Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.886

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,2,1,2,3,2,1,1,1,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.886']
Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.373', '6.434', '6.878', '6.886', '6.952', '6.1160']
Outer characteristic polynomial of the knot is: t^7+53t^5+66t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.886']
2-strand cable arrow polynomial of the knot is: -592*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 736*K1**3*K3 - 1088*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 3696*K1**2*K2 - 1200*K1**2*K3**2 - 96*K1**2*K3*K5 - 160*K1**2*K4**2 - 96*K1**2*K4*K6 - 3736*K1**2 + 64*K1*K2**3*K3 - 832*K1*K2**2*K3 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4464*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2040*K1*K3*K4 + 544*K1*K4*K5 + 120*K1*K5*K6 + 24*K1*K6*K7 - 64*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 40*K2**2*K4**2 + 752*K2**2*K4 - 8*K2**2*K6**2 - 2722*K2**2 - 32*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 528*K2*K3*K5 - 32*K2*K4**2*K6 + 216*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 - 80*K3**2*K4**2 + 128*K3**2*K6 - 1896*K3**2 + 104*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 882*K4**2 - 252*K5**2 - 118*K6**2 - 20*K7**2 - 4*K8**2 + 2908
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.886']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4645', 'vk6.4924', 'vk6.6085', 'vk6.6582', 'vk6.8102', 'vk6.8494', 'vk6.9484', 'vk6.9851', 'vk6.20632', 'vk6.22059', 'vk6.28114', 'vk6.29555', 'vk6.39542', 'vk6.41765', 'vk6.46149', 'vk6.47791', 'vk6.48677', 'vk6.48868', 'vk6.49419', 'vk6.49652', 'vk6.50687', 'vk6.50872', 'vk6.51162', 'vk6.51379', 'vk6.57532', 'vk6.58720', 'vk6.62224', 'vk6.63170', 'vk6.67034', 'vk6.67907', 'vk6.69659', 'vk6.70340']
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 O1O2O3O4U2U1O5O6U5U3U6U4
R3 orbit {'O1O2O3O4U2U1O5O6U5U3U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U2U6O5O6U4U3
Gauss code of K* O1O2O3O4U5U6U2U4O6O5U1U3
Gauss code of -K* O1O2O3O4U2U4O5O6U1U3U6U5
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 -2 -2 0 3 -1 2],[ 2 0 0 2 3 0 1],[ 2 0 0 1 2 0 1],[ 0 -2 -1 0 2 0 2],[-3 -3 -2 -2 0 -1 1],[ 1 0 0 0 1 0 1],[-2 -1 -1 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 -1 -2 -2],[-3 0 1 -2 -1 -2 -3],[-2 -1 0 -2 -1 -1 -1],[ 0 2 2 0 0 -1 -2],[ 1 1 1 0 0 0 0],[ 2 2 1 1 0 0 0],[ 2 3 1 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,2,2,-1,2,1,2,3,2,1,1,1,0,1,2,0,0,0]
Phi over symmetry [-3,-2,0,1,2,2,-1,2,1,2,3,2,1,1,1,0,1,2,0,0,0]
Phi of -K [-2,-2,-1,0,2,3,0,1,0,3,2,1,1,3,3,1,2,3,0,1,2]
Phi of K* [-3,-2,0,1,2,2,2,1,3,2,3,0,2,3,3,1,0,1,1,1,0]
Phi of -K* [-2,-2,-1,0,2,3,0,0,1,1,2,0,2,1,3,0,1,1,2,2,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+31t^4+23t^2
Outer characteristic polynomial t^7+53t^5+66t^3+4t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + K2 + K3 + 2*K4 + 2
2-strand cable arrow polynomial -592*K1**4 + 480*K1**3*K2*K3 + 32*K1**3*K3*K4 - 736*K1**3*K3 - 1088*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 3696*K1**2*K2 - 1200*K1**2*K3**2 - 96*K1**2*K3*K5 - 160*K1**2*K4**2 - 96*K1**2*K4*K6 - 3736*K1**2 + 64*K1*K2**3*K3 - 832*K1*K2**2*K3 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4464*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2040*K1*K3*K4 + 544*K1*K4*K5 + 120*K1*K5*K6 + 24*K1*K6*K7 - 64*K2**4 - 32*K2**3*K6 - 336*K2**2*K3**2 - 40*K2**2*K4**2 + 752*K2**2*K4 - 8*K2**2*K6**2 - 2722*K2**2 - 32*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 528*K2*K3*K5 - 32*K2*K4**2*K6 + 216*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 - 80*K3**2*K4**2 + 128*K3**2*K6 - 1896*K3**2 + 104*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 882*K4**2 - 252*K5**2 - 118*K6**2 - 20*K7**2 - 4*K8**2 + 2908
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
Contact