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

Flat knot 6.851

Min(phi) over symmetries of the knot is: [-3,-1,0,0,1,3,0,0,2,3,4,0,0,2,2,-1,1,0,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.851']
Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 4*K1*K3 - 5*K1 + 6*K2 + K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.851']
Outer characteristic polynomial of the knot is: t^7+56t^5+97t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.851']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 384*K1**4*K2 - 2176*K1**4 + 128*K1**3*K2**3*K3 + 1056*K1**3*K2*K3 - 704*K1**3*K3 - 384*K1**2*K2**4 + 608*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7728*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 10808*K1**2*K2 - 864*K1**2*K3**2 - 48*K1**2*K4**2 - 8072*K1**2 + 2080*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 1056*K1*K2**2*K5 + 224*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 11144*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1712*K1*K3*K4 + 448*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 96*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2736*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 64*K2**2*K3*K7 - 272*K2**2*K4**2 - 32*K2**2*K4*K8 + 3432*K2**2*K4 - 144*K2**2*K5**2 - 16*K2**2*K6**2 - 6322*K2**2 - 64*K2*K3**2*K4 + 1896*K2*K3*K5 + 280*K2*K4*K6 + 112*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 3636*K3**2 + 24*K3*K4*K7 - 1304*K4**2 - 508*K5**2 - 62*K6**2 - 32*K7**2 - 2*K8**2 + 6952
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.851']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11227', 'vk6.11308', 'vk6.12492', 'vk6.12605', 'vk6.18225', 'vk6.18560', 'vk6.24690', 'vk6.25107', 'vk6.30897', 'vk6.31022', 'vk6.32085', 'vk6.32206', 'vk6.36819', 'vk6.37280', 'vk6.44060', 'vk6.44399', 'vk6.51973', 'vk6.52070', 'vk6.52858', 'vk6.52907', 'vk6.56030', 'vk6.56304', 'vk6.60576', 'vk6.60914', 'vk6.63629', 'vk6.63676', 'vk6.64061', 'vk6.64108', 'vk6.65695', 'vk6.65989', 'vk6.68741', 'vk6.68949']
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 O1O2O3O4U1U3O5O6U5U4U2U6
R3 orbit {'O1O2O3O4U1U3O5O6U5U4U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U1U6O5O6U2U4
Gauss code of K* O1O2O3O4U5U3U6U2O5O6U1U4
Gauss code of -K* O1O2O3O4U1U4O5O6U3U5U2U6
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 1 -1 3],[ 3 0 3 1 2 0 2],[ 0 -3 0 -1 1 0 3],[ 0 -1 1 0 1 0 1],[-1 -2 -1 -1 0 0 2],[ 1 0 0 0 0 0 1],[-3 -2 -3 -1 -2 -1 0]]
Primitive based matrix [[ 0 3 1 0 0 -1 -3],[-3 0 -2 -1 -3 -1 -2],[-1 2 0 -1 -1 0 -2],[ 0 1 1 0 1 0 -1],[ 0 3 1 -1 0 0 -3],[ 1 1 0 0 0 0 0],[ 3 2 2 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,1,3,2,1,3,1,2,1,1,0,2,-1,0,1,0,3,0]
Phi over symmetry [-3,-1,0,0,1,3,0,0,2,3,4,0,0,2,2,-1,1,0,1,2,2]
Phi of -K [-3,-1,0,0,1,3,2,0,2,2,4,1,1,2,3,1,0,0,0,2,0]
Phi of K* [-3,-1,0,0,1,3,0,0,2,3,4,0,0,2,2,-1,1,0,1,2,2]
Phi of -K* [-3,-1,0,0,1,3,0,1,3,2,2,0,0,0,1,1,1,1,1,3,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+36t^4+29t^2
Outer characteristic polynomial t^7+56t^5+97t^3+10t
Flat arrow polynomial 12*K1**3 + 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 4*K1*K3 - 5*K1 + 6*K2 + K3 + K4 + 6
2-strand cable arrow polynomial -192*K1**4*K2**2 + 384*K1**4*K2 - 2176*K1**4 + 128*K1**3*K2**3*K3 + 1056*K1**3*K2*K3 - 704*K1**3*K3 - 384*K1**2*K2**4 + 608*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7728*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 10808*K1**2*K2 - 864*K1**2*K3**2 - 48*K1**2*K4**2 - 8072*K1**2 + 2080*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 1056*K1*K2**2*K5 + 224*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 11144*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1712*K1*K3*K4 + 448*K1*K4*K5 + 32*K1*K5*K6 + 8*K1*K6*K7 - 96*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2736*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 64*K2**2*K3*K7 - 272*K2**2*K4**2 - 32*K2**2*K4*K8 + 3432*K2**2*K4 - 144*K2**2*K5**2 - 16*K2**2*K6**2 - 6322*K2**2 - 64*K2*K3**2*K4 + 1896*K2*K3*K5 + 280*K2*K4*K6 + 112*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 3636*K3**2 + 24*K3*K4*K7 - 1304*K4**2 - 508*K5**2 - 62*K6**2 - 32*K7**2 - 2*K8**2 + 6952
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{6}, {3, 5}, {1, 4}, {2}]]
If K is slice False
Contact