L11a39

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L11a38.gif

L11a38

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L11a40

Contents

L11a39.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a39's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X16,8,17,7 X20,13,21,14 X22,18,5,17 X18,22,19,21 X14,19,15,20 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -7, 9, -3, 5, -6, 7, -4, 6, -5}
A Braid Representative
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A Morse Link Presentation L11a39 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(3 v^2-8 v+3\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -13 q^{9/2}+16 q^{7/2}-\frac{1}{q^{7/2}}-18 q^{5/2}+\frac{2}{q^{5/2}}+18 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-4 q^{13/2}+8 q^{11/2}-15 \sqrt{q}+\frac{10}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-7} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +4 z^3 a^{-3} +a^3 z+5 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-z^3 a^{-1} -2 a z-2 z a^{-1} -a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -10 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +6 z^8 a^{-6} -13 z^6 a^{-6} +5 z^4 a^{-6} -z^2 a^{-6} +4 z^9 a^{-5} +z^7 a^{-5} -24 z^5 a^{-5} +24 z^3 a^{-5} -9 z a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +12 z^8 a^{-4} -34 z^6 a^{-4} +26 z^4 a^{-4} -7 z^2 a^{-4} + a^{-4} +7 z^9 a^{-3} -4 z^7 a^{-3} +a^3 z^5-23 z^5 a^{-3} -3 a^3 z^3+36 z^3 a^{-3} +3 a^3 z-17 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +z^{10} a^{-2} +10 z^8 a^{-2} +2 a^2 z^6-27 z^6 a^{-2} -3 a^2 z^4+29 z^4 a^{-2} -12 z^2 a^{-2} +a^2+3 a^{-2} +3 z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -2 a z^5-12 z^5 a^{-1} -3 a z^3+19 z^3 a^{-1} +3 a z-10 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +4 z^8-5 z^6+7 z^4-7 z^2+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 \
-4-3-2-101234567χ
16           1-1
14          3 3
12         51 -4
10        83  5
8       85   -3
6      108    2
4     88     0
2    710      -3
0   510       5
-2  15        -4
-4 15         4
-6 1          -1
-81           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}_2 {\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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L11a38

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L11a40