6 + 600*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 - 1008*K3**2 + 16*K3*K4*K7 - 56*K4**4 - 32*K4**2*K5**2 - 8*K4**2*K6**2 + 32*K4**2*K8 - 1344*K4**2 + 8*K4*K5*K9 - 504*K5**2 - 174*K6**2 - 4*K7**2 - 2*K8**2 + 2072 |
| 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}], [{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}, {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}, {3}, {1, 2}]] |
| If K is slice | False |