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Flat knot 6.1683

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,0,1,2,2,1,1,1,2,0,-1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1683']
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+57t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1683']
2-strand cable arrow polynomial of the knot is: -64K16 - 768K1*4K2*2 + 5152K1*4K2 - 6288K14 - 384K1*3K2*2K3 + 1024K13K2K3 + 32K1*3K3K4 - 2688K1*3K3 + 384K12K2*5 - 960K1*2K2*4 - 384K1*2K2*3K4 + 2688K12K2*3 + 992K1*2K2*2K4 - 8320K12K2*2 + 448K1*2K2K32 + 96K1*2K2K42 - 736K1*2K2K4 + 10288K1*2K2 - 1200K12K3*2 - 128K1*2K3K5 - 176K1*2K4*2 - 4404K1*2 + 672K1K23K3 - 1760K1K2*2K3 - 160K1K2*2K5 - 736K1K2K3K4 + 7624K1K2K3 + 1440K1K3K4 + 336K1K4K5 - 288K2*6 + 352K2*4K4 - 1248K24 - 32K2*3K6 - 416K22K3*2 - 112K2*2K4*2 + 1144K2*2K4 - 3454K22 + 360K2K3K5 + 24K2K4K6 - 1836K3*2 - 528K4*2 - 128K5*2 - 2K6**2 + 4142
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1683']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16346', 'vk6.16356', 'vk6.16389', 'vk6.16399', 'vk6.19179', 'vk6.19187', 'vk6.19472', 'vk6.19480', 'vk6.22776', 'vk6.22782', 'vk6.25978', 'vk6.25984', 'vk6.26368', 'vk6.26374', 'vk6.34631', 'vk6.34647', 'vk6.34717', 'vk6.34739', 'vk6.38077', 'vk6.38085', 'vk6.42353', 'vk6.42362', 'vk6.44563', 'vk6.44569', 'vk6.54613', 'vk6.54615', 'vk6.56521', 'vk6.56542', 'vk6.59144', 'vk6.59154', 'vk6.66257', 'vk6.66265']
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,-1,0,1,2,2,1,1,1,2,0,-1,0,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1683']

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+57t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1683']

2-strand cable arrow polynomial of the knot is: -64*K1**6 - 768*K1**4*K2**2 + 5152*K1**4*K2 - 6288*K1**4 - 384*K1**3*K2**2*K3 + 1024*K1**3*K2*K3 + 32*K1**3*K3*K4 - 2688*K1**3*K3 + 384*K1**2*K2**5 - 960*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2688*K1**2*K2**3 + 992*K1**2*K2**2*K4 - 8320*K1**2*K2**2 + 448*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 736*K1**2*K2*K4 + 10288*K1**2*K2 - 1200*K1**2*K3**2 - 128*K1**2*K3*K5 - 176*K1**2*K4**2 - 4404*K1**2 + 672*K1*K2**3*K3 - 1760*K1*K2**2*K3 - 160*K1*K2**2*K5 - 736*K1*K2*K3*K4 + 7624*K1*K2*K3 + 1440*K1*K3*K4 + 336*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 + 1144*K2**2*K4 - 3454*K2**2 + 360*K2*K3*K5 + 24*K2*K4*K6 - 1836*K3**2 - 528*K4**2 - 128*K5**2 - 2*K6**2 + 4142

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1683']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16346', 'vk6.16356', 'vk6.16389', 'vk6.16399', 'vk6.19179', 'vk6.19187', 'vk6.19472', 'vk6.19480', 'vk6.22776', 'vk6.22782', 'vk6.25978', 'vk6.25984', 'vk6.26368', 'vk6.26374', 'vk6.34631', 'vk6.34647', 'vk6.34717', 'vk6.34739', 'vk6.38077', 'vk6.38085', 'vk6.42353', 'vk6.42362', 'vk6.44563', 'vk6.44569', 'vk6.54613', 'vk6.54615', 'vk6.56521', 'vk6.56542', 'vk6.59144', 'vk6.59154', 'vk6.66257', 'vk6.66265']

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	O1O2O3U4O5U1U3U6O4O6U5U2
R3 orbit	{'O1O2O3U4O5U1U3U6O4O6U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4O5O6U5U1U3O4U6
Gauss code of K*	O1O2O3U4U3O5O6U1U6U2O4U5
Gauss code of -K*	O1O2O3U4O5U2U6U3O6O4U1U5
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 -2 1 1 -2 1 1],[ 2 0 2 1 1 1 3],[-1 -2 0 0 -2 0 0],[-1 -1 0 0 -1 0 0],[ 2 -1 2 1 0 2 2],[-1 -1 0 0 -2 0 -1],[-1 -3 0 0 -2 1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 1 0 0 -2 -3],[-1 -1 0 0 0 -2 -1],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -2 -2],[ 2 2 2 1 2 0 -1],[ 2 3 1 1 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,-1,0,0,2,3,0,0,2,1,0,1,1,2,2,1]
Phi over symmetry	[-2,-2,1,1,1,1,-1,0,1,2,2,1,1,1,2,0,-1,0,0,0,0]
Phi of -K	[-2,-2,1,1,1,1,-1,0,1,2,2,1,1,1,2,0,-1,0,0,0,0]
Phi of K*	[-1,-1,-1,-1,2,2,-1,0,0,1,2,0,0,1,0,0,1,1,2,2,-1]
Phi of -K*	[-2,-2,1,1,1,1,-1,1,2,2,2,1,1,2,3,0,0,0,0,-1,0]
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+29t^2
Outer characteristic polynomial	t^7+42t^5+57t^3+6t
Flat arrow polynomial	4K13 - 4K1*2 - 4K1K2 - K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-64K16 - 768K1*4K2*2 + 5152K1*4K2 - 6288K14 - 384K1*3K2*2K3 + 1024K13K2K3 + 32K1*3K3K4 - 2688K1*3K3 + 384K12K2*5 - 960K1*2K2*4 - 384K1*2K2*3K4 + 2688K12K2*3 + 992K1*2K2*2K4 - 8320K12K2*2 + 448K1*2K2K32 + 96K1*2K2K42 - 736K1*2K2K4 + 10288K1*2K2 - 1200K12K3*2 - 128K1*2K3K5 - 176K1*2K4*2 - 4404K1*2 + 672K1K23K3 - 1760K1K2*2K3 - 160K1K2*2K5 - 736K1K2K3K4 + 7624K1K2K3 + 1440K1K3K4 + 336K1K4K5 - 288K2*6 + 352K2*4K4 - 1248K24 - 32K2*3K6 - 416K22K3*2 - 112K2*2K4*2 + 1144K2*2K4 - 3454K22 + 360K2K3K5 + 24K2K4K6 - 1836K3*2 - 528K4*2 - 128K5*2 - 2K6**2 + 4142
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice	False

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