10 3

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10_2

10_4

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

Visit 10_3's page at Knotilus!

Visit 10 3's page at the original Knot Atlas!

[edit 10 3 Quick Notes]

[edit 10 3 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₁₆₂₇ X_11,16,12,17 X_5,13,6,12 X_3,15,4,14 X_13,5,14,4 X_15,3,16,2 X_7,20,8,1 X_9,18,10,19 X_17,10,18,11 X_19,8,20,9
Gauss code	-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7
Dowker-Thistlethwaite code	6 14 12 20 18 16 4 2 10 8
Conway Notation	[64]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 13, width is 6,

Braid index is 6

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

[edit Notes on presentations of 10 3]

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 3"];

In[4]:= PD[K]

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

Out[4]= X₁₆₂₇ X_11,16,12,17 X_5,13,6,12 X_3,15,4,14 X_13,5,14,4 X_15,3,16,2 X_7,20,8,1 X_9,18,10,19 X_17,10,18,11 X_19,8,20,9

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [64]

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

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	1
Bridge index	2
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-7][-5]
Hyperbolic Volume	5.7321
A-Polynomial	See Data:10 3/A-polynomial

[edit Notes for 10 3's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for 10 3's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-6 t + 13-6 t -1$
Conway polynomial	$1-6 z 2$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$q 4 - q 3 + 2 q 2 -3 q + 4-4 q -1 + 3 q -2 -3 q -3 + 2 q -4 - q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$a 6 - z 2 a 4 -2 z 2 a 2 - a 2 -2 z 2 - z 2 a -2 + a -4$
Kauffman polynomial (db, data sources)	$a 3 z 9 + a z 9 + a 4 z 8 + 2 a 2 z 8 + z 8 + a 5 z 7 -6 a 3 z 7 -6 a z 7 + z 7 a -1 + a 6 z 6 -4 a 4 z 6 -10 a 2 z 6 + z 6 a -2 -4 z 6 -4 a 5 z 5 + 15 a 3 z 5 + 15 a z 5 -3 z 5 a -1 + z 5 a -3 -5 a 6 z 4 + 4 a 4 z 4 + 18 a 2 z 4 -2 z 4 a -2 + z 4 a -4 + 6 z 4 + 3 a 5 z 3 -18 a 3 z 3 -15 a z 3 + 4 z 3 a -1 -2 z 3 a -3 + 6 a 6 z 2 -2 a 4 z 2 -12 a 2 z 2 + z 2 a -2 -3 z 2 a -4 + 6 a 3 z + 6 a z - a 6 + a 2 + a -4$
The A2 invariant	$q 20 + q 18 + q 14 - q 10 - q 6 - q 4 + q -2 - q -4 + q -8 + q -12 + q -14$
The G2 invariant	Data:10 3/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_3.

A1 Invariants.

Weight	Invariant
1	$q 13 + q 9 - q 7 - q 3 + q -1 - q -3 + q -5 + q -9$
2	$q 38 + q 32 - q 30 -2 q 28 + q 26 -2 q 22 + q 20 + q 18 - q 16 + 2 q 12 + q 6 + 2 q -2 - q -4 - q -6 + 2 q -8 - q -10 - q -12 - q -16 + q -20 + q -26$
3	$q 75 - q 65 - q 63 + q 59 - q 57 -2 q 55 + 3 q 51 + 2 q 49 -2 q 47 -2 q 45 + q 43 + 3 q 41 - q 37 - q 35 + q 33 + 2 q 31 + q 29 -3 q 27 - q 25 + 2 q 23 + 2 q 21 -2 q 19 -2 q 17 + q 15 - q 11 - q 9 + q 7 + q 3 - q - q -1 + 2 q -3 + 2 q -5 -2 q -9 + 3 q -13 + q -15 -2 q -19 + 2 q -23 + 2 q -25 - q -27 -2 q -29 + q -33 -2 q -37 - q -39 + q -43 + q -51$

A2 Invariants.

Weight	Invariant
1,0	$q 20 + q 18 + q 14 - q 10 - q 6 - q 4 + q -2 - q -4 + q -8 + q -12 + q -14$
2,0	$q 52 + q 50 + q 48 + q 44 + q 42 - q 40 -3 q 38 -2 q 36 -2 q 30 - q 28 + q 26 + q 24 - q 20 + 2 q 18 + 2 q 16 + 2 q 10 + q 8 + q 6 + q 4 + 1 + q -2 + q -4 -2 q -6 -2 q -8 + 2 q -10 -3 q -14 -2 q -16 - q -22 + q -26 + q -28 + q -32 + q -34 + q -36$

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 3"];

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

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

Out[4]= $-6 t + 13-6 t -1$

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

Out[5]= $1-6 z 2$

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]= { 25, 0 }

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

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

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

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

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

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

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

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

Out[10]= $a 3 z 9 + a z 9 + a 4 z 8 + 2 a 2 z 8 + z 8 + a 5 z 7 -6 a 3 z 7 -6 a z 7 + z 7 a -1 + a 6 z 6 -4 a 4 z 6 -10 a 2 z 6 + z 6 a -2 -4 z 6 -4 a 5 z 5 + 15 a 3 z 5 + 15 a z 5 -3 z 5 a -1 + z 5 a -3 -5 a 6 z 4 + 4 a 4 z 4 + 18 a 2 z 4 -2 z 4 a -2 + z 4 a -4 + 6 z 4 + 3 a 5 z 3 -18 a 3 z 3 -15 a z 3 + 4 z 3 a -1 -2 z 3 a -3 + 6 a 6 z 2 -2 a 4 z 2 -12 a 2 z 2 + z 2 a -2 -3 z 2 a -4 + 6 a 3 z + 6 a z - a 6 + a 2 + a -4$

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

Same Alexander/Conway Polynomial: {}

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 3"];

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]= { $-6 t + 13-6 t -1$ , $q 4 - q 3 + 2 q 2 -3 q + 4-4 q -1 + 3 q -2 -3 q -3 + 2 q -4 - q -5 + q -6$ }

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]= {}

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:

(-6, 3)

[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 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-3

-1

-5

-7

-1

-9

-11

-13

Integral Khovanov Homology

(db, data source)

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

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n + 1

-dimensional representation of

s l (2)

)

$n$	$J n$
2	$q 12 - q 11 + 2 q 9 -2 q 8 - q 7 + 3 q 6 -3 q 5 - q 4 + 6 q 3 -6 q 2 - q + 9-8 q -1 - q -2 + 9 q -3 -7 q -4 -2 q -5 + 9 q -6 -5 q -7 -4 q -8 + 8 q -9 -3 q -10 -4 q -11 + 5 q -12 - q -13 -3 q -14 + 2 q -15 - q -17 + q -18$
3	$q 24 - q 23 + 2 q 20 -2 q 19 - q 18 - q 17 + 4 q 16 - q 15 -2 q 14 -3 q 13 + 5 q 12 + 2 q 11 -2 q 10 -5 q 9 + 3 q 8 + 4 q 7 - q 6 -3 q 5 + 2 q 3 + q 2 - q - q -1 + q -2 + q -3 - q -4 - q -6 + q -7 + q -9 -4 q -10 + q -11 + 4 q -12 + q -13 -7 q -14 - q -15 + 8 q -16 + 2 q -17 -8 q -18 -3 q -19 + 8 q -20 + 3 q -21 -5 q -22 -5 q -23 + 5 q -24 + 3 q -25 - q -26 -4 q -27 + 2 q -28 + q -29 -2 q -31 + q -32 - q -35 + q -36$

[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.