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

Flat knot 6.18

Min(phi) over symmetries of the knot is: [-5,-2,0,0,3,4,1,2,4,3,5,1,2,2,3,0,1,2,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.18']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 - 12*K1**2 - 12*K1*K2 - 2*K1*K3 - 2*K1*K4 - 4*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.18']
Outer characteristic polynomial of the knot is: t^7+145t^5+53t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.18']
2-strand cable arrow polynomial of the knot is: -2848*K1**4 + 1856*K1**3*K2*K3 + 32*K1**3*K3*K4 - 864*K1**3*K3 - 128*K1**2*K2**4 + 992*K1**2*K2**3 - 1088*K1**2*K2**2*K3**2 - 12064*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1856*K1**2*K2*K4 + 13792*K1**2*K2 - 1184*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 8264*K1**2 + 256*K1*K2**3*K3**3 + 4960*K1*K2**3*K3 + 800*K1*K2**2*K3*K4 - 2656*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 1408*K1*K2**2*K5 + 32*K1*K2**2*K6*K7 + 384*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 13112*K1*K2*K3 - 224*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 2016*K1*K3*K4 + 296*K1*K4*K5 + 48*K1*K5*K6 + 16*K1*K6*K7 - 96*K2**6 - 768*K2**4*K3**2 - 32*K2**4*K4**2 + 416*K2**4*K4 - 32*K2**4*K6**2 - 4512*K2**4 + 608*K2**3*K3*K5 + 160*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 160*K2**3*K6 - 128*K2**2*K3**4 - 3136*K2**2*K3**2 - 64*K2**2*K3*K7 - 800*K2**2*K4**2 - 32*K2**2*K4*K8 + 4248*K2**2*K4 - 288*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 5080*K2**2 + 1696*K2*K3*K5 + 472*K2*K4*K6 + 104*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 - 3212*K3**2 - 1070*K4**2 - 244*K5**2 - 64*K6**2 - 16*K7**2 - 4*K8**2 + 6664
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.18']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73274', 'vk6.73415', 'vk6.74013', 'vk6.74551', 'vk6.75182', 'vk6.75414', 'vk6.76027', 'vk6.76761', 'vk6.78139', 'vk6.78377', 'vk6.78990', 'vk6.79543', 'vk6.79972', 'vk6.80127', 'vk6.80508', 'vk6.80980', 'vk6.81880', 'vk6.82149', 'vk6.82176', 'vk6.82594', 'vk6.83579', 'vk6.83754', 'vk6.84040', 'vk6.84618', 'vk6.84945', 'vk6.85580', 'vk6.85701', 'vk6.85924', 'vk6.86731', 'vk6.87654', 'vk6.88939', 'vk6.89966']
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 O1O2O3O4O5O6U1U3U4U6U2U5
R3 orbit {'O1O2O3O4O5O6U1U3U4U6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U5U1U3U4U6
Gauss code of K* O1O2O3O4O5O6U1U5U2U3U6U4
Gauss code of -K* O1O2O3O4O5O6U3U1U4U5U2U6
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 0 -2 0 4 3],[ 5 0 4 1 2 5 3],[ 0 -4 0 -2 0 3 2],[ 2 -1 2 0 1 3 2],[ 0 -2 0 -1 0 2 1],[-4 -5 -3 -3 -2 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 4 3 0 0 -2 -5],[-4 0 0 -2 -3 -3 -5],[-3 0 0 -1 -2 -2 -3],[ 0 2 1 0 0 -1 -2],[ 0 3 2 0 0 -2 -4],[ 2 3 2 1 2 0 -1],[ 5 5 3 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,0,0,2,5,0,2,3,3,5,1,2,2,3,0,1,2,2,4,1]
Phi over symmetry [-5,-2,0,0,3,4,1,2,4,3,5,1,2,2,3,0,1,2,2,3,0]
Phi of -K [-5,-2,0,0,3,4,2,1,3,5,4,0,1,3,3,0,1,1,2,2,1]
Phi of K* [-4,-3,0,0,2,5,1,1,2,3,4,1,2,3,5,0,0,1,1,3,2]
Phi of -K* [-5,-2,0,0,3,4,1,2,4,3,5,1,2,2,3,0,1,2,2,3,0]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^3+t^2
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+91t^4+12t^2
Outer characteristic polynomial t^7+145t^5+53t^3+4t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 - 12*K1**2 - 12*K1*K2 - 2*K1*K3 - 2*K1*K4 - 4*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -2848*K1**4 + 1856*K1**3*K2*K3 + 32*K1**3*K3*K4 - 864*K1**3*K3 - 128*K1**2*K2**4 + 992*K1**2*K2**3 - 1088*K1**2*K2**2*K3**2 - 12064*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1856*K1**2*K2*K4 + 13792*K1**2*K2 - 1184*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 8264*K1**2 + 256*K1*K2**3*K3**3 + 4960*K1*K2**3*K3 + 800*K1*K2**2*K3*K4 - 2656*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 1408*K1*K2**2*K5 + 32*K1*K2**2*K6*K7 + 384*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 13112*K1*K2*K3 - 224*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 2016*K1*K3*K4 + 296*K1*K4*K5 + 48*K1*K5*K6 + 16*K1*K6*K7 - 96*K2**6 - 768*K2**4*K3**2 - 32*K2**4*K4**2 + 416*K2**4*K4 - 32*K2**4*K6**2 - 4512*K2**4 + 608*K2**3*K3*K5 + 160*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 160*K2**3*K6 - 128*K2**2*K3**4 - 3136*K2**2*K3**2 - 64*K2**2*K3*K7 - 800*K2**2*K4**2 - 32*K2**2*K4*K8 + 4248*K2**2*K4 - 288*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 5080*K2**2 + 1696*K2*K3*K5 + 472*K2*K4*K6 + 104*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 - 3212*K3**2 - 1070*K4**2 - 244*K5**2 - 64*K6**2 - 16*K7**2 - 4*K8**2 + 6664
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact