L11n124

L11n123

L11n125

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n124 at Knotilus!

Link Presentations

[edit Notes on L11n124's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X9,22,10,5 X7,19,8,18 X17,9,18,8 X19,13,20,12 X11,21,12,20 X15,10,16,11 X21,16,22,17 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 5, -3, 8, -7, 6, -11, 2, -8, 9, -5, 4, -6, 7, -9, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$3q^{9/2}-6q^{7/2}+{\frac {1}{q^{7/2}}}+8q^{5/2}-{\frac {4}{q^{5/2}}}-9q^{3/2}+{\frac {6}{q^{3/2}}}-q^{11/2}+9{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-1}+3az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-a^{3}z+5az-9za^{-1}+6za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-10z^{8}a^{-2}-3z^{8}a^{-4}-7z^{8}-7az^{7}-5z^{7}a^{-1}+z^{7}a^{-3}-z^{7}a^{-5}-2a^{2}z^{6}+39z^{6}a^{-2}+12z^{6}a^{-4}+25z^{6}+25az^{5}+42z^{5}a^{-1}+21z^{5}a^{-3}+4z^{5}a^{-5}+a^{2}z^{4}-42z^{4}a^{-2}-14z^{4}a^{-4}-27z^{4}-4a^{3}z^{3}-32az^{3}-56z^{3}a^{-1}-34z^{3}a^{-3}-6z^{3}a^{-5}-a^{4}z^{2}+17z^{2}a^{-2}+6z^{2}a^{-4}+12z^{2}+3a^{3}z+15az+26za^{-1}+18za^{-3}+4za^{-5}-a^{2}-3a^{-2}-a^{-4}-2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 12                   1 1 10                 2   -2 8               4 1   3 6             4 2     -2 4           5 4       1 2       1 5 4         0 0       5 5           0 -2     3 6             3 -4   1 3               -2 -6   3                 3 -8 1                   -1

Integral Khovanov Homology (db, data source)

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

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). Back to the top.

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