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Flat knot 6.36

Min(phi) over symmetries of the knot is: [-5,-1,0,1,2,3,1,4,2,5,3,2,1,2,2,0,1,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.36']
Arrow polynomial of the knot is: 16*K1**5 - 8*K1**3*K2 - 8*K1**3 - 2*K1**2 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.36']
Outer characteristic polynomial of the knot is: t^7+118t^5+161t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.36']
2-strand cable arrow polynomial of the knot is: -32*K1**4 + 192*K1**2*K2**3 - 928*K1**2*K2**2 + 872*K1**2*K2 - 820*K1**2 + 224*K1*K2**3*K3 + 936*K1*K2*K3 + 32*K1*K3*K4 + 16*K1*K4*K5 - 512*K2**10 + 512*K2**8*K4 - 768*K2**8 - 128*K2**6*K4**2 + 640*K2**6*K4 - 1728*K2**6 - 192*K2**4*K4**2 + 1216*K2**4*K4 - 72*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 304*K2**2*K3**2 - 112*K2**2*K4**2 + 528*K2**2*K4 - 32*K2**2*K5**2 + 8*K2**2 + 152*K2*K3*K5 + 8*K2*K4*K6 + 8*K2*K5*K7 - 312*K3**2 - 90*K4**2 - 36*K5**2 + 656
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.36']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70492', 'vk6.70494', 'vk6.70545', 'vk6.70549', 'vk6.70686', 'vk6.70694', 'vk6.70799', 'vk6.70803', 'vk6.70967', 'vk6.70971', 'vk6.71038', 'vk6.71045', 'vk6.71184', 'vk6.71188', 'vk6.71267', 'vk6.71269', 'vk6.74040', 'vk6.74043', 'vk6.74604', 'vk6.74611', 'vk6.76095', 'vk6.76108', 'vk6.79035', 'vk6.79041', 'vk6.81087', 'vk6.81091', 'vk6.81220', 'vk6.81225', 'vk6.87885', 'vk6.87889', 'vk6.88999', 'vk6.89009']
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 O1O2O3O4O5O6U1U4U5U6U2U3
R3 orbit {'O1O2O3O4O5O6U1U4U5U6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U5U1U2U3U6
Gauss code of K* O1O2O3O4O5O6U1U5U6U2U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U5U1U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 0 2 -1 1 3],[ 5 0 4 5 1 2 3],[ 0 -4 0 1 -2 0 2],[-2 -5 -1 0 -2 0 2],[ 1 -1 2 2 0 1 2],[-1 -2 0 0 -1 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 2 1 0 -1 -5],[-3 0 -2 -1 -2 -2 -3],[-2 2 0 0 -1 -2 -5],[-1 1 0 0 0 -1 -2],[ 0 2 1 0 0 -2 -4],[ 1 2 2 1 2 0 -1],[ 5 3 5 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,1,5,2,1,2,2,3,0,1,2,5,0,1,2,2,4,1]
Phi over symmetry [-5,-1,0,1,2,3,1,4,2,5,3,2,1,2,2,0,1,2,0,1,2]
Phi of -K [-5,-1,0,1,2,3,3,1,4,2,5,-1,1,1,2,1,1,1,1,1,-1]
Phi of K* [-3,-2,-1,0,1,5,-1,1,1,2,5,1,1,1,2,1,1,4,-1,1,3]
Phi of -K* [-5,-1,0,1,2,3,1,4,2,5,3,2,1,2,2,0,1,2,0,1,2]
Symmetry type of based matrix c
u-polynomial t^5-t^3-t^2
Normalized Jones-Krushkal polynomial 3z+7
Enhanced Jones-Krushkal polynomial -8w^5z+16w^4z-8w^3z+8w^3+3w^2z-w
Inner characteristic polynomial t^6+78t^4+26t^2
Outer characteristic polynomial t^7+118t^5+161t^3
Flat arrow polynomial 16*K1**5 - 8*K1**3*K2 - 8*K1**3 - 2*K1**2 + K2 + 2
2-strand cable arrow polynomial -32*K1**4 + 192*K1**2*K2**3 - 928*K1**2*K2**2 + 872*K1**2*K2 - 820*K1**2 + 224*K1*K2**3*K3 + 936*K1*K2*K3 + 32*K1*K3*K4 + 16*K1*K4*K5 - 512*K2**10 + 512*K2**8*K4 - 768*K2**8 - 128*K2**6*K4**2 + 640*K2**6*K4 - 1728*K2**6 - 192*K2**4*K4**2 + 1216*K2**4*K4 - 72*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 304*K2**2*K3**2 - 112*K2**2*K4**2 + 528*K2**2*K4 - 32*K2**2*K5**2 + 8*K2**2 + 152*K2*K3*K5 + 8*K2*K4*K6 + 8*K2*K5*K7 - 312*K3**2 - 90*K4**2 - 36*K5**2 + 656
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]]
If K is slice False
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