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Flat knot 6.110

Min(phi) over symmetries of the knot is: [-3,-3,-1,1,3,3,-1,1,1,3,4,1,1,2,3,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.110']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.110', '6.328', '6.334', '6.842']
Outer characteristic polynomial of the knot is: t^7+111t^5+76t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.110']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 224*K1**4*K2 - 640*K1**4 + 384*K1**3*K2*K3 - 800*K1**3*K3 + 384*K1**2*K2**3 - 1904*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 3816*K1**2*K2 - 352*K1**2*K3**2 - 64*K1**2*K3*K5 - 320*K1**2*K4**2 - 32*K1**2*K4*K6 - 3596*K1**2 + 288*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 416*K1*K2*K3*K4 + 4120*K1*K2*K3 - 192*K1*K2*K4*K5 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 1856*K1*K3*K4 + 512*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 472*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 576*K2**2*K3**2 + 32*K2**2*K4**3 - 752*K2**2*K4**2 + 1480*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 2528*K2**2 - 96*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 648*K2*K3*K5 - 32*K2*K4**2*K6 + 416*K2*K4*K6 + 72*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 56*K3**2*K6 - 1680*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1080*K4**2 - 256*K5**2 - 48*K6**2 - 4*K7**2 - 2*K8**2 + 2888
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.110']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11505', 'vk6.11824', 'vk6.12839', 'vk6.13160', 'vk6.20279', 'vk6.21608', 'vk6.27551', 'vk6.29115', 'vk6.31266', 'vk6.31633', 'vk6.32404', 'vk6.32837', 'vk6.38954', 'vk6.41197', 'vk6.45731', 'vk6.47428', 'vk6.52264', 'vk6.52511', 'vk6.53085', 'vk6.53419', 'vk6.57116', 'vk6.58302', 'vk6.61706', 'vk6.62852', 'vk6.63787', 'vk6.63905', 'vk6.64219', 'vk6.64429', 'vk6.66743', 'vk6.67623', 'vk6.69399', 'vk6.70125']
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 O1O2O3O4O5O6U3U2U5U6U1U4
R3 orbit {'O1O2O3O4O5O6U3U2U5U6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U6U1U2U5U4
Gauss code of K* O1O2O3O4O5O6U5U2U1U6U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U1U6U5U2
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 -1 -3 -3 3 1 3],[ 1 0 -2 -2 3 1 3],[ 3 2 0 0 4 2 3],[ 3 2 0 0 3 1 2],[-3 -3 -4 -3 0 -1 1],[-1 -1 -2 -1 1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 1 -1 -3 -3],[-3 0 1 -1 -3 -3 -4],[-3 -1 0 -1 -3 -2 -3],[-1 1 1 0 -1 -1 -2],[ 1 3 3 1 0 -2 -2],[ 3 3 2 1 2 0 0],[ 3 4 3 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,3,3,-1,1,3,3,4,1,3,2,3,1,1,2,2,2,0]
Phi over symmetry [-3,-3,-1,1,3,3,-1,1,1,3,4,1,1,2,3,1,2,3,0,0,0]
Phi of -K [-3,-3,-1,1,3,3,0,0,2,2,3,0,3,3,4,1,1,1,1,1,-1]
Phi of K* [-3,-3,-1,1,3,3,-1,1,1,3,4,1,1,2,3,1,2,3,0,0,0]
Phi of -K* [-3,-3,-1,1,3,3,0,2,1,2,3,2,2,3,4,1,3,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+73t^4+28t^2
Outer characteristic polynomial t^7+111t^5+76t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -256*K1**4*K2**2 + 224*K1**4*K2 - 640*K1**4 + 384*K1**3*K2*K3 - 800*K1**3*K3 + 384*K1**2*K2**3 - 1904*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 1024*K1**2*K2*K4 + 3816*K1**2*K2 - 352*K1**2*K3**2 - 64*K1**2*K3*K5 - 320*K1**2*K4**2 - 32*K1**2*K4*K6 - 3596*K1**2 + 288*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 64*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 416*K1*K2*K3*K4 + 4120*K1*K2*K3 - 192*K1*K2*K4*K5 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 1856*K1*K3*K4 + 512*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 472*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 576*K2**2*K3**2 + 32*K2**2*K4**3 - 752*K2**2*K4**2 + 1480*K2**2*K4 - 144*K2**2*K5**2 - 8*K2**2*K6**2 - 2528*K2**2 - 96*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 648*K2*K3*K5 - 32*K2*K4**2*K6 + 416*K2*K4*K6 + 72*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 56*K3**2*K6 - 1680*K3**2 + 48*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1080*K4**2 - 256*K5**2 - 48*K6**2 - 4*K7**2 - 2*K8**2 + 2888
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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