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Flat knot 6.1161

Min(phi) over symmetries of the knot is: [-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,1,0,0,-1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1161', '7.24433']
Arrow polynomial of the knot is: 4K12K2 - 4K12 - 2K1K2 - 4K1K3 + K1 + 2K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+31t^5+56t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1161', '6.1888']
2-strand cable arrow polynomial of the knot is: -2624K14 + 1408K1*3K2K3 + 128K1*3K3K4 - 672K1*3K3 + 384K12K2*2K4 - 5392K12K2*2 - 1216K1*2K2K4 + 6144K1*2K2 - 1280K12K3*2 - 32K1*2K3K5 - 352K1*2K4*2 - 2124K1*2 + 1440K1K23K3 + 320K1K2*2K3K4 - 960K1K22K3 + 32K1K2*2K4K5 - 832K1K22K5 - 512K1K2K3K4 - 32K1K2K3K6 + 6208K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 + 1760K1K3K4 + 392K1K4K5 + 24K1K5K6 + 8K1K6K7 - 32K2*4K4*2 + 480K2*4K4 - 1792K24 + 64K2*3K4K6 - 224K2*3K6 - 1280K22K3*2 - 552K2*2K4*2 - 32K2*2K4K8 + 1824K2*2K4 - 16K22K5*2 - 16K2*2K6*2 - 1822K2*2 - 96K2K32K4 + 920K2K3K5 + 384K2K4K6 + 16K2K5K7 + 16K2K6K8 + 24K32K6 - 1464K32 - 648K4*2 - 160K5*2 - 58K6*2 - 4K7*2 - 2K8**2 + 2408
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1161']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.52', 'vk6.105', 'vk6.202', 'vk6.253', 'vk6.380', 'vk6.793', 'vk6.800', 'vk6.1257', 'vk6.1348', 'vk6.1397', 'vk6.1537', 'vk6.2018', 'vk6.2405', 'vk6.2428', 'vk6.2677', 'vk6.2976', 'vk6.10454', 'vk6.10469', 'vk6.10673', 'vk6.10862', 'vk6.14653', 'vk6.16262', 'vk6.19170', 'vk6.25738', 'vk6.25902', 'vk6.30149', 'vk6.30356', 'vk6.30485', 'vk6.33415', 'vk6.33571', 'vk6.53783', 'vk6.63417']
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: [-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,1,0,0,-1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1161', '7.24433']

Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']

Outer characteristic polynomial of the knot is: t^7+31t^5+56t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1161', '6.1888']

2-strand cable arrow polynomial of the knot is: -2624*K1**4 + 1408*K1**3*K2*K3 + 128*K1**3*K3*K4 - 672*K1**3*K3 + 384*K1**2*K2**2*K4 - 5392*K1**2*K2**2 - 1216*K1**2*K2*K4 + 6144*K1**2*K2 - 1280*K1**2*K3**2 - 32*K1**2*K3*K5 - 352*K1**2*K4**2 - 2124*K1**2 + 1440*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 832*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6208*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1760*K1*K3*K4 + 392*K1*K4*K5 + 24*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4*K4**2 + 480*K2**4*K4 - 1792*K2**4 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 1280*K2**2*K3**2 - 552*K2**2*K4**2 - 32*K2**2*K4*K8 + 1824*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 1822*K2**2 - 96*K2*K3**2*K4 + 920*K2*K3*K5 + 384*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1464*K3**2 - 648*K4**2 - 160*K5**2 - 58*K6**2 - 4*K7**2 - 2*K8**2 + 2408

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1161']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.52', 'vk6.105', 'vk6.202', 'vk6.253', 'vk6.380', 'vk6.793', 'vk6.800', 'vk6.1257', 'vk6.1348', 'vk6.1397', 'vk6.1537', 'vk6.2018', 'vk6.2405', 'vk6.2428', 'vk6.2677', 'vk6.2976', 'vk6.10454', 'vk6.10469', 'vk6.10673', 'vk6.10862', 'vk6.14653', 'vk6.16262', 'vk6.19170', 'vk6.25738', 'vk6.25902', 'vk6.30149', 'vk6.30356', 'vk6.30485', 'vk6.33415', 'vk6.33571', 'vk6.53783', 'vk6.63417']

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	O1O2O3O4U1U4U3O5O6U5U6U2
R3 orbit	{'O1O2O3O4U1U4U3O5O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U5U6O5O6U2U1U4
Gauss code of K*	O1O2O3U4U5O4O5O6U1U6U3U2
Gauss code of -K*	O1O2O3U2U3O4O5O6U5U4U1U6
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 1 1 1 -1 1],[ 3 0 3 2 1 0 0],[-1 -3 0 0 0 -1 1],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[ 1 0 1 0 0 0 1],[-1 0 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -1 -3],[-1 0 1 0 0 -1 -3],[-1 -1 0 0 0 -1 0],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[ 1 1 1 0 0 0 0],[ 3 3 0 1 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,1,3,-1,0,0,1,3,0,0,1,0,0,0,1,0,2,0]
Phi over symmetry	[-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,1,0,0,-1,0,0,0]
Phi of -K	[-3,-1,1,1,1,1,2,1,2,3,4,1,2,2,1,0,0,-1,0,0,0]
Phi of K*	[-1,-1,-1,-1,1,3,-1,0,0,1,4,0,0,1,1,0,2,2,2,3,2]
Phi of -K*	[-3,-1,1,1,1,1,0,0,1,2,3,1,0,0,1,0,0,-1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	7z^2+24z+21
Enhanced Jones-Krushkal polynomial	-4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial	t^6+17t^4+24t^2
Outer characteristic polynomial	t^7+31t^5+56t^3+8t
Flat arrow polynomial	4K12K2 - 4K12 - 2K1K2 - 4K1K3 + K1 + 2K2 + K3 + K4 + 2
2-strand cable arrow polynomial	-2624K14 + 1408K1*3K2K3 + 128K1*3K3K4 - 672K1*3K3 + 384K12K2*2K4 - 5392K12K2*2 - 1216K1*2K2K4 + 6144K1*2K2 - 1280K12K3*2 - 32K1*2K3K5 - 352K1*2K4*2 - 2124K1*2 + 1440K1K23K3 + 320K1K2*2K3K4 - 960K1K22K3 + 32K1K2*2K4K5 - 832K1K22K5 - 512K1K2K3K4 - 32K1K2K3K6 + 6208K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 + 1760K1K3K4 + 392K1K4K5 + 24K1K5K6 + 8K1K6K7 - 32K2*4K4*2 + 480K2*4K4 - 1792K24 + 64K2*3K4K6 - 224K2*3K6 - 1280K22K3*2 - 552K2*2K4*2 - 32K2*2K4K8 + 1824K2*2K4 - 16K22K5*2 - 16K2*2K6*2 - 1822K2*2 - 96K2K32K4 + 920K2K3K5 + 384K2K4K6 + 16K2K5K7 + 16K2K6K8 + 24K32K6 - 1464K32 - 648K4*2 - 160K5*2 - 58K6*2 - 4K7*2 - 2K8**2 + 2408
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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