4 1

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3_1

5_1

Contents

Image:4 1.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,-3,4,-2/goTop.html at Knotilus!

Visit 4 1's page at the original Knot Atlas!

[edit 4 1 Quick Notes]

4_1 is also known as "the Figure Eight knot", as some people think it looks like a figure `8' in one of its common projections. See e.g. [1] .

For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.

[edit 4 1 Further Notes and Views]
Square depiction
Square depiction
Alternate square depiction
Alternate square depiction
3D depiction
3D depiction
In "figure 8" form
In "figure 8" form
A Neli-Kolam with 3x2 dot array[1]
A Neli-Kolam with 3x2 dot array[1]
In curved symmetrical form
In curved symmetrical form
Quasi-Celtic depiction
Quasi-Celtic depiction
Symmetrical from parametric equation
Symmetrical from parametric equation
Thurston's Trick [2]
Thurston's Trick [2]

[edit] Non-prime (compound) versions

Pseudo-Celtic ornamental knot pattern, with three figure-8 knots along a closed triangular loop.

(alternative)

Coat of arms surrounded by figure-8 knots

[edit] Knot presentations

Planar diagram presentation X4251 X8615 X6374 X2738
Gauss code 1, -4, 3, -1, 2, -3, 4, -2
Dowker-Thistlethwaite code 4 6 8 2
Conway Notation [22]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif

Length is 4, width is 3,

Braid index is 3

Image:4 1_ML.gif Image:4 1_AP.gif
[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]

[edit Notes on presentations of 4 1]

Knot 4_1.
Knot 4_1.
A graph, knot 4_1.
A graph, knot 4_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["4 1"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8615 X6374 X2738
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 2, -3, 4, -2
In[6]:= DTCode[K]
Out[6]= 4 6 8 2

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [22]
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(3,{−1,2,−1,2})
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]= { 3, 4, 3 }
In[11]:= Show[BraidPlot[br]]
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
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."
Image:4 1_ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 5}, {6, 4}, {5, 2}, {1, 3}, {2, 6}, {4, 1}]
In[14]:= Draw[ap]
Image:4 1_AP.gif
Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index 3
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-3]
Hyperbolic Volume 2.02988
A-Polynomial See Data:4 1/A-polynomial

[edit Notes for 4 1's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus ConcordanceGenus(Knot(4,1))
Rasmussen s-Invariant 0

[edit Notes for 4 1's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t−t−1 + 3
Conway polynomial 1−z2
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 5, 0 }
Jones polynomial q2 + q−2−q−q−1 + 1
HOMFLY-PT polynomial (db, data sources) a2 + a−2−z2−1
Kauffman polynomial (db, data sources) a2z2 + z2a−2−a2−a−2 + az3 + z3a−1−az−za−1 + 2z2−1
The A2 invariant q8 + q6−1 + q−6 + q−8
The G2 invariant q38 + q34−q30 + q28 + q26 + q24 + q18 + q16−q10−q4−1−q−4−q−10 + q−16 + q−18 + q−24 + q−26 + q−28−q−30 + q−34 + q−38
Further Quantum Invariants
Further quantum knot invariants for 4_1.

A1 Invariants.

Weight Invariant
1 q5 + q−5
2 q14−q10 + q2 + 1 + q−2−q−10 + q−14
3 q27−q23−q21 + q17 + q11 + q9 + q−9 + q−11 + q−17−q−21−q−23 + q−27
4 q44−q40−q38−q36 + q34 + q32 + q30−q26 + q24 + q22−q18−q16 + q4 + q2 + 1 + q−2 + q−4−q−16−q−18 + q−22 + q−24−q−26 + q−30 + q−32 + q−34−q−36−q−38−q−40 + q−44
5 q65−q61−q59−q57 + q53 + 2q51 + q49−q45−q43 + q39 + q37−q35−2q33−q31 + q27 + q25 + q17 + q15 + q13 + q−13 + q−15 + q−17 + q−25 + q−27−q−31−2q−33−q−35 + q−37 + q−39−q−43−q−45 + q−49 + 2q−51 + q−53−q−57−q−59−q−61 + q−65
6 q90−q86−q84−q82 + 2q76 + 2q74 + q72−q68−2q66−2q64 + q62 + q60 + q58−q54−2q52−2q50 + q48 + 2q46 + 2q44 + q42−q38−q36 + q34 + q32 + q30−q26−q24−q22 + q6 + q4 + q2 + 1 + q−2 + q−4 + q−6−q−22−q−24−q−26 + q−30 + q−32 + q−34−q−36−q−38 + q−42 + 2q−44 + 2q−46 + q−48−2q−50−2q−52−q−54 + q−58 + q−60 + q−62−2q−64−2q−66−q−68 + q−72 + 2q−74 + 2q−76−q−82−q−84−q−86 + q−90

A2 Invariants.

Weight Invariant
1,0 q8 + q6−1 + q−6 + q−8
1,1 q20 + 2q16−2q10−2q8 + 4q2 + 2 + 4q−2−2q−8−2q−10 + 2q−16 + q−20
2,0 q20 + q18 + q16−q14−q12−q10−q8 + q4 + 2q2 + 2 + 2q−2 + q−4−q−8−q−10−q−12−q−14 + q−16 + q−18 + q−20
3,0 q36 + q34 + q32−2q28−2q26−2q24 + q18 + 2q16 + 3q14 + 3q12 + 2q10 + q8−q4−2q2−2−2q−2−q−4 + q−8 + 2q−10 + 3q−12 + 3q−14 + 2q−16 + q−18−2q−24−2q−26−2q−28 + q−32 + q−34 + q−36

A3 Invariants.

Weight Invariant
0,1,0 q16 + q12 + q10 + q−10 + q−12 + q−16
1,0,0 q11 + q9 + q7−q−q−1 + q−7 + q−9 + q−11
1,0,1 q26 + 2q22 + 2q20 + q18 + 2q16−2q14−2q12−4q10−4q8 + 2q4 + 6q2 + 7 + 6q−2 + 2q−4−4q−8−4q−10−2q−12−2q−14 + 2q−16 + q−18 + 2q−20 + 2q−22 + q−26

A4 Invariants.

Weight Invariant
0,1,0,0 q22 + q20 + q18 + q16 + q14−q10−q8 + q2 + 2 + q−2−q−8−q−10 + q−14 + q−16 + q−18 + q−20 + q−22
1,0,0,0 q14 + q12 + q10 + q8−q2−1−q−2 + q−8 + q−10 + q−12 + q−14

B2 Invariants.

Weight Invariant
0,1 q16 + q12 + q10−2 + q−10 + q−12 + q−16
1,0 q26 + q18 + 1 + q−18 + q−26

B3 Invariants.

Weight Invariant
1,0,0 q38 + q30 + q26 + q22−q2 + 1−q−2 + q−22 + q−26 + q−30 + q−38

B4 Invariants.

Weight Invariant
1,0,0,0 q50 + q42 + q38 + q34 + q30 + q26−q6−q2 + 1−q−2−q−6 + q−26 + q−30 + q−34 + q−38 + q−42 + q−50

C3 Invariants.

Weight Invariant
1,0,0 q22 + q18 + q16 + q14 + q12−q2−2−q−2 + q−12 + q−14 + q−16 + q−18 + q−22

C4 Invariants.

Weight Invariant
1,0,0,0 q28 + q24 + q22 + q20 + q18 + q16 + q14−q4−q2−2−q−2−q−4 + q−14 + q−16 + q−18 + q−20 + q−22 + q−24 + q−28

D4 Invariants.

Weight Invariant
0,1,0,0 q38 + q34 + 3q32 + 2q30 + q28 + 4q26 + q24−2q18−5q16−3q14−3q12−4q10 + 3q6 + 5q4 + 6q2 + 8 + 6q−2 + 5q−4 + 3q−6−4q−10−3q−12−3q−14−5q−16−2q−18 + q−24 + 4q−26 + q−28 + 2q−30 + 3q−32 + q−34 + q−38
1,0,0,0 q22 + q18 + q16 + q14 + q12−q2−q−2 + q−12 + q−14 + q−16 + q−18 + q−22

G2 Invariants.

Weight Invariant
1,0 q38 + q34−q30 + q28 + q26 + q24 + q18 + q16−q10−q4−1−q−4−q−10 + q−16 + q−18 + q−24 + q−26 + q−28−q−30 + q−34 + q−38

.

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["4 1"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t−t−1 + 3
In[5]:= Conway[K][z]
Out[5]= 1−z2
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]= { 5, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q2 + q−2−q−q−1 + 1
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= a2 + a−2−z2−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a2z2 + z2a−2−a2−a−2 + az3 + z3a−1−az−za−1 + 2z2−1

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n19,}

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.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["4 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]= { −t−t−1 + 3, q2 + q−2−q−q−1 + 1 }
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]= {K11n19,}

[edit] Vassiliev invariants

V2 and V3: (-1, 0)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
−4 0 8 \frac{34}{3} \frac{14}{3} 0 0 0 0 -\frac{32}{3} 0 -\frac{136}{3} -\frac{56}{3} -\frac{1231}{30} \frac{142}{15} -\frac{1742}{45} \frac{79}{18} -\frac{271}{30}

V2,1 through V6,9 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 trqj 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−2r = s + 1 or j−2r = s−1, where s = 0 is the signature of 4 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-1012χ
5    11
3     0
1  11 0
-1 11  0
-3     0
-51    1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −2 {\mathbb Z}
r = −1 {\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z} {\mathbb Z}
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 sl(2))

 n  Jn
2 q6−q5−q4 + 2q3−q2−q + 3−q−1−q−2 + 2q−3−q−4−q−5 + q−6
3 q12−q11−q10 + 2q8−2q6 + 3q4−3q2 + 3−3q−2 + 3q−4−2q−6 + 2q−8−q−10−q−11 + q−12
4 q20−q19−q18 + 3q15−q14−q13−q12−q11 + 5q10−q9−2q8−2q7−q6 + 6q5−q4−2q3−2q2−q + 7−q−1−2q−2−2q−3−q−4 + 6q−5−q−6−2q−7−2q−8−q−9 + 5q−10−q−11−q−12−q−13−q−14 + 3q−15−q−18−q−19 + q−20
5 q30−q29−q28 + q25 + 2q24−2q22−q21−q20 + q19 + 3q18 + q17−2q16−3q15−2q14 + 2q13 + 4q12 + 2q11−2q10−4q9−2q8 + 2q7 + 5q6 + 2q5−2q4−5q3−2q2 + 2q + 5 + 2q−1−2q−2−5q−3−2q−4 + 2q−5 + 5q−6 + 2q−7−2q−8−4q−9−2q−10 + 2q−11 + 4q−12 + 2q−13−2q−14−3q−15−2q−16 + q−17 + 3q−18 + q−19−q−20−q−21−2q−22 + 2q−24 + q−25−q−28−q−29 + q−30
6 q42−q41−q40 + q37 + 3q35−q34−2q33−q32−q31 + 6q28−q27−2q26−2q25−2q24−q23 + 9q21−2q19−3q18−3q17−2q16 + 11q14−2q12−4q11−4q10−2q9 + 12q7−2q5−4q4−4q3−2q2 + 13−2q−2−4q−3−4q−4−2q−5 + 12q−7−2q−9−4q−10−4q−11−2q−12 + 11q−14−2q−16−3q−17−3q−18−2q−19 + 9q−21−q−23−2q−24−2q−25−2q−26−q−27 + 6q−28−q−31−q−32−2q−33−q−34 + 3q−35 + q−37−q−40−q−41 + q−42
7 q56−q55−q54 + q51 + q49 + 2q48−q47−2q46−q45−2q44 + q43 + q41 + 5q40−2q38−2q37−4q36 + 2q33 + 7q32 + q31−q30−2q29−7q28−2q27 + 2q25 + 9q24 + 2q23−3q21−9q20−3q19 + 3q17 + 10q16 + 3q15−3q13−10q12−3q11 + 3q9 + 11q8 + 3q7−3q5−11q4−3q3 + 3q + 11 + 3q−1−3q−3−11q−4−3q−5 + 3q−7 + 11q−8 + 3q−9−3q−11−10q−12−3q−13 + 3q−15 + 10q−16 + 3q−17−3q−19−9q−20−3q−21 + 2q−23 + 9q−24 + 2q−25−2q−27−7q−28−2q−29−q−30 + q−31 + 7q−32 + 2q−33−4q−36−2q−37−2q−38 + 5q−40 + q−41 + q−43−2q−44−q−45−2q−46−q−47 + 2q−48 + q−49 + q−51−q−54−q−55 + q−56

[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.

[edit] Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

3_1

5_1

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