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Flat knot 6.1219

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,1,2,1,0,2,1,0,1,1,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.1219']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 6K12 - 4K1K2 - 4K1K3 - 4K1 + 3*K2 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1219']
Outer characteristic polynomial of the knot is: t^7+33t^5+78t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1219']
2-strand cable arrow polynomial of the knot is: -640K14K2*2 + 896K1*4K2 - 1280K14 + 128K1*3K2*3K3 - 128K13K2*2K3 + 992K13K2K3 - 288K1*3K3 - 256K12K2*4 + 1184K1*2K2*3 - 320K1*2K2*2K3*2 - 7424K1*2K2*2 - 864K1*2K2K4 + 6160K1*2K2 - 416K12K3*2 - 4220K1*2 + 2272K1K23K3 + 384K1K2*2K3K4 - 992K1K22K3 - 576K1K2*2K5 + 96K1K2K33 - 160K1K2K3K4 - 32K1K2K3K6 + 8168K1K2K3 - 160K1K2K4K5 + 1008K1K3K4 + 56K1K4K5 + 40K1K5K6 - 64K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 2424K24 + 224K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 - 2352K22K3*2 - 32K2*2K3K7 - 400K2*2K4*2 - 32K2*2K4K8 + 1984K2*2K4 - 64K22K5*2 - 16K2*2K6*2 - 2732K2*2 - 128K2K32K4 + 1368K2K3K5 + 360K2K4K6 + 8K2K5K7 + 16K2K6K8 - 32K34 + 48K3*2K6 - 2328K32 - 646K4*2 - 212K5*2 - 92K6*2 - 2K8**2 + 3854
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1219']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4748', 'vk6.5075', 'vk6.6290', 'vk6.6729', 'vk6.8255', 'vk6.8704', 'vk6.9641', 'vk6.9956', 'vk6.20406', 'vk6.21763', 'vk6.27752', 'vk6.29282', 'vk6.39180', 'vk6.41416', 'vk6.45916', 'vk6.47549', 'vk6.48780', 'vk6.48991', 'vk6.49592', 'vk6.49795', 'vk6.50796', 'vk6.51011', 'vk6.51283', 'vk6.51478', 'vk6.57267', 'vk6.58492', 'vk6.61919', 'vk6.63020', 'vk6.66880', 'vk6.67758', 'vk6.69512', 'vk6.70226']
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: [-2,-2,0,1,1,2,0,0,1,1,2,1,0,2,1,0,1,1,-1,0,2]

Flat knots (up to 7 crossings) with same phi are :['6.1219']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 6*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 3*K2 + K4 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1219']

Outer characteristic polynomial of the knot is: t^7+33t^5+78t^3+14t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1219']

2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 896*K1**4*K2 - 1280*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 992*K1**3*K2*K3 - 288*K1**3*K3 - 256*K1**2*K2**4 + 1184*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 7424*K1**2*K2**2 - 864*K1**2*K2*K4 + 6160*K1**2*K2 - 416*K1**2*K3**2 - 4220*K1**2 + 2272*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 - 576*K1*K2**2*K5 + 96*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8168*K1*K2*K3 - 160*K1*K2*K4*K5 + 1008*K1*K3*K4 + 56*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2424*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 2352*K2**2*K3**2 - 32*K2**2*K3*K7 - 400*K2**2*K4**2 - 32*K2**2*K4*K8 + 1984*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 2732*K2**2 - 128*K2*K3**2*K4 + 1368*K2*K3*K5 + 360*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 48*K3**2*K6 - 2328*K3**2 - 646*K4**2 - 212*K5**2 - 92*K6**2 - 2*K8**2 + 3854

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1219']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4748', 'vk6.5075', 'vk6.6290', 'vk6.6729', 'vk6.8255', 'vk6.8704', 'vk6.9641', 'vk6.9956', 'vk6.20406', 'vk6.21763', 'vk6.27752', 'vk6.29282', 'vk6.39180', 'vk6.41416', 'vk6.45916', 'vk6.47549', 'vk6.48780', 'vk6.48991', 'vk6.49592', 'vk6.49795', 'vk6.50796', 'vk6.51011', 'vk6.51283', 'vk6.51478', 'vk6.57267', 'vk6.58492', 'vk6.61919', 'vk6.63020', 'vk6.66880', 'vk6.67758', 'vk6.69512', 'vk6.70226']

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	O1O2O3O4U3U1U4O5O6U2U5U6
R3 orbit	{'O1O2O3O4U3U1U4O5O6U2U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U6U3O5O6U1U4U2
Gauss code of K*	O1O2O3U4U5O6O4O5U2U6U1U3
Gauss code of -K*	O1O2O3U1U2O4O5O6U4U6U3U5
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 -2 -1 -1 2 0 2],[ 2 0 2 0 2 1 1],[ 1 -2 0 -1 1 1 2],[ 1 0 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[ 0 -1 -1 0 0 0 1],[-2 -1 -2 0 0 -1 0]]
Primitive based matrix	[[ 0 2 2 0 -1 -1 -2],[-2 0 0 0 -1 -1 -2],[-2 0 0 -1 0 -2 -1],[ 0 0 1 0 0 -1 -1],[ 1 1 0 0 0 1 0],[ 1 1 2 1 -1 0 -2],[ 2 2 1 1 0 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,1,1,2,0,0,1,1,2,1,0,2,1,0,1,1,-1,0,2]
Phi over symmetry	[-2,-2,0,1,1,2,0,0,1,1,2,1,0,2,1,0,1,1,-1,0,2]
Phi of -K	[-2,-1,-1,0,2,2,-1,1,1,2,3,1,0,2,1,1,2,3,2,1,0]
Phi of K*	[-2,-2,0,1,1,2,0,1,1,3,3,2,2,2,2,0,1,1,-1,-1,1]
Phi of -K*	[-2,-1,-1,0,2,2,0,2,1,1,2,1,0,0,1,1,2,1,1,0,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+19t^4+27t^2+1
Outer characteristic polynomial	t^7+33t^5+78t^3+14t
Flat arrow polynomial	8K13 + 4K1*2K2 - 6K12 - 4K1K2 - 4K1K3 - 4K1 + 3*K2 + K4 + 3
2-strand cable arrow polynomial	-640K14K2*2 + 896K1*4K2 - 1280K14 + 128K1*3K2*3K3 - 128K13K2*2K3 + 992K13K2K3 - 288K1*3K3 - 256K12K2*4 + 1184K1*2K2*3 - 320K1*2K2*2K3*2 - 7424K1*2K2*2 - 864K1*2K2K4 + 6160K1*2K2 - 416K12K3*2 - 4220K1*2 + 2272K1K23K3 + 384K1K2*2K3K4 - 992K1K22K3 - 576K1K2*2K5 + 96K1K2K33 - 160K1K2K3K4 - 32K1K2K3K6 + 8168K1K2K3 - 160K1K2K4K5 + 1008K1K3K4 + 56K1K4K5 + 40K1K5K6 - 64K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 2424K24 + 224K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 - 2352K22K3*2 - 32K2*2K3K7 - 400K2*2K4*2 - 32K2*2K4K8 + 1984K2*2K4 - 64K22K5*2 - 16K2*2K6*2 - 2732K2*2 - 128K2K32K4 + 1368K2K3K5 + 360K2K4K6 + 8K2K5K7 + 16K2K6K8 - 32K34 + 48K3*2K6 - 2328K32 - 646K4*2 - 212K5*2 - 92K6*2 - 2K8**2 + 3854
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}, {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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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