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Flat knot 6.16

Min(phi) over symmetries of the knot is: [-5,-2,-1,0,4,4,1,3,2,4,5,1,1,2,3,1,3,4,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.16']
Arrow polynomial of the knot is: -2*K1**2 + 4*K1*K2**2 - 6*K1*K2 - 2*K1*K4 + K1 + K2 + 3*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.16', '6.51', '6.55']
Outer characteristic polynomial of the knot is: t^7+163t^5+155t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.16']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2 - 864*K1**4 + 384*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 2384*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 1312*K1**2*K2*K4 + 4712*K1**2*K2 - 352*K1**2*K3**2 - 64*K1**2*K3*K5 - 224*K1**2*K4**2 - 4068*K1**2 + 288*K1*K2**3*K3 + 1120*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 64*K1*K2**2*K6*K7 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 5184*K1*K2*K3 - 96*K1*K2*K4*K5 - 96*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1944*K1*K3*K4 + 608*K1*K4*K5 + 8*K1*K5*K6 + 32*K1*K6*K7 - 72*K2**4 - 896*K2**2*K3**2 + 64*K2**2*K3*K4*K7 - 32*K2**2*K3*K7 - 32*K2**2*K4**4 + 64*K2**2*K4**3 + 32*K2**2*K4**2*K8 - 888*K2**2*K4**2 - 32*K2**2*K4*K8 + 1800*K2**2*K4 - 32*K2**2*K5**2 - 32*K2**2*K6**2 - 64*K2**2*K7**2 - 8*K2**2*K8**2 - 3378*K2**2 - 128*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 680*K2*K3*K5 - 32*K2*K4**2*K6 + 472*K2*K4*K6 + 144*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1852*K3**2 + 32*K3*K4*K7 - 16*K4**4 + 16*K4**2*K8 - 1214*K4**2 - 344*K5**2 - 54*K6**2 - 32*K7**2 - 4*K8**2 + 3360
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.16']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81843', 'vk6.81893', 'vk6.82065', 'vk6.82083', 'vk6.82561', 'vk6.82611', 'vk6.82776', 'vk6.82785', 'vk6.82836', 'vk6.82848', 'vk6.82946', 'vk6.83054', 'vk6.83063', 'vk6.83285', 'vk6.83322', 'vk6.83360', 'vk6.83521', 'vk6.84539', 'vk6.84640', 'vk6.84909', 'vk6.84953', 'vk6.85821', 'vk6.86096', 'vk6.86130', 'vk6.86152', 'vk6.86837', 'vk6.88452', 'vk6.88885', 'vk6.89032', 'vk6.89685', 'vk6.89933', 'vk6.90012']
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 O1O2O3O4O5O6U1U3U4U2U6U5
R3 orbit {'O1O2O3O4O5O6U1U3U4U2U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U1U5U3U4U6
Gauss code of K* O1O2O3O4O5O6U1U4U2U3U6U5
Gauss code of -K* O1O2O3O4O5O6U2U1U4U5U3U6
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 -1 -2 0 4 4],[ 5 0 3 1 2 5 4],[ 1 -3 0 -1 1 4 3],[ 2 -1 1 0 1 3 2],[ 0 -2 -1 -1 0 2 1],[-4 -5 -4 -3 -2 0 0],[-4 -4 -3 -2 -1 0 0]]
Primitive based matrix [[ 0 4 4 0 -1 -2 -5],[-4 0 0 -1 -3 -2 -4],[-4 0 0 -2 -4 -3 -5],[ 0 1 2 0 -1 -1 -2],[ 1 3 4 1 0 -1 -3],[ 2 2 3 1 1 0 -1],[ 5 4 5 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-4,0,1,2,5,0,1,3,2,4,2,4,3,5,1,1,2,1,3,1]
Phi over symmetry [-5,-2,-1,0,4,4,1,3,2,4,5,1,1,2,3,1,3,4,1,2,0]
Phi of -K [-5,-2,-1,0,4,4,2,1,3,4,5,0,1,3,4,0,1,2,2,3,0]
Phi of K* [-4,-4,0,1,2,5,0,2,1,3,4,3,2,4,5,0,1,3,0,1,2]
Phi of -K* [-5,-2,-1,0,4,4,1,3,2,4,5,1,1,2,3,1,3,4,1,2,0]
Symmetry type of based matrix c
u-polynomial t^5-2t^4+t^2+t
Normalized Jones-Krushkal polynomial 7z^2+26z+25
Enhanced Jones-Krushkal polynomial 7w^3z^2+26w^2z+25w
Inner characteristic polynomial t^6+101t^4+34t^2+1
Outer characteristic polynomial t^7+163t^5+155t^3+6t
Flat arrow polynomial -2*K1**2 + 4*K1*K2**2 - 6*K1*K2 - 2*K1*K4 + K1 + K2 + 3*K3 + 2
2-strand cable arrow polynomial 128*K1**4*K2 - 864*K1**4 + 384*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 2384*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 1312*K1**2*K2*K4 + 4712*K1**2*K2 - 352*K1**2*K3**2 - 64*K1**2*K3*K5 - 224*K1**2*K4**2 - 4068*K1**2 + 288*K1*K2**3*K3 + 1120*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 64*K1*K2**2*K6*K7 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 5184*K1*K2*K3 - 96*K1*K2*K4*K5 - 96*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1944*K1*K3*K4 + 608*K1*K4*K5 + 8*K1*K5*K6 + 32*K1*K6*K7 - 72*K2**4 - 896*K2**2*K3**2 + 64*K2**2*K3*K4*K7 - 32*K2**2*K3*K7 - 32*K2**2*K4**4 + 64*K2**2*K4**3 + 32*K2**2*K4**2*K8 - 888*K2**2*K4**2 - 32*K2**2*K4*K8 + 1800*K2**2*K4 - 32*K2**2*K5**2 - 32*K2**2*K6**2 - 64*K2**2*K7**2 - 8*K2**2*K8**2 - 3378*K2**2 - 128*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 680*K2*K3*K5 - 32*K2*K4**2*K6 + 472*K2*K4*K6 + 144*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1852*K3**2 + 32*K3*K4*K7 - 16*K4**4 + 16*K4**2*K8 - 1214*K4**2 - 344*K5**2 - 54*K6**2 - 32*K7**2 - 4*K8**2 + 3360
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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