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Flat knot 6.328

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,-1,1,1,3,4,0,1,2,3,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.328']
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+70t^5+57t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.328']
2-strand cable arrow polynomial of the knot is: -416*K1**4 + 352*K1**3*K2*K3 + 96*K1**3*K3*K4 - 192*K1**3*K3 + 160*K1**2*K2**3 - 1232*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 2696*K1**2*K2 - 704*K1**2*K3**2 - 480*K1**2*K4**2 - 3556*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 + 128*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 + 4216*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2576*K1*K3*K4 + 736*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 248*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 352*K2**2*K3**2 + 32*K2**2*K4**3 - 560*K2**2*K4**2 + 1712*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3192*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 712*K2*K3*K5 - 32*K2*K4**2*K6 + 456*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 48*K3**2*K6 - 2200*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1568*K4**2 - 376*K5**2 - 112*K6**2 - 4*K7**2 - 2*K8**2 + 3448
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.328']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11199', 'vk6.11210', 'vk6.11216', 'vk6.12390', 'vk6.12400', 'vk6.12407', 'vk6.12417', 'vk6.14502', 'vk6.14522', 'vk6.15723', 'vk6.15742', 'vk6.16153', 'vk6.16163', 'vk6.30788', 'vk6.30806', 'vk6.30818', 'vk6.31990', 'vk6.32006', 'vk6.34077', 'vk6.34189', 'vk6.34470', 'vk6.34513', 'vk6.51928', 'vk6.51946', 'vk6.51960', 'vk6.54156', 'vk6.54166', 'vk6.54352', 'vk6.54372', 'vk6.63605', 'vk6.63616', 'vk6.63622']
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 O1O2O3O4O5U2U5O6U1U6U4U3
R3 orbit {'O1O2O3O4O5U2U5O6U1U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U6U5O6U1U4
Gauss code of K* O1O2O3O4U1U5U4U3U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U2U1U6U4
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 -3 2 2 1 1],[ 3 0 -1 4 3 1 1],[ 3 1 0 3 2 1 0],[-2 -4 -3 0 0 0 0],[-2 -3 -2 0 0 0 0],[-1 -1 -1 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -3 -3],[-2 0 0 0 0 -2 -3],[-2 0 0 0 0 -3 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 3 2 3 0 1 0 1],[ 3 3 4 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,3,3,0,0,0,2,3,0,0,3,4,0,0,1,1,1,-1]
Phi over symmetry [-3,-3,1,1,2,2,-1,1,1,3,4,0,1,2,3,0,0,0,0,0,0]
Phi of -K [-3,-3,1,1,2,2,-1,3,4,2,3,3,3,1,2,0,1,1,1,1,0]
Phi of K* [-2,-2,-1,-1,3,3,0,1,1,1,2,1,1,2,3,0,3,3,3,4,-1]
Phi of -K* [-3,-3,1,1,2,2,-1,1,1,3,4,0,1,2,3,0,0,0,0,0,0]
Symmetry type of based matrix c
u-polynomial 2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+16z+21
Enhanced Jones-Krushkal polynomial 3w^3z^2-8w^3z+24w^2z+21w
Inner characteristic polynomial t^6+42t^4+17t^2
Outer characteristic polynomial t^7+70t^5+57t^3+7t
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 -416*K1**4 + 352*K1**3*K2*K3 + 96*K1**3*K3*K4 - 192*K1**3*K3 + 160*K1**2*K2**3 - 1232*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 2696*K1**2*K2 - 704*K1**2*K3**2 - 480*K1**2*K4**2 - 3556*K1**2 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 + 128*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 + 4216*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2576*K1*K3*K4 + 736*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 248*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 352*K2**2*K3**2 + 32*K2**2*K4**3 - 560*K2**2*K4**2 + 1712*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3192*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 712*K2*K3*K5 - 32*K2*K4**2*K6 + 456*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 48*K3**2*K6 - 2200*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1568*K4**2 - 376*K5**2 - 112*K6**2 - 4*K7**2 - 2*K8**2 + 3448
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
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