L11a172

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L11a171.gif

L11a171

L11a173.gif

L11a173

Contents

L11a172.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a172 at Knotilus!


Link Presentations

[edit Notes on L11a172's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X22,16,7,15 X6,21,1,22 X18,11,19,12 X14,6,15,5 X20,13,21,14 X12,19,13,20 X4,18,5,17
Gauss code {1, -2, 3, -11, 8, -6}, {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -7, 10, -9, 6, -5}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L11a172 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^6-t(1) t(2)^6-2 t(1)^2 t(2)^5+3 t(1) t(2)^5-t(2)^5+3 t(1)^2 t(2)^4-5 t(1) t(2)^4+2 t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-3 t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^3} (db)
Jones polynomial \frac{13}{q^{9/2}}-\frac{15}{q^{7/2}}-q^{5/2}+\frac{14}{q^{5/2}}+3 q^{3/2}-\frac{13}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{10}{q^{11/2}}-6 \sqrt{q}+\frac{9}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^5 z^7-5 a^5 z^5-8 a^5 z^3-4 a^5 z+a^3 z^9+7 a^3 z^7+18 a^3 z^5+20 a^3 z^3+8 a^3 z+a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-5 a z-a z^{-1} (db)
Kauffman polynomial a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+5 a^8 z^6-5 a^8 z^4+a^8 z^2+7 a^7 z^7-12 a^7 z^5+10 a^7 z^3-2 a^7 z+7 a^6 z^8-14 a^6 z^6+12 a^6 z^4-2 a^6 z^2+5 a^5 z^9-9 a^5 z^7+5 a^5 z^5-2 a^5 z^3+2 a^5 z+2 a^4 z^{10}+2 a^4 z^8-16 a^4 z^6+16 a^4 z^4-6 a^4 z^2+9 a^3 z^9-33 a^3 z^7+46 a^3 z^5-36 a^3 z^3+11 a^3 z-a^3 z^{-1} +2 a^2 z^{10}-2 a^2 z^8-9 a^2 z^6+11 a^2 z^4-5 a^2 z^2+a^2+4 a z^9-16 a z^7+z^7 a^{-1} +22 a z^5-4 z^5 a^{-1} -17 a z^3+4 z^3 a^{-1} +7 a z-a z^{-1} +3 z^8-12 z^6+13 z^4-3 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 \
-7-6-5-4-3-2-101234χ
6           11
4          2 -2
2         41 3
0        52  -3
-2       84   4
-4      76    -1
-6     87     1
-8    68      2
-10   47       -3
-12  26        4
-14 14         -3
-16 2          2
-181           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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L11a171

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L11a173