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Flat knot 6.215

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.215']
Arrow polynomial of the knot is: -2K1K2 + K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354']
Outer characteristic polynomial of the knot is: t^7+59t^5+59t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.215']
2-strand cable arrow polynomial of the knot is: -64K13K3 - 1568K12K2*2 + 96K1*2K2K32 - 64K1*2K2K4 + 2968K1*2K2 - 128K12K3*2 - 3376K1*2 + 288K1K23K3 + 96K1K2*2K3K4 - 768K1K22K3 - 64K1K2*2K5 - 320K1K2K3K4 + 4040K1K2K3 + 616K1K3K4 + 152K1K4K5 - 672K2*4 - 704K2*2K3*2 - 72K2*2K4*2 + 1192K2*2K4 - 2670K22 + 576K2K3K5 + 16K2K4K6 - 1624K3*2 - 532K4*2 - 152K5*2 - 2K6**2 + 2738
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.215']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4230', 'vk6.4309', 'vk6.5497', 'vk6.5611', 'vk6.7598', 'vk6.7689', 'vk6.9095', 'vk6.9174', 'vk6.18383', 'vk6.18721', 'vk6.24840', 'vk6.25297', 'vk6.37028', 'vk6.37476', 'vk6.44193', 'vk6.44512', 'vk6.48542', 'vk6.48597', 'vk6.49241', 'vk6.49353', 'vk6.50325', 'vk6.50384', 'vk6.51068', 'vk6.51099', 'vk6.56156', 'vk6.56383', 'vk6.60685', 'vk6.61034', 'vk6.65820', 'vk6.66072', 'vk6.68813', 'vk6.69021']
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,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,1,-2]

Flat knots (up to 7 crossings) with same phi are :['6.215']

Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354']

Outer characteristic polynomial of the knot is: t^7+59t^5+59t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.215']

2-strand cable arrow polynomial of the knot is: -64*K1**3*K3 - 1568*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 64*K1**2*K2*K4 + 2968*K1**2*K2 - 128*K1**2*K3**2 - 3376*K1**2 + 288*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 64*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 4040*K1*K2*K3 + 616*K1*K3*K4 + 152*K1*K4*K5 - 672*K2**4 - 704*K2**2*K3**2 - 72*K2**2*K4**2 + 1192*K2**2*K4 - 2670*K2**2 + 576*K2*K3*K5 + 16*K2*K4*K6 - 1624*K3**2 - 532*K4**2 - 152*K5**2 - 2*K6**2 + 2738

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.215']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4230', 'vk6.4309', 'vk6.5497', 'vk6.5611', 'vk6.7598', 'vk6.7689', 'vk6.9095', 'vk6.9174', 'vk6.18383', 'vk6.18721', 'vk6.24840', 'vk6.25297', 'vk6.37028', 'vk6.37476', 'vk6.44193', 'vk6.44512', 'vk6.48542', 'vk6.48597', 'vk6.49241', 'vk6.49353', 'vk6.50325', 'vk6.50384', 'vk6.51068', 'vk6.51099', 'vk6.56156', 'vk6.56383', 'vk6.60685', 'vk6.61034', 'vk6.65820', 'vk6.66072', 'vk6.68813', 'vk6.69021']

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	O1O2O3O4O5U2O6U5U4U6U1U3
R3 orbit	{'O1O2O3O4O5U2O6U5U4U6U1U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U5U6U2U1O6U4
Gauss code of K*	O1O2O3O4O5U4U6U5U2U1O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U5U4U1U6U2
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 -1 -3 2 0 0 2],[ 1 0 -2 2 0 0 2],[ 3 2 0 3 2 1 2],[-2 -2 -3 0 -1 -1 2],[ 0 0 -2 1 0 0 2],[ 0 0 -1 1 0 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 2 -1 -1 -2 -3],[-2 -2 0 -1 -2 -2 -2],[ 0 1 1 0 0 0 -1],[ 0 1 2 0 0 0 -2],[ 1 2 2 0 0 0 -2],[ 3 3 2 1 2 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,-2,1,1,2,3,1,2,2,2,0,0,1,0,2,2]
Phi over symmetry	[-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,1,-2]
Phi of -K	[-3,-1,0,0,2,2,0,1,2,2,3,1,1,1,1,0,1,0,1,1,-2]
Phi of K*	[-2,-2,0,0,1,3,-2,0,1,1,3,1,1,1,2,0,1,1,1,2,0]
Phi of -K*	[-3,-1,0,0,2,2,2,1,2,2,3,0,0,2,2,0,1,1,2,1,-2]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	7z^2+24z+21
Enhanced Jones-Krushkal polynomial	7w^3z^2-4w^3z+28w^2z+21w
Inner characteristic polynomial	t^6+41t^4+15t^2
Outer characteristic polynomial	t^7+59t^5+59t^3+8t
Flat arrow polynomial	-2K1K2 + K1 + K3 + 1
2-strand cable arrow polynomial	-64K13K3 - 1568K12K2*2 + 96K1*2K2K32 - 64K1*2K2K4 + 2968K1*2K2 - 128K12K3*2 - 3376K1*2 + 288K1K23K3 + 96K1K2*2K3K4 - 768K1K22K3 - 64K1K2*2K5 - 320K1K2K3K4 + 4040K1K2K3 + 616K1K3K4 + 152K1K4K5 - 672K2*4 - 704K2*2K3*2 - 72K2*2K4*2 + 1192K2*2K4 - 2670K22 + 576K2K3K5 + 16K2K4K6 - 1624K3*2 - 532K4*2 - 152K5*2 - 2K6**2 + 2738
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}, {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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