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Flat knot 6.1179

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1179']
Arrow polynomial of the knot is: -6K1K2 + 3K1 - 4K2*2 + 3K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.881', '6.967', '6.1179']
Outer characteristic polynomial of the knot is: t^7+43t^5+46t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1179']
2-strand cable arrow polynomial of the knot is: -1392K14 + 736K1*3K2K3 + 256K1*3K3K4 - 480K1*3K3 - 1824K12K2*2 - 928K1*2K2K4 + 3880K1*2K2 - 2144K12K3*2 - 256K1*2K3K5 - 272K1*2K4*2 - 3900K1*2 + 128K1K23K3 - 352K1K2*2K3 - 64K1K2*2K5 + 128K1K2K33 + 64K1K2K3K42 - 256K1K2K3K4 - 64K1K2K3K6 + 5552K1K2K3 - 64K1K2K4K5 - 64K1K3*2K5 - 64K1K3K4K6 + 3456K1K3K4 + 536K1K4K5 + 64K1K5K6 + 16K1K6K7 - 64K24 - 224K2*2K3*2 - 120K2*2K4*2 + 688K2*2K4 - 2910K22 - 32K2K32K4 - 32K2K3K4K5 + 416K2K3K5 + 232K2K4K6 + 8K2K5K7 - 144K3*4 - 112K3*2K4*2 + 160K3*2K6 - 2596K32 + 184K3K4K7 - 16K44 + 32K4*2K8 - 1404K42 - 260K5*2 - 122K6*2 - 60K7*2 - 12K8**2 + 3694
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1179']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4648', 'vk6.4929', 'vk6.6088', 'vk6.6583', 'vk6.8109', 'vk6.8507', 'vk6.9495', 'vk6.9856', 'vk6.20625', 'vk6.22054', 'vk6.28111', 'vk6.29554', 'vk6.39527', 'vk6.41752', 'vk6.46138', 'vk6.47782', 'vk6.48688', 'vk6.48889', 'vk6.49438', 'vk6.49661', 'vk6.50702', 'vk6.50901', 'vk6.51189', 'vk6.51392', 'vk6.57509', 'vk6.58699', 'vk6.62205', 'vk6.63153', 'vk6.67019', 'vk6.67894', 'vk6.69648', 'vk6.70331']
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,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,0,1,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1179']

Arrow polynomial of the knot is: -6*K1*K2 + 3*K1 - 4*K2**2 + 3*K3 + 2*K4 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.881', '6.967', '6.1179']

Outer characteristic polynomial of the knot is: t^7+43t^5+46t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1179']

2-strand cable arrow polynomial of the knot is: -1392*K1**4 + 736*K1**3*K2*K3 + 256*K1**3*K3*K4 - 480*K1**3*K3 - 1824*K1**2*K2**2 - 928*K1**2*K2*K4 + 3880*K1**2*K2 - 2144*K1**2*K3**2 - 256*K1**2*K3*K5 - 272*K1**2*K4**2 - 3900*K1**2 + 128*K1*K2**3*K3 - 352*K1*K2**2*K3 - 64*K1*K2**2*K5 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5552*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 3456*K1*K3*K4 + 536*K1*K4*K5 + 64*K1*K5*K6 + 16*K1*K6*K7 - 64*K2**4 - 224*K2**2*K3**2 - 120*K2**2*K4**2 + 688*K2**2*K4 - 2910*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 416*K2*K3*K5 + 232*K2*K4*K6 + 8*K2*K5*K7 - 144*K3**4 - 112*K3**2*K4**2 + 160*K3**2*K6 - 2596*K3**2 + 184*K3*K4*K7 - 16*K4**4 + 32*K4**2*K8 - 1404*K4**2 - 260*K5**2 - 122*K6**2 - 60*K7**2 - 12*K8**2 + 3694

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1179']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4648', 'vk6.4929', 'vk6.6088', 'vk6.6583', 'vk6.8109', 'vk6.8507', 'vk6.9495', 'vk6.9856', 'vk6.20625', 'vk6.22054', 'vk6.28111', 'vk6.29554', 'vk6.39527', 'vk6.41752', 'vk6.46138', 'vk6.47782', 'vk6.48688', 'vk6.48889', 'vk6.49438', 'vk6.49661', 'vk6.50702', 'vk6.50901', 'vk6.51189', 'vk6.51392', 'vk6.57509', 'vk6.58699', 'vk6.62205', 'vk6.63153', 'vk6.67019', 'vk6.67894', 'vk6.69648', 'vk6.70331']

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	O1O2O3O4U1U5U2O6O5U6U4U3
R3 orbit	{'O1O2O3O4U1U5U2O6O5U6U4U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U1U5O6O5U3U6U4
Gauss code of K*	O1O2O3U4U2O4O5O6U1U3U6U5
Gauss code of -K*	O1O2O3U4U3O5O4O6U2U1U5U6
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 0 2 2 0 -1],[ 3 0 1 3 2 2 0],[ 0 -1 0 1 0 0 -1],[-2 -3 -1 0 0 -1 -1],[-2 -2 0 0 0 -1 -1],[ 0 -2 0 1 1 0 -1],[ 1 0 1 1 1 1 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 0 0 -1 -1 -2],[-2 0 0 -1 -1 -1 -3],[ 0 0 1 0 0 -1 -1],[ 0 1 1 0 0 -1 -2],[ 1 1 1 1 1 0 0],[ 3 2 3 1 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,0,0,1,1,2,1,1,1,3,0,1,1,1,2,0]
Phi over symmetry	[-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,0,1,1,1,0]
Phi of -K	[-3,-1,0,0,2,2,2,1,2,2,3,0,0,2,2,0,1,1,1,2,0]
Phi of K*	[-2,-2,0,0,1,3,0,1,1,2,2,1,2,2,3,0,0,1,0,2,2]
Phi of -K*	[-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,0,1,1,1,0]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	5z^2+24z+29
Enhanced Jones-Krushkal polynomial	5w^3z^2+24w^2z+29w
Inner characteristic polynomial	t^6+25t^4+18t^2+1
Outer characteristic polynomial	t^7+43t^5+46t^3+6t
Flat arrow polynomial	-6K1K2 + 3K1 - 4K2*2 + 3K3 + 2*K4 + 3
2-strand cable arrow polynomial	-1392K14 + 736K1*3K2K3 + 256K1*3K3K4 - 480K1*3K3 - 1824K12K2*2 - 928K1*2K2K4 + 3880K1*2K2 - 2144K12K3*2 - 256K1*2K3K5 - 272K1*2K4*2 - 3900K1*2 + 128K1K23K3 - 352K1K2*2K3 - 64K1K2*2K5 + 128K1K2K33 + 64K1K2K3K42 - 256K1K2K3K4 - 64K1K2K3K6 + 5552K1K2K3 - 64K1K2K4K5 - 64K1K3*2K5 - 64K1K3K4K6 + 3456K1K3K4 + 536K1K4K5 + 64K1K5K6 + 16K1K6K7 - 64K24 - 224K2*2K3*2 - 120K2*2K4*2 + 688K2*2K4 - 2910K22 - 32K2K32K4 - 32K2K3K4K5 + 416K2K3K5 + 232K2K4K6 + 8K2K5K7 - 144K3*4 - 112K3*2K4*2 + 160K3*2K6 - 2596K32 + 184K3K4K7 - 16K44 + 32K4*2K8 - 1404K42 - 260K5*2 - 122K6*2 - 60K7*2 - 12K8**2 + 3694
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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