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Flat knot 6.151

Min(phi) over symmetries of the knot is: [-4,-2,-2,2,3,3,0,1,3,3,4,0,2,1,2,3,2,3,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.151']
Arrow polynomial of the knot is: -2K12 - 4K1K2 + 2K1 - 2K22 + K2 + 2K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+131t^5+162t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.151']
2-strand cable arrow polynomial of the knot is: -96K13K3 + 128K12K2*2K4 - 1040K12K2*2 - 448K1*2K2K4 + 2616K1*2K2 - 320K12K3*2 - 176K1*2K4*2 - 3900K1*2 - 768K1K22K3 - 160K1K2*2K5 + 32K1K2K33 - 352K1K2K3K4 + 3976K1K2K3 - 96K1K2K4K5 + 2272K1K3K4 + 592K1K4K5 + 64K1K5K6 - 72K2*4 - 64K2*2K3*2 - 80K2*2K4*2 + 1248K2*2K4 - 3164K22 - 96K2K32K4 - 32K2K3K4K5 + 440K2K3K5 + 160K2K4K6 + 16K2K5K7 - 32K3*4 - 32K3*2K4*2 + 40K3*2K6 - 2232K32 + 32K3K4K7 - 8K44 + 8K4*2K8 - 1458K42 - 356K5*2 - 60K6*2 - 8K7*2 - 2K8**2 + 3426
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.151']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81584', 'vk6.81664', 'vk6.81674', 'vk6.81902', 'vk6.81908', 'vk6.82099', 'vk6.82270', 'vk6.82276', 'vk6.82344', 'vk6.82349', 'vk6.82619', 'vk6.82623', 'vk6.82863', 'vk6.82872', 'vk6.83159', 'vk6.83162', 'vk6.83374', 'vk6.83382', 'vk6.84156', 'vk6.84657', 'vk6.84968', 'vk6.84971', 'vk6.85970', 'vk6.85973', 'vk6.86173', 'vk6.86183', 'vk6.86424', 'vk6.88130', 'vk6.89047', 'vk6.89051', 'vk6.89718', 'vk6.90042']
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,-2,-2,2,3,3,0,1,3,3,4,0,2,1,2,3,2,3,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.151']

Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']

Outer characteristic polynomial of the knot is: t^7+131t^5+162t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.151']

2-strand cable arrow polynomial of the knot is: -96*K1**3*K3 + 128*K1**2*K2**2*K4 - 1040*K1**2*K2**2 - 448*K1**2*K2*K4 + 2616*K1**2*K2 - 320*K1**2*K3**2 - 176*K1**2*K4**2 - 3900*K1**2 - 768*K1*K2**2*K3 - 160*K1*K2**2*K5 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 3976*K1*K2*K3 - 96*K1*K2*K4*K5 + 2272*K1*K3*K4 + 592*K1*K4*K5 + 64*K1*K5*K6 - 72*K2**4 - 64*K2**2*K3**2 - 80*K2**2*K4**2 + 1248*K2**2*K4 - 3164*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 440*K2*K3*K5 + 160*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2232*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1458*K4**2 - 356*K5**2 - 60*K6**2 - 8*K7**2 - 2*K8**2 + 3426

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.151']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81584', 'vk6.81664', 'vk6.81674', 'vk6.81902', 'vk6.81908', 'vk6.82099', 'vk6.82270', 'vk6.82276', 'vk6.82344', 'vk6.82349', 'vk6.82619', 'vk6.82623', 'vk6.82863', 'vk6.82872', 'vk6.83159', 'vk6.83162', 'vk6.83374', 'vk6.83382', 'vk6.84156', 'vk6.84657', 'vk6.84968', 'vk6.84971', 'vk6.85970', 'vk6.85973', 'vk6.86173', 'vk6.86183', 'vk6.86424', 'vk6.88130', 'vk6.89047', 'vk6.89051', 'vk6.89718', 'vk6.90042']

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	O1O2O3O4O5U1O6U3U2U5U6U4
R3 orbit	{'O1O2O3O4O5U1O6U3U2U5U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U6U1U4U3O6U5
Gauss code of K*	O1O2O3O4O5U6U2U1U5U3O6U4
Gauss code of -K*	O1O2O3O4O5U2O6U3U1U5U4U6
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 -2 -2 3 2 3],[ 4 0 2 1 4 3 3],[ 2 -2 0 0 4 2 3],[ 2 -1 0 0 3 1 2],[-3 -4 -4 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix	[[ 0 3 3 2 -2 -2 -4],[-3 0 1 -1 -3 -4 -4],[-3 -1 0 -1 -2 -3 -3],[-2 1 1 0 -1 -2 -3],[ 2 3 2 1 0 0 -1],[ 2 4 3 2 0 0 -2],[ 4 4 3 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-2,2,2,4,-1,1,3,4,4,1,2,3,3,1,2,3,0,1,2]
Phi over symmetry	[-4,-2,-2,2,3,3,0,1,3,3,4,0,2,1,2,3,2,3,0,0,-1]
Phi of -K	[-4,-2,-2,2,3,3,0,1,3,3,4,0,2,1,2,3,2,3,0,0,-1]
Phi of K*	[-3,-3,-2,2,2,4,-1,0,2,3,4,0,1,2,3,2,3,3,0,0,1]
Phi of -K*	[-4,-2,-2,2,3,3,1,2,3,3,4,0,1,2,3,2,3,4,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial	3z^2+16z+21
Enhanced Jones-Krushkal polynomial	-2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w
Inner characteristic polynomial	t^6+85t^4+49t^2
Outer characteristic polynomial	t^7+131t^5+162t^3+7t
Flat arrow polynomial	-2K12 - 4K1K2 + 2K1 - 2K22 + K2 + 2K3 + K4 + 3
2-strand cable arrow polynomial	-96K13K3 + 128K12K2*2K4 - 1040K12K2*2 - 448K1*2K2K4 + 2616K1*2K2 - 320K12K3*2 - 176K1*2K4*2 - 3900K1*2 - 768K1K22K3 - 160K1K2*2K5 + 32K1K2K33 - 352K1K2K3K4 + 3976K1K2K3 - 96K1K2K4K5 + 2272K1K3K4 + 592K1K4K5 + 64K1K5K6 - 72K2*4 - 64K2*2K3*2 - 80K2*2K4*2 + 1248K2*2K4 - 3164K22 - 96K2K32K4 - 32K2K3K4K5 + 440K2K3K5 + 160K2K4K6 + 16K2K5K7 - 32K3*4 - 32K3*2K4*2 + 40K3*2K6 - 2232K32 + 32K3K4K7 - 8K44 + 8K4*2K8 - 1458K42 - 356K5*2 - 60K6*2 - 8K7*2 - 2K8**2 + 3426
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice	False

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