L11n422

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L11n421.gif

L11n421

L11n423.gif

L11n423

Contents

L11n422.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n422 at Knotilus!


Link Presentations

[edit Notes on L11n422's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X19,5,20,4 X12,22,7,21
Gauss code {1, -4, -3, 10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, 4, 5, -10, -6, 11, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n422 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2 w^2-2 u^2 v^2 w-u^2 v w^2+2 u^2 v w-u^2 v-u v^2 w^2+u v^2 w+3 u v w^2-3 u v w-u w^2+u w+v w^3-2 v w^2+v w+2 w^2-w}{u v w^{3/2}} (db)
Jones polynomial q^9-2 q^8+5 q^7-6 q^6+8 q^5-8 q^4+8 q^3-5 q^2+4 q-1 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +6 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} + a^{-2} +3 a^{-4} -6 a^{-6} +2 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} (db)
Kauffman polynomial z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +5 z^8 a^{-6} +3 z^8 a^{-8} +z^7 a^{-3} +z^7 a^{-5} +2 z^7 a^{-7} +2 z^7 a^{-9} -5 z^6 a^{-4} -17 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +z^5 a^{-3} -8 z^5 a^{-5} -15 z^5 a^{-7} -6 z^5 a^{-9} +4 z^4 a^{-2} +11 z^4 a^{-4} +25 z^4 a^{-6} +14 z^4 a^{-8} -4 z^4 a^{-10} +z^3 a^{-1} +14 z^3 a^{-5} +18 z^3 a^{-7} +3 z^3 a^{-9} -3 z^2 a^{-2} -10 z^2 a^{-4} -23 z^2 a^{-6} -12 z^2 a^{-8} +4 z^2 a^{-10} -9 z a^{-5} -9 z a^{-7} - a^{-2} +5 a^{-4} +11 a^{-6} +5 a^{-8} - a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} z^{-2} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-1012345678χ
19         11
17        21-1
15       3  3
13      32  -1
11     53   2
9    33    0
7   55     0
5  25      3
3 23       -1
1 3        3
-11         -1
Integral Khovanov Homology

(db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L11n421.gif

L11n421

L11n423.gif

L11n423