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Flat knot 6.1125

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,0,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1125']
Arrow polynomial of the knot is: 4K13 - 4K1*2 - 4K1K2 - K1 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928']
Outer characteristic polynomial of the knot is: t^7+42t^5+45t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1125']
2-strand cable arrow polynomial of the knot is: -64K16 - 768K1*4K2*2 + 5024K1*4K2 - 6640K14 - 384K1*3K2*2K3 + 1024K13K2K3 - 2336K1*3K3 + 384K12K2*5 - 960K1*2K2*4 - 384K1*2K2*3K4 + 2688K12K2*3 + 960K1*2K2*2K4 - 7968K12K2*2 + 448K1*2K2K32 + 96K1*2K2K42 - 736K1*2K2K4 + 10032K1*2K2 - 976K12K3*2 - 144K1*2K4*2 - 3660K1*2 + 672K1K23K3 - 1696K1K2*2K3 - 96K1K2*2K5 - 544K1K2K3K4 + 6544K1K2K3 + 1024K1K3K4 + 88K1K4K5 - 288K2*6 + 352K2*4K4 - 1248K24 - 32K2*3K6 - 416K22K3*2 - 112K2*2K4*2 + 936K2*2K4 - 3006K22 + 192K2K3K5 + 24K2K4K6 - 1388K3*2 - 288K4*2 - 16K5*2 - 2K6**2 + 3662
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1125']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16345', 'vk6.16354', 'vk6.16386', 'vk6.16396', 'vk6.19182', 'vk6.19193', 'vk6.19477', 'vk6.19486', 'vk6.22769', 'vk6.22779', 'vk6.25979', 'vk6.25994', 'vk6.26371', 'vk6.26379', 'vk6.34636', 'vk6.34639', 'vk6.34722', 'vk6.34726', 'vk6.38076', 'vk6.38087', 'vk6.42354', 'vk6.42357', 'vk6.44556', 'vk6.44575', 'vk6.54612', 'vk6.54621', 'vk6.56528', 'vk6.56539', 'vk6.59137', 'vk6.59159', 'vk6.66256', 'vk6.66259']
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

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,0,-1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1125']

Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928']

Outer characteristic polynomial of the knot is: t^7+42t^5+45t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1125']

2-strand cable arrow polynomial of the knot is: -64*K1**6 - 768*K1**4*K2**2 + 5024*K1**4*K2 - 6640*K1**4 - 384*K1**3*K2**2*K3 + 1024*K1**3*K2*K3 - 2336*K1**3*K3 + 384*K1**2*K2**5 - 960*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2688*K1**2*K2**3 + 960*K1**2*K2**2*K4 - 7968*K1**2*K2**2 + 448*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 736*K1**2*K2*K4 + 10032*K1**2*K2 - 976*K1**2*K3**2 - 144*K1**2*K4**2 - 3660*K1**2 + 672*K1*K2**3*K3 - 1696*K1*K2**2*K3 - 96*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 6544*K1*K2*K3 + 1024*K1*K3*K4 + 88*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1248*K2**4 - 32*K2**3*K6 - 416*K2**2*K3**2 - 112*K2**2*K4**2 + 936*K2**2*K4 - 3006*K2**2 + 192*K2*K3*K5 + 24*K2*K4*K6 - 1388*K3**2 - 288*K4**2 - 16*K5**2 - 2*K6**2 + 3662

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1125']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16345', 'vk6.16354', 'vk6.16386', 'vk6.16396', 'vk6.19182', 'vk6.19193', 'vk6.19477', 'vk6.19486', 'vk6.22769', 'vk6.22779', 'vk6.25979', 'vk6.25994', 'vk6.26371', 'vk6.26379', 'vk6.34636', 'vk6.34639', 'vk6.34722', 'vk6.34726', 'vk6.38076', 'vk6.38087', 'vk6.42354', 'vk6.42357', 'vk6.44556', 'vk6.44575', 'vk6.54612', 'vk6.54621', 'vk6.56528', 'vk6.56539', 'vk6.59137', 'vk6.59159', 'vk6.66256', 'vk6.66259']

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	O1O2O3O4U3U5O6U2U4O5U1U6
R3 orbit	{'O1O2O3O4U3U5O6U2U4O5U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U4O6U1U3O5U6U2
Gauss code of K*	O1O2U3O4O5U2O6O3U6U4U1U5
Gauss code of -K*	O1O2U3O4O5U1O3O6U4U6U5U2
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 -1 -1 2 -1 2],[ 1 0 0 -1 3 -1 2],[ 1 0 0 0 2 0 1],[ 1 1 0 0 1 0 1],[-2 -3 -2 -1 0 -2 0],[ 1 1 0 0 2 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 2 -1 -1 -1 -1],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -1 -2 -2 -3],[ 1 1 1 0 0 0 1],[ 1 1 2 0 0 0 0],[ 1 2 2 0 0 0 1],[ 1 2 3 -1 0 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,1,1,1,1,0,1,1,2,2,1,2,2,3,0,0,-1,0,0,-1]
Phi over symmetry	[-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,0,-1,0,0,0]
Phi of -K	[-1,-1,-1,-1,2,2,-1,0,0,1,1,0,1,0,1,0,1,2,2,2,0]
Phi of K*	[-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,0,-1,0,0,0]
Phi of -K*	[-1,-1,-1,-1,2,2,-1,-1,0,2,3,0,0,1,1,0,2,2,1,2,0]
Symmetry type of based matrix	c
u-polynomial	-2t^2+4t
Normalized Jones-Krushkal polynomial	6z^2+26z+29
Enhanced Jones-Krushkal polynomial	-2w^4z^2+8w^3z^2+26w^2z+29w
Inner characteristic polynomial	t^6+30t^4+23t^2+1
Outer characteristic polynomial	t^7+42t^5+45t^3+7t
Flat arrow polynomial	4K13 - 4K1*2 - 4K1K2 - K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-64K16 - 768K1*4K2*2 + 5024K1*4K2 - 6640K14 - 384K1*3K2*2K3 + 1024K13K2K3 - 2336K1*3K3 + 384K12K2*5 - 960K1*2K2*4 - 384K1*2K2*3K4 + 2688K12K2*3 + 960K1*2K2*2K4 - 7968K12K2*2 + 448K1*2K2K32 + 96K1*2K2K42 - 736K1*2K2K4 + 10032K1*2K2 - 976K12K3*2 - 144K1*2K4*2 - 3660K1*2 + 672K1K23K3 - 1696K1K2*2K3 - 96K1K2*2K5 - 544K1K2K3K4 + 6544K1K2K3 + 1024K1K3K4 + 88K1K4K5 - 288K2*6 + 352K2*4K4 - 1248K24 - 32K2*3K6 - 416K22K3*2 - 112K2*2K4*2 + 936K2*2K4 - 3006K22 + 192K2K3K5 + 24K2K4K6 - 1388K3*2 - 288K4*2 - 16K5*2 - 2K6**2 + 3662
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice	False

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