10 1

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9_49

10_2

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 The Coloured Jones Polynomials
9 Computer Talk
10 Modifying This Page

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_1's page at Knotilus!

Visit 10 1's page at the original Knot Atlas!

[edit 10 1 Quick Notes]

[edit 10 1 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₁₄₂₅ X_11,14,12,15 X_3,13,4,12 X_13,3,14,2 X_5,20,6,1 X_7,18,8,19 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X_19,6,20,7
Gauss code	-1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5
Dowker-Thistlethwaite code	4 12 20 18 16 14 2 10 8 6
Conway Notation	[82]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 13, width is 6,

Braid index is 6

[{12, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {11, 2}, {10, 12}, {1, 11}]

[edit Notes on presentations of 10 1]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 1"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₁₄₂₅ X_11,14,12,15 X_3,13,4,12 X_13,3,14,2 X_5,20,6,1 X_7,18,8,19 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X_19,6,20,7

In[5]:= GaussCode[K]

Out[5]= -1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5

In[6]:= DTCode[K]

Out[6]= 4 12 20 18 16 14 2 10 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [82]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= $BR(6,{-1,-1,-2,1,-2,-3,2,-3,-4,3,5,-4,5})$

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]= { 6, 13, 6 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{12, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {11, 2}, {10, 12}, {1, 11}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	1
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-9][-3]
Hyperbolic Volume	3.5262
A-Polynomial	See Data:10 1/A-polynomial

[edit Notes for 10 1's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$1$
Rasmussen s-Invariant	0

[edit Notes for 10 1's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-4 t + 9-4 t -1$
Conway polynomial	$1-4 z 2$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q 2 - q + 2-2 q -1 + 2 q -2 -2 q -3 + 2 q -4 -2 q -5 + q -6 - q -7 + q -8$
HOMFLY-PT polynomial (db, data sources)	$a 8 - z 2 a 6 - a 6 - z 2 a 4 - z 2 a 2 - z 2 + a -2$
Kauffman polynomial (db, data sources)	$a 7 z 9 + a 5 z 9 + a 8 z 8 + 2 a 6 z 8 + a 4 z 8 -7 a 7 z 7 -6 a 5 z 7 + a 3 z 7 -7 a 8 z 6 -12 a 6 z 6 -4 a 4 z 6 + a 2 z 6 + 16 a 7 z 5 + 12 a 5 z 5 -3 a 3 z 5 + a z 5 + 15 a 8 z 4 + 21 a 6 z 4 + 3 a 4 z 4 -2 a 2 z 4 + z 4 -14 a 7 z 3 -11 a 5 z 3 + a 3 z 3 - a z 3 + z 3 a -1 -10 a 8 z 2 -11 a 6 z 2 + z 2 a -2 + 4 a 7 z + 4 a 5 z + a 8 + a 6 - a -2$
The A2 invariant	$q 26 + q 24 - q 18 - q 16 + q -2 + q -6 + q -8$
The G2 invariant	Data:10 1/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_1.

A1 Invariants.

Weight	Invariant
1	$q 17 - q 11 + q -1 + q -5$
2	$q 50 - q 46 - q 40 + q 36 + q 16 + q 14 - q 4 - q 2 + q -2 + q -8 + q -14$
3	$q 99 - q 95 - q 93 + q 89 - q 85 + q 81 + q 79 - q 75 + q 49 + q 47 - q 43 - q 37 - q 35 - q 29 + q 25 + q 23 + q 15 + q 13 + q 11 - q 9 + q 5 + q 3 - q - q -1 + q -3 + q -5 - q -7 - q -9 + q -19 + q -27$

A2 Invariants.

Weight	Invariant
1,0	$q 26 + q 24 - q 18 - q 16 + q -2 + q -6 + q -8$
2,0	$q 68 + q 66 + q 64 - q 62 - q 60 - q 58 - q 56 - q 54 - q 52 + q 50 + q 48 + q 46 + q 24 + 2 q 22 + q 20 + q 18 + q 16 - q 10 -2 q 8 -2 q 6 - q 4 + q -4 + q -10 + q -12 + q -16 + q -18 + q -20$

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 1"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $-4 t + 9-4 t -1$

In[5]:= Conway[K][z]

Out[5]= $1-4 z 2$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= ${1}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 17, 0 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $q 2 - q + 2-2 q -1 + 2 q -2 -2 q -3 + 2 q -4 -2 q -5 + q -6 - q -7 + q -8$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $a 8 - z 2 a 6 - a 6 - z 2 a 4 - z 2 a 2 - z 2 + a -2$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $a 7 z 9 + a 5 z 9 + a 8 z 8 + 2 a 6 z 8 + a 4 z 8 -7 a 7 z 7 -6 a 5 z 7 + a 3 z 7 -7 a 8 z 6 -12 a 6 z 6 -4 a 4 z 6 + a 2 z 6 + 16 a 7 z 5 + 12 a 5 z 5 -3 a 3 z 5 + a z 5 + 15 a 8 z 4 + 21 a 6 z 4 + 3 a 4 z 4 -2 a 2 z 4 + z 4 -14 a 7 z 3 -11 a 5 z 3 + a 3 z 3 - a z 3 + z 3 a -1 -10 a 8 z 2 -11 a 6 z 2 + z 2 a -2 + 4 a 7 z + 4 a 5 z + a 8 + a 6 - a -2$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_3,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {}

Computer Talk

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 1"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { $-4 t + 9-4 t -1$ , $q 2 - q + 2-2 q -1 + 2 q -2 -2 q -3 + 2 q -4 -2 q -5 + q -6 - q -7 + q -8$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {8_3,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(-4, 6)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

-16

48

128

$\frac{232}{3}$

$\frac{200}{3}$

-768

-1088

-256

-272

$-\frac{2048}{3}$

1152

$-\frac{3712}{3}$

$-\frac{3200}{3}$

$\frac{30898}{15}$

$\frac{5768}{15}$

$-\frac{6248}{45}$

$\frac{4046}{9}$

$-\frac{4142}{15}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

0 is the signature of 10 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

-3

-5

-7

-9

-11

-1

-13

-15

-17

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 1$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n + 1

-dimensional representation of

s l (2)

)

$n$	$J n$
2	$q 6 - q 5 + 2 q 3 -2 q 2 + 3-3 q -1 - q -2 + 3 q -3 -2 q -4 - q -5 + 3 q -6 -2 q -7 + 3 q -9 -3 q -10 + 3 q -12 -3 q -13 + 3 q -15 -3 q -16 + 3 q -18 -2 q -19 - q -20 + 2 q -21 - q -22 - q -23 + q -24$
3	$q 12 - q 11 + 2 q 8 -2 q 7 + 2 q 4 -3 q 3 + 2 q + 2-5 q -1 + 4 q -3 + 2 q -4 -6 q -5 - q -6 + 6 q -7 + 2 q -8 -6 q -9 -2 q -10 + 6 q -11 + 2 q -12 -5 q -13 -2 q -14 + 5 q -15 + q -16 -4 q -17 -2 q -18 + 4 q -19 + q -20 -3 q -21 -2 q -22 + 3 q -23 + 2 q -24 -2 q -25 -2 q -26 + 2 q -27 + 2 q -28 -2 q -29 -2 q -30 + 2 q -31 + 2 q -32 -2 q -33 -2 q -34 + 2 q -35 + 2 q -36 -2 q -37 -2 q -38 + q -39 + 3 q -40 - q -41 -2 q -42 + 2 q -44 - q -46 - q -47 + q -48$

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.