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Flat knot 6.146

Min(phi) over symmetries of the knot is: [-4,-3,0,2,2,3,0,2,2,4,4,2,1,3,3,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.146']
Arrow polynomial of the knot is: -8K12 - 4K1K2 + 2K1 - 2K22 + 4K2 + 2*K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.146']
Outer characteristic polynomial of the knot is: t^7+117t^5+97t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.146']
2-strand cable arrow polynomial of the knot is: 192K14K2 - 1136K14 + 128K1*3K2K3 + 32K1*2K2*3 - 1120K1*2K2*2 + 3296K1*2K2 - 736K12K3*2 - 128K1*2K4*2 - 3088K1*2 + 64K1K2K3*3 + 2488K1K2K3 + 32K1K3*3K4 + 1152K1K3K4 + 272K1K4K5 + 16K1K5K6 - 96K2*4 - 192K2*2K3*2 - 16K2*2K4*2 + 408K2*2K4 - 2416K22 + 272K2K3K5 + 16K2K4K6 - 208K3*4 - 96K3*2K4*2 + 104K3*2K6 - 1176K32 + 40K3K4K7 - 8K44 + 8K4*2K8 - 656K42 - 208K5*2 - 24K6*2 - 2K8**2 + 2784
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.146']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73286', 'vk6.73314', 'vk6.73427', 'vk6.73455', 'vk6.74081', 'vk6.74104', 'vk6.74650', 'vk6.74675', 'vk6.75427', 'vk6.75455', 'vk6.76115', 'vk6.76138', 'vk6.78167', 'vk6.78187', 'vk6.78397', 'vk6.78417', 'vk6.79083', 'vk6.79106', 'vk6.79988', 'vk6.80008', 'vk6.80139', 'vk6.80159', 'vk6.80587', 'vk6.80610', 'vk6.83796', 'vk6.83799', 'vk6.85111', 'vk6.85117', 'vk6.86597', 'vk6.86611', 'vk6.87382', 'vk6.87394']
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,-3,0,2,2,3,0,2,2,4,4,2,1,3,3,0,1,1,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.146']

Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + 4*K2 + 2*K3 + K4 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.146']

Outer characteristic polynomial of the knot is: t^7+117t^5+97t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.146']

2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 1136*K1**4 + 128*K1**3*K2*K3 + 32*K1**2*K2**3 - 1120*K1**2*K2**2 + 3296*K1**2*K2 - 736*K1**2*K3**2 - 128*K1**2*K4**2 - 3088*K1**2 + 64*K1*K2*K3**3 + 2488*K1*K2*K3 + 32*K1*K3**3*K4 + 1152*K1*K3*K4 + 272*K1*K4*K5 + 16*K1*K5*K6 - 96*K2**4 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 408*K2**2*K4 - 2416*K2**2 + 272*K2*K3*K5 + 16*K2*K4*K6 - 208*K3**4 - 96*K3**2*K4**2 + 104*K3**2*K6 - 1176*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 656*K4**2 - 208*K5**2 - 24*K6**2 - 2*K8**2 + 2784

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.146']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73286', 'vk6.73314', 'vk6.73427', 'vk6.73455', 'vk6.74081', 'vk6.74104', 'vk6.74650', 'vk6.74675', 'vk6.75427', 'vk6.75455', 'vk6.76115', 'vk6.76138', 'vk6.78167', 'vk6.78187', 'vk6.78397', 'vk6.78417', 'vk6.79083', 'vk6.79106', 'vk6.79988', 'vk6.80008', 'vk6.80139', 'vk6.80159', 'vk6.80587', 'vk6.80610', 'vk6.83796', 'vk6.83799', 'vk6.85111', 'vk6.85117', 'vk6.86597', 'vk6.86611', 'vk6.87382', 'vk6.87394']

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	O1O2O3O4O5U1O6U2U4U6U5U3
R3 orbit	{'O1O2O3O4O5U1O6U2U4U6U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U1U6U2U4O6U5
Gauss code of K*	O1O2O3O4O5U6U1U5U2U4O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U2U4U1U5U6
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -4 -3 2 0 3 2],[ 4 0 1 4 2 3 2],[ 3 -1 0 4 1 3 2],[-2 -4 -4 0 -2 1 1],[ 0 -2 -1 2 0 2 1],[-3 -3 -3 -1 -2 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 2 2 0 -3 -4],[-3 0 0 -1 -2 -3 -3],[-2 0 0 -1 -1 -2 -2],[-2 1 1 0 -2 -4 -4],[ 0 2 1 2 0 -1 -2],[ 3 3 2 4 1 0 -1],[ 4 3 2 4 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-2,0,3,4,0,1,2,3,3,1,1,2,2,2,4,4,1,2,1]
Phi over symmetry	[-4,-3,0,2,2,3,0,2,2,4,4,2,1,3,3,0,1,1,-1,0,1]
Phi of -K	[-4,-3,0,2,2,3,0,2,2,4,4,2,1,3,3,0,1,1,-1,0,1]
Phi of K*	[-3,-2,-2,0,3,4,0,1,1,3,4,1,0,1,2,1,3,4,2,2,0]
Phi of -K*	[-4,-3,0,2,2,3,1,2,2,4,3,1,2,4,3,1,2,2,-1,0,1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	14z+29
Enhanced Jones-Krushkal polynomial	14w^2z+29w
Inner characteristic polynomial	t^6+75t^4+19t^2
Outer characteristic polynomial	t^7+117t^5+97t^3
Flat arrow polynomial	-8K12 - 4K1K2 + 2K1 - 2K22 + 4K2 + 2*K3 + K4 + 6
2-strand cable arrow polynomial	192K14K2 - 1136K14 + 128K1*3K2K3 + 32K1*2K2*3 - 1120K1*2K2*2 + 3296K1*2K2 - 736K12K3*2 - 128K1*2K4*2 - 3088K1*2 + 64K1K2K3*3 + 2488K1K2K3 + 32K1K3*3K4 + 1152K1K3K4 + 272K1K4K5 + 16K1K5K6 - 96K2*4 - 192K2*2K3*2 - 16K2*2K4*2 + 408K2*2K4 - 2416K22 + 272K2K3K5 + 16K2K4K6 - 208K3*4 - 96K3*2K4*2 + 104K3*2K6 - 1176K32 + 40K3K4K7 - 8K44 + 8K4*2K8 - 656K42 - 208K5*2 - 24K6*2 - 2K8**2 + 2784
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}], [{2, 6}, {5}, {3, 4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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