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

Flat knot 6.1154

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,2,3,3,-1,0,0,0,1,0,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1154']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 6*K2 + K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.961', '6.1154']
Outer characteristic polynomial of the knot is: t^7+40t^5+72t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1154']
2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1088*K1**4*K2 - 3072*K1**4 + 768*K1**3*K2*K3 - 1056*K1**3*K3 - 256*K1**2*K2**4 + 896*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7120*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 672*K1**2*K2*K4 + 10368*K1**2*K2 - 1152*K1**2*K3**2 - 160*K1**2*K3*K5 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 6912*K1**2 + 1056*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 + 128*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 10056*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 - 64*K1*K3**2*K5 + 1720*K1*K3*K4 + 416*K1*K4*K5 + 80*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1632*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1472*K2**2*K3**2 - 32*K2**2*K3*K7 - 232*K2**2*K4**2 + 2176*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5434*K2**2 + 1464*K2*K3*K5 + 232*K2*K4*K6 + 40*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 3052*K3**2 + 8*K3*K4*K7 - 924*K4**2 - 380*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 5868
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1154']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4426', 'vk6.4521', 'vk6.5808', 'vk6.5935', 'vk6.7875', 'vk6.7984', 'vk6.9295', 'vk6.9414', 'vk6.10173', 'vk6.10244', 'vk6.10391', 'vk6.17881', 'vk6.17944', 'vk6.18279', 'vk6.18616', 'vk6.24388', 'vk6.25171', 'vk6.30060', 'vk6.30121', 'vk6.36889', 'vk6.37349', 'vk6.43815', 'vk6.44110', 'vk6.44435', 'vk6.48629', 'vk6.50534', 'vk6.50621', 'vk6.51141', 'vk6.51672', 'vk6.55832', 'vk6.56070', 'vk6.65496']
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 O1O2O3O4U1U3U5O6O5U4U6U2
R3 orbit {'O1O2O3O4U1U3U5O6O5U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U1O6O5U6U2U4
Gauss code of K* O1O2O3U4U3O5O4O6U1U6U2U5
Gauss code of -K* O1O2O3U4U2O4O5O6U3U5U1U6
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 1 0 1 1 0],[ 3 0 3 1 2 3 1],[-1 -3 0 -1 0 0 0],[ 0 -1 1 0 1 0 1],[-1 -2 0 -1 0 -1 0],[-1 -3 0 0 1 0 0],[ 0 -1 0 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 0 -3],[-1 -1 0 0 0 -1 -2],[-1 0 0 0 0 -1 -3],[ 0 0 0 0 0 -1 -1],[ 0 0 1 1 1 0 -1],[ 3 3 2 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,0,0,0,3,0,0,1,2,0,1,3,1,1,1]
Phi over symmetry [-3,0,0,1,1,1,1,1,2,3,3,-1,0,0,0,1,0,1,-1,0,0]
Phi of -K [-3,0,0,1,1,1,2,2,1,1,2,-1,0,1,0,1,1,1,0,0,-1]
Phi of K* [-1,-1,-1,0,0,3,-1,0,0,1,2,0,1,1,1,0,1,1,1,2,2]
Phi of -K* [-3,0,0,1,1,1,1,1,2,3,3,-1,0,0,0,1,0,1,-1,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+28t^4+40t^2+4
Outer characteristic polynomial t^7+40t^5+72t^3+10t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 6*K2 + K3 + K4 + 6
2-strand cable arrow polynomial -320*K1**4*K2**2 + 1088*K1**4*K2 - 3072*K1**4 + 768*K1**3*K2*K3 - 1056*K1**3*K3 - 256*K1**2*K2**4 + 896*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 7120*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 672*K1**2*K2*K4 + 10368*K1**2*K2 - 1152*K1**2*K3**2 - 160*K1**2*K3*K5 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 6912*K1**2 + 1056*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 + 128*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 10056*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 - 64*K1*K3**2*K5 + 1720*K1*K3*K4 + 416*K1*K4*K5 + 80*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1632*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1472*K2**2*K3**2 - 32*K2**2*K3*K7 - 232*K2**2*K4**2 + 2176*K2**2*K4 - 80*K2**2*K5**2 - 48*K2**2*K6**2 - 5434*K2**2 + 1464*K2*K3*K5 + 232*K2*K4*K6 + 40*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 3052*K3**2 + 8*K3*K4*K7 - 924*K4**2 - 380*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 5868
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
Contact