5 2

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5_1

6_1

Contents

Image:5 2.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,1,-3,4,-5,2,-4,3/goTop.html at Knotilus!

Visit 5 2's page at the original Knot Atlas!

5_2 is also known as the 3-twist knot.


3D depiction
3D depiction
Simple square depiction
Simple square depiction

[edit] Knot presentations

Planar diagram presentation X1425 X3849 X5,10,6,1 X9,6,10,7 X7283
Gauss code -1, 5, -2, 1, -3, 4, -5, 2, -4, 3
Dowker-Thistlethwaite code 4 8 10 2 6
Conway Notation [32]


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

Length is 6, width is 3,

Braid index is 3

Image:5 2_ML.gif Image:5 2_AP.gif
[{7, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 6}, {5, 7}, {6, 1}]

[edit Notes on presentations of 5 2]

Knot 5_2.
Knot 5_2.
A graph, knot 5_2.
A graph, knot 5_2.

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index {3,4}
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][1]
Hyperbolic Volume 2.82812
A-Polynomial See Data:5 2/A-polynomial

[edit Notes for 5 2's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus 1
Topological 4 genus 1
Concordance genus ConcordanceGenus(Knot(5,2))
Rasmussen s-Invariant -2

[edit Notes for 5 2's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial 2t + 2t−1−3
Conway polynomial 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 7, -2 }
Jones polynomial q−6 + q−5q−4 + 2q−3q−2 + q−1
HOMFLY-PT polynomial (db, data sources) a6 + a4z2 + a4 + a2z2 + a2
Kauffman polynomial (db, data sources) a7z3−2a7z + a6z4−2a6z2 + a6 + 2a5z3−2a5z + a4z4a4z2 + a4 + a3z3 + a2z2a2
The A2 invariant q20q18 + q12 + q10 + q8 + q6 + q2
The G2 invariant q100 + q96q94q92 + q90q88q84q82q78q76q74q72q68q66 + q64 + q60 + q56 + q54 + 2q50q48 + 2q46 + q44 + q40 + q34 + 2q24 + q20 + q14 + q10

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n57,}

[edit] Vassiliev invariants

V2 and V3: (2, -3)
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
8 −24 32 \frac{268}{3} \frac{44}{3} −192 −368 −64 −56 \frac{256}{3} 288 \frac{2144}{3} \frac{352}{3} \frac{22951}{15} -\frac{28}{5} \frac{29764}{45} \frac{137}{9} \frac{1351}{15}

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 = -2 is the signature of 5 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10χ
-1     11
-3    110
-5   1  1
-7   1  1
-9 11   0
-11      0
-131     -1
Integral Khovanov Homology

(db, data source)

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

[edit] The Coloured Jones Polynomials

[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).

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5_1

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