L11a97

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L11a96.gif

L11a96

L11a98.gif

L11a98

Contents

L11a97.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a97 at Knotilus!


Link Presentations

[edit Notes on L11a97's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X20,17,21,18 X18,13,19,14 X14,19,15,20 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X8,15,9,16
Gauss code {1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, 4, -5, 11, -2, 3, -4, 5, -3, 6, -9}
A Braid Representative
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A Morse Link Presentation L11a97 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5+6 t(1) t(2)^4-6 t(2)^4-13 t(1) t(2)^3+13 t(2)^3+13 t(1) t(2)^2-13 t(2)^2-6 t(1) t(2)+6 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{16}{q^{11/2}}-\frac{23}{q^{13/2}}+\frac{25}{q^{15/2}}-\frac{25}{q^{17/2}}+\frac{22}{q^{19/2}}-\frac{15}{q^{21/2}}+\frac{9}{q^{23/2}}-\frac{4}{q^{25/2}}+\frac{1}{q^{27/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z a^{13}+3 z^3 a^{11}+2 z a^{11}-2 a^{11} z^{-1} -2 z^5 a^9+7 z a^9+5 a^9 z^{-1} -4 z^5 a^7-11 z^3 a^7-10 z a^7-3 a^7 z^{-1} -z^5 a^5-z^3 a^5 (db)
Kauffman polynomial -z^6 a^{16}+2 z^4 a^{16}-z^2 a^{16}-4 z^7 a^{15}+9 z^5 a^{15}-6 z^3 a^{15}+z a^{15}-7 z^8 a^{14}+15 z^6 a^{14}-11 z^4 a^{14}+4 z^2 a^{14}-a^{14}-6 z^9 a^{13}+3 z^7 a^{13}+14 z^5 a^{13}-13 z^3 a^{13}+2 z a^{13}-2 z^{10} a^{12}-17 z^8 a^{12}+48 z^6 a^{12}-41 z^4 a^{12}+14 z^2 a^{12}-14 z^9 a^{11}+15 z^7 a^{11}+12 z^5 a^{11}-14 z^3 a^{11}+4 z a^{11}-2 a^{11} z^{-1} -2 z^{10} a^{10}-22 z^8 a^{10}+52 z^6 a^{10}-34 z^4 a^{10}+2 z^2 a^{10}+5 a^{10}-8 z^9 a^9-2 z^7 a^9+25 z^5 a^9-24 z^3 a^9+13 z a^9-5 a^9 z^{-1} -12 z^8 a^8+16 z^6 a^8-3 z^4 a^8-7 z^2 a^8+5 a^8-10 z^7 a^7+17 z^5 a^7-16 z^3 a^7+10 z a^7-3 a^7 z^{-1} -4 z^6 a^6+3 z^4 a^6-z^5 a^5+z^3 a^5 (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          41-3
-8         7  7
-10        94  -5
-12       147   7
-14      1210    -2
-16     1313     0
-18    912      3
-20   613       -7
-22  39        6
-24 16         -5
-26 3          3
-281           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-8 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-7 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-6 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-5 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\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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L11a96

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L11a98