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Flat knot 6.510

Min(phi) over symmetries of the knot is: [-4,-1,-1,2,2,2,0,1,2,3,5,0,0,0,1,1,2,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.510']
Arrow polynomial of the knot is: -4K1K2 - 2K1K3 + 2K1 + K2 + 2K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.108', '6.157', '6.283', '6.399', '6.445', '6.510']
Outer characteristic polynomial of the knot is: t^7+99t^5+87t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.510']
2-strand cable arrow polynomial of the knot is: -384K14 + 416K1*3K2K3 + 32K1*3K3K4 - 768K1*3K3 - 256K12K2*2K3*2 - 1184K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 320K12K2K4 + 3800K1*2K2 - 736K12K3*2 - 112K1*2K4*2 - 4160K1*2 + 416K1K23K3 + 160K1K2*2K3K4 - 896K1K22K3 - 32K1K2*2K5 - 288K1K2K3K4 + 4400K1K2K3 - 32K1K32K5 + 1392K1K3K4 + 272K1K4K5 + 64K1K5K6 - 224K2*4 - 512K2*2K3*2 - 48K2*2K4*2 + 952K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3168K2*2 + 472K2K3K5 + 88K2K4K6 + 32K2K5K7 + 8K2K6K8 + 48K3*2K6 - 1944K32 - 858K4*2 - 268K5*2 - 88K6*2 - 12K7*2 - 2K8**2 + 3346
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.510']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4667', 'vk6.4956', 'vk6.6129', 'vk6.6618', 'vk6.8142', 'vk6.8546', 'vk6.9520', 'vk6.9877', 'vk6.20365', 'vk6.21706', 'vk6.27669', 'vk6.29213', 'vk6.39109', 'vk6.41363', 'vk6.45861', 'vk6.47522', 'vk6.48699', 'vk6.48904', 'vk6.49463', 'vk6.49684', 'vk6.50723', 'vk6.50924', 'vk6.51202', 'vk6.51405', 'vk6.57234', 'vk6.58459', 'vk6.61852', 'vk6.62987', 'vk6.66853', 'vk6.67721', 'vk6.69485', 'vk6.70207']
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: [-4,-1,-1,2,2,2,0,1,2,3,5,0,0,0,1,1,2,2,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.510']

Arrow polynomial of the knot is: -4*K1*K2 - 2*K1*K3 + 2*K1 + K2 + 2*K3 + K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.108', '6.157', '6.283', '6.399', '6.445', '6.510']

Outer characteristic polynomial of the knot is: t^7+99t^5+87t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.510']

2-strand cable arrow polynomial of the knot is: -384*K1**4 + 416*K1**3*K2*K3 + 32*K1**3*K3*K4 - 768*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1184*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 3800*K1**2*K2 - 736*K1**2*K3**2 - 112*K1**2*K4**2 - 4160*K1**2 + 416*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 4400*K1*K2*K3 - 32*K1*K3**2*K5 + 1392*K1*K3*K4 + 272*K1*K4*K5 + 64*K1*K5*K6 - 224*K2**4 - 512*K2**2*K3**2 - 48*K2**2*K4**2 + 952*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3168*K2**2 + 472*K2*K3*K5 + 88*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 48*K3**2*K6 - 1944*K3**2 - 858*K4**2 - 268*K5**2 - 88*K6**2 - 12*K7**2 - 2*K8**2 + 3346

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.510']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4667', 'vk6.4956', 'vk6.6129', 'vk6.6618', 'vk6.8142', 'vk6.8546', 'vk6.9520', 'vk6.9877', 'vk6.20365', 'vk6.21706', 'vk6.27669', 'vk6.29213', 'vk6.39109', 'vk6.41363', 'vk6.45861', 'vk6.47522', 'vk6.48699', 'vk6.48904', 'vk6.49463', 'vk6.49684', 'vk6.50723', 'vk6.50924', 'vk6.51202', 'vk6.51405', 'vk6.57234', 'vk6.58459', 'vk6.61852', 'vk6.62987', 'vk6.66853', 'vk6.67721', 'vk6.69485', 'vk6.70207']

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	O1O2O3O4O5U1U6U5O6U2U4U3
R3 orbit	{'O1O2O3O4O5U1U6U5O6U2U4U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U2U4O6U1U6U5
Gauss code of K*	O1O2O3U2O4O5O6U1U4U6U5U3
Gauss code of -K*	O1O2O3U4O5O4O6U5U2U1U3U6
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 -4 -1 2 2 2 -1],[ 4 0 2 4 3 1 3],[ 1 -2 0 2 1 1 0],[-2 -4 -2 0 0 1 -3],[-2 -3 -1 0 0 1 -3],[-2 -1 -1 -1 -1 0 -2],[ 1 -3 0 3 3 2 0]]
Primitive based matrix	[[ 0 2 2 2 -1 -1 -4],[-2 0 1 0 -1 -3 -3],[-2 -1 0 -1 -1 -2 -1],[-2 0 1 0 -2 -3 -4],[ 1 1 1 2 0 0 -2],[ 1 3 2 3 0 0 -3],[ 4 3 1 4 2 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-2,1,1,4,-1,0,1,3,3,1,1,2,1,2,3,4,0,2,3]
Phi over symmetry	[-4,-1,-1,2,2,2,0,1,2,3,5,0,0,0,1,1,2,2,0,-1,-1]
Phi of -K	[-4,-1,-1,2,2,2,0,1,2,3,5,0,0,0,1,1,2,2,0,-1,-1]
Phi of K*	[-2,-2,-2,1,1,4,-1,-1,1,2,5,0,0,1,2,0,2,3,0,0,1]
Phi of -K*	[-4,-1,-1,2,2,2,2,3,1,3,4,0,1,1,2,2,3,3,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-3t^2+2t
Normalized Jones-Krushkal polynomial	5z^2+22z+25
Enhanced Jones-Krushkal polynomial	5w^3z^2+22w^2z+25w
Inner characteristic polynomial	t^6+69t^4+40t^2+1
Outer characteristic polynomial	t^7+99t^5+87t^3+5t
Flat arrow polynomial	-4K1K2 - 2K1K3 + 2K1 + K2 + 2K3 + K4 + 1
2-strand cable arrow polynomial	-384K14 + 416K1*3K2K3 + 32K1*3K3K4 - 768K1*3K3 - 256K12K2*2K3*2 - 1184K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 320K12K2K4 + 3800K1*2K2 - 736K12K3*2 - 112K1*2K4*2 - 4160K1*2 + 416K1K23K3 + 160K1K2*2K3K4 - 896K1K22K3 - 32K1K2*2K5 - 288K1K2K3K4 + 4400K1K2K3 - 32K1K32K5 + 1392K1K3K4 + 272K1K4K5 + 64K1K5K6 - 224K2*4 - 512K2*2K3*2 - 48K2*2K4*2 + 952K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3168K2*2 + 472K2K3K5 + 88K2K4K6 + 32K2K5K7 + 8K2K6K8 + 48K3*2K6 - 1944K32 - 858K4*2 - 268K5*2 - 88K6*2 - 12K7*2 - 2K8**2 + 3346
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice	False

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