L11a23

L11a22

L11a24

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a23 at Knotilus!

Link Presentations

[edit Notes on L11a23's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{5}+4t(1)t(2)^{4}-6t(2)^{4}-9t(1)t(2)^{3}+10t(2)^{3}+10t(1)t(2)^{2}-9t(2)^{2}-6t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {13}{q^{21/2}}}+{\frac {17}{q^{19/2}}}-{\frac {20}{q^{17/2}}}+{\frac {20}{q^{15/2}}}-{\frac {18}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+3a^{11}z^{3}+4a^{11}z+a^{11}z^{-1}-2a^{9}z^{5}-2a^{9}z^{3}+3a^{9}z+2a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-7a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-z^{6}a^{16}+3z^{4}a^{16}-3z^{2}a^{16}+a^{16}-3z^{7}a^{15}+7z^{5}a^{15}-5z^{3}a^{15}+za^{15}-5z^{8}a^{14}+10z^{6}a^{14}-5z^{4}a^{14}+z^{2}a^{14}-4z^{9}a^{13}+z^{7}a^{13}+11z^{5}a^{13}-6z^{3}a^{13}-za^{13}+a^{13}z^{-1}-z^{10}a^{12}-13z^{8}a^{12}+34z^{6}a^{12}-29z^{4}a^{12}+14z^{2}a^{12}-3a^{12}-8z^{9}a^{11}+5z^{7}a^{11}+11z^{5}a^{11}-5z^{3}a^{11}-3za^{11}+a^{11}z^{-1}-z^{10}a^{10}-14z^{8}a^{10}+30z^{6}a^{10}-20z^{4}a^{10}+3z^{2}a^{10}-4z^{9}a^{9}-5z^{7}a^{9}+19z^{5}a^{9}-18z^{3}a^{9}+8za^{9}-2a^{9}z^{-1}-6z^{8}a^{8}+4z^{6}a^{8}+5z^{4}a^{8}-8z^{2}a^{8}+3a^{8}-6z^{7}a^{7}+11z^{5}a^{7}-12z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 7 3     -4 -12               11 5       6 -14             10 8         -2 -16           10 10           0 -18         7 10             3 -20       6 10               -4 -22     2 7                 5 -24   1 6                   -5 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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). Back to the top.

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