Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 6.19

Min(phi) over symmetries of the knot is: [-5,-2,0,1,3,3,1,2,5,3,4,1,3,2,3,1,1,2,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.19']
Arrow polynomial of the knot is: 4K12K3 - 2K12 - 6K1K2 - 2K1K4 + 2K1 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.19', '6.22', '6.41']
Outer characteristic polynomial of the knot is: t^7+140t^5+143t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.19']
2-strand cable arrow polynomial of the knot is: -304K14 + 640K1*3K2K3 - 352K1*3K3 - 512K12K2*2K3*2 - 2752K1*2K2*2 + 256K1*2K2K32 + 32K1*2K2K3K5 - 320K12K2K4 + 3472K1*2K2 - 1232K12K3*2 - 32K1*2K3K5 - 3692K1*2 + 1824K1K23K3 + 352K1K2*2K3K4 - 1152K1K22K3 + 64K1K2*2K5K6 - 384K1K22K5 + 640K1K2K33 - 384K1K2K3K4 - 352K1K2K3K6 + 6064K1K2K3 - 64K1K2K4K5 - 32K1K2K5K6 + 1728K1K3K4 + 128K1K4K5 + 72K1K5K6 - 512K2*4K3*2 - 32K2*4K6*2 - 888K2*4 + 512K2*3K3K5 + 64K2*3K4K6 + 32K2*3K6K8 - 256K2*2K3*4 + 224K2*2K3*2K6 - 2000K22K3*2 - 64K2*2K3K7 - 152K2*2K4*2 - 32K2*2K4K8 + 1080K2*2K4 - 128K22K5*2 - 128K2*2K6*2 - 8K2*2K8*2 - 2652K2*2 - 256K2K32K4 + 1160K2K3K5 + 336K2K4K6 + 32K2K5K7 + 32K2K6K8 - 128K34 + 160K3*2K6 - 2264K32 - 714K4*2 - 188K5*2 - 132K6*2 - 4K8**2 + 3196
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.19']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19916', 'vk6.19962', 'vk6.21141', 'vk6.21221', 'vk6.26835', 'vk6.26961', 'vk6.28611', 'vk6.28701', 'vk6.38269', 'vk6.38371', 'vk6.40399', 'vk6.40532', 'vk6.45142', 'vk6.45238', 'vk6.46992', 'vk6.47049', 'vk6.56695', 'vk6.56756', 'vk6.57779', 'vk6.57869', 'vk6.61095', 'vk6.61221', 'vk6.62351', 'vk6.62441', 'vk6.66383', 'vk6.66466', 'vk6.67143', 'vk6.67249', 'vk6.69040', 'vk6.69112', 'vk6.69832', 'vk6.69885']
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: [-5,-2,0,1,3,3,1,2,5,3,4,1,3,2,3,1,1,2,2,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.19']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.19', '6.22', '6.41']

Outer characteristic polynomial of the knot is: t^7+140t^5+143t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.19']

2-strand cable arrow polynomial of the knot is: -304*K1**4 + 640*K1**3*K2*K3 - 352*K1**3*K3 - 512*K1**2*K2**2*K3**2 - 2752*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 3472*K1**2*K2 - 1232*K1**2*K3**2 - 32*K1**2*K3*K5 - 3692*K1**2 + 1824*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 + 64*K1*K2**2*K5*K6 - 384*K1*K2**2*K5 + 640*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 352*K1*K2*K3*K6 + 6064*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1728*K1*K3*K4 + 128*K1*K4*K5 + 72*K1*K5*K6 - 512*K2**4*K3**2 - 32*K2**4*K6**2 - 888*K2**4 + 512*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K6*K8 - 256*K2**2*K3**4 + 224*K2**2*K3**2*K6 - 2000*K2**2*K3**2 - 64*K2**2*K3*K7 - 152*K2**2*K4**2 - 32*K2**2*K4*K8 + 1080*K2**2*K4 - 128*K2**2*K5**2 - 128*K2**2*K6**2 - 8*K2**2*K8**2 - 2652*K2**2 - 256*K2*K3**2*K4 + 1160*K2*K3*K5 + 336*K2*K4*K6 + 32*K2*K5*K7 + 32*K2*K6*K8 - 128*K3**4 + 160*K3**2*K6 - 2264*K3**2 - 714*K4**2 - 188*K5**2 - 132*K6**2 - 4*K8**2 + 3196

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.19']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19916', 'vk6.19962', 'vk6.21141', 'vk6.21221', 'vk6.26835', 'vk6.26961', 'vk6.28611', 'vk6.28701', 'vk6.38269', 'vk6.38371', 'vk6.40399', 'vk6.40532', 'vk6.45142', 'vk6.45238', 'vk6.46992', 'vk6.47049', 'vk6.56695', 'vk6.56756', 'vk6.57779', 'vk6.57869', 'vk6.61095', 'vk6.61221', 'vk6.62351', 'vk6.62441', 'vk6.66383', 'vk6.66466', 'vk6.67143', 'vk6.67249', 'vk6.69040', 'vk6.69112', 'vk6.69832', 'vk6.69885']

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	O1O2O3O4O5O6U1U3U4U6U5U2
R3 orbit	{'O1O2O3O4O5O6U1U3U4U6U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U2U1U3U4U6
Gauss code of K*	O1O2O3O4O5O6U1U6U2U3U5U4
Gauss code of -K*	O1O2O3O4O5O6U3U2U4U5U1U6
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 -5 1 -2 0 3 3],[ 5 0 5 1 2 4 3],[-1 -5 0 -3 -1 2 2],[ 2 -1 3 0 1 3 2],[ 0 -2 1 -1 0 2 1],[-3 -4 -2 -3 -2 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 3 1 0 -2 -5],[-3 0 0 -2 -1 -2 -3],[-3 0 0 -2 -2 -3 -4],[-1 2 2 0 -1 -3 -5],[ 0 1 2 1 0 -1 -2],[ 2 2 3 3 1 0 -1],[ 5 3 4 5 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-1,0,2,5,0,2,1,2,3,2,2,3,4,1,3,5,1,2,1]
Phi over symmetry	[-5,-2,0,1,3,3,1,2,5,3,4,1,3,2,3,1,1,2,2,2,0]
Phi of -K	[-5,-2,0,1,3,3,2,3,1,4,5,1,0,2,3,0,1,2,0,0,0]
Phi of K*	[-3,-3,-1,0,2,5,0,0,1,2,4,0,2,3,5,0,0,1,1,3,2]
Phi of -K*	[-5,-2,0,1,3,3,1,2,5,3,4,1,3,2,3,1,1,2,2,2,0]
Symmetry type of based matrix	c
u-polynomial	t^5-2t^3+t^2-t
Normalized Jones-Krushkal polynomial	7z^2+26z+25
Enhanced Jones-Krushkal polynomial	7w^3z^2+26w^2z+25w
Inner characteristic polynomial	t^6+92t^4+30t^2
Outer characteristic polynomial	t^7+140t^5+143t^3+5t
Flat arrow polynomial	4K12K3 - 2K12 - 6K1K2 - 2K1K4 + 2K1 + K2 + 2*K3 + 2
2-strand cable arrow polynomial	-304K14 + 640K1*3K2K3 - 352K1*3K3 - 512K12K2*2K3*2 - 2752K1*2K2*2 + 256K1*2K2K32 + 32K1*2K2K3K5 - 320K12K2K4 + 3472K1*2K2 - 1232K12K3*2 - 32K1*2K3K5 - 3692K1*2 + 1824K1K23K3 + 352K1K2*2K3K4 - 1152K1K22K3 + 64K1K2*2K5K6 - 384K1K22K5 + 640K1K2K33 - 384K1K2K3K4 - 352K1K2K3K6 + 6064K1K2K3 - 64K1K2K4K5 - 32K1K2K5K6 + 1728K1K3K4 + 128K1K4K5 + 72K1K5K6 - 512K2*4K3*2 - 32K2*4K6*2 - 888K2*4 + 512K2*3K3K5 + 64K2*3K4K6 + 32K2*3K6K8 - 256K2*2K3*4 + 224K2*2K3*2K6 - 2000K22K3*2 - 64K2*2K3K7 - 152K2*2K4*2 - 32K2*2K4K8 + 1080K2*2K4 - 128K22K5*2 - 128K2*2K6*2 - 8K2*2K8*2 - 2652K2*2 - 256K2K32K4 + 1160K2K3K5 + 336K2K4K6 + 32K2K5K7 + 32K2K6K8 - 128K34 + 160K3*2K6 - 2264K32 - 714K4*2 - 188K5*2 - 132K6*2 - 4K8**2 + 3196
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

Contact