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Flat knot 6.705

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,3,3,0,2,2,2,1,-1,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.705']
Arrow polynomial of the knot is: -4K12 - 2K1K2 + K1 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+58t^5+153t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.705']
2-strand cable arrow polynomial of the knot is: 160K14K2 - 928K14 + 64K1*3K2K3 + 32K1*3K3K4 - 640K1*2K2*2 + 1776K1*2K2 - 352K12K3*2 - 96K1*2K4*2 - 1408K1*2 + 1640K1K2K3 + 432K1K3K4 + 80K1K4K5 - 16K24 - 32K2*2K3*2 - 8K2*2K4*2 + 80K2*2K4 - 1246K22 + 24K2K3K5 + 8K2K4K6 - 728K3*2 - 176K4*2 - 24K5*2 - 2K6**2 + 1358
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.705']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11014', 'vk6.11095', 'vk6.12184', 'vk6.12293', 'vk6.18187', 'vk6.18524', 'vk6.24643', 'vk6.25074', 'vk6.30587', 'vk6.30684', 'vk6.31857', 'vk6.31905', 'vk6.36781', 'vk6.37231', 'vk6.44024', 'vk6.44366', 'vk6.51825', 'vk6.51894', 'vk6.52697', 'vk6.52793', 'vk6.55994', 'vk6.56268', 'vk6.60530', 'vk6.60873', 'vk6.63508', 'vk6.63554', 'vk6.63990', 'vk6.64036', 'vk6.65658', 'vk6.65941', 'vk6.68706', 'vk6.68916', 'vk6.83163', 'vk6.83595', 'vk6.84129', 'vk6.84345', 'vk6.86470', 'vk6.86492', 'vk6.88725', 'vk6.88923']
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,-2,1,1,1,2,0,1,2,3,3,0,2,2,2,1,-1,0,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.705']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']

Outer characteristic polynomial of the knot is: t^7+58t^5+153t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.705']

2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 928*K1**4 + 64*K1**3*K2*K3 + 32*K1**3*K3*K4 - 640*K1**2*K2**2 + 1776*K1**2*K2 - 352*K1**2*K3**2 - 96*K1**2*K4**2 - 1408*K1**2 + 1640*K1*K2*K3 + 432*K1*K3*K4 + 80*K1*K4*K5 - 16*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 80*K2**2*K4 - 1246*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 728*K3**2 - 176*K4**2 - 24*K5**2 - 2*K6**2 + 1358

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.705']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11014', 'vk6.11095', 'vk6.12184', 'vk6.12293', 'vk6.18187', 'vk6.18524', 'vk6.24643', 'vk6.25074', 'vk6.30587', 'vk6.30684', 'vk6.31857', 'vk6.31905', 'vk6.36781', 'vk6.37231', 'vk6.44024', 'vk6.44366', 'vk6.51825', 'vk6.51894', 'vk6.52697', 'vk6.52793', 'vk6.55994', 'vk6.56268', 'vk6.60530', 'vk6.60873', 'vk6.63508', 'vk6.63554', 'vk6.63990', 'vk6.64036', 'vk6.65658', 'vk6.65941', 'vk6.68706', 'vk6.68916', 'vk6.83163', 'vk6.83595', 'vk6.84129', 'vk6.84345', 'vk6.86470', 'vk6.86492', 'vk6.88725', 'vk6.88923']

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	O1O2O3O4U5O6U3U1O5U2U4U6
R3 orbit	{'O1O2O3O4U5O6U3U1O5U2U4U6', 'O1O2O3O4U5U2O5U6U1U3O6U4'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4U5U1U3O6U4U2O5U6
Gauss code of K*	O1O2O3U4U1U5U2O6U3O5O4U6
Gauss code of -K*	O1O2O3U4O5O6U1O4U2U6U3U5
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 -2 -1 -1 2 -1 3],[ 2 0 0 0 2 1 3],[ 1 0 0 1 2 0 2],[ 1 0 -1 0 0 1 1],[-2 -2 -2 0 0 -2 0],[ 1 -1 0 -1 2 0 3],[-3 -3 -2 -1 0 -3 0]]
Primitive based matrix	[[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -1 -2 -3 -3],[-2 0 0 0 -2 -2 -2],[ 1 1 0 0 -1 1 0],[ 1 2 2 1 0 0 0],[ 1 3 2 -1 0 0 -1],[ 2 3 2 0 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,1,1,1,2,0,1,2,3,3,0,2,2,2,1,-1,0,0,0,1]
Phi over symmetry	[-3,-2,1,1,1,2,0,1,2,3,3,0,2,2,2,1,-1,0,0,0,1]
Phi of -K	[-2,-1,-1,-1,2,3,0,1,1,2,2,0,1,1,1,-1,1,2,3,3,1]
Phi of K*	[-3,-2,1,1,1,2,1,1,2,3,2,1,1,3,2,0,-1,0,1,1,1]
Phi of -K*	[-2,-1,-1,-1,2,3,0,0,1,2,3,-1,1,0,1,0,2,2,2,3,0]
Symmetry type of based matrix	c
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	10z+21
Enhanced Jones-Krushkal polynomial	-2w^3z+12w^2z+21w
Inner characteristic polynomial	t^6+38t^4+91t^2
Outer characteristic polynomial	t^7+58t^5+153t^3
Flat arrow polynomial	-4K12 - 2K1K2 + K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	160K14K2 - 928K14 + 64K1*3K2K3 + 32K1*3K3K4 - 640K1*2K2*2 + 1776K1*2K2 - 352K12K3*2 - 96K1*2K4*2 - 1408K1*2 + 1640K1K2K3 + 432K1K3K4 + 80K1K4K5 - 16K24 - 32K2*2K3*2 - 8K2*2K4*2 + 80K2*2K4 - 1246K22 + 24K2K3K5 + 8K2K4K6 - 728K3*2 - 176K4*2 - 24K5*2 - 2K6**2 + 1358
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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