L11n234

L11n233

L11n235

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n234 at Knotilus!

Link Presentations

[edit Notes on L11n234's Link Presentations]

Planar diagram presentation	X10,1,11,2 X5,20,6,21 X3,15,4,14 X15,5,16,4 X19,22,20,9 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X21,6,22,7
Gauss code	{1, -9, -3, 4, -2, 11, -6, 10}, {9, -1, -7, 8, -10, 3, -4, 6, -8, 7, -5, 2, -11, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+5t(1)t(2)-t(1)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}-q^{7/2}+{\frac {8}{q^{7/2}}}+3q^{5/2}-{\frac {12}{q^{5/2}}}-7q^{3/2}+{\frac {12}{q^{3/2}}}+{\frac {2}{q^{11/2}}}+10{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z-a^{3}z^{5}-2a^{3}z^{3}-2a^{3}z+az^{7}+5az^{5}-z^{5}a^{-1}+10az^{3}-3z^{3}a^{-1}+7az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-6a^{2}z^{8}-4z^{8}-a^{5}z^{7}-5a^{3}z^{7}-9az^{7}-5z^{7}a^{-1}+6a^{2}z^{6}-3z^{6}a^{-2}+3z^{6}-3a^{5}z^{5}+9a^{3}z^{5}+22az^{5}+9z^{5}a^{-1}-z^{5}a^{-3}-3a^{6}z^{4}+a^{2}z^{4}+5z^{4}a^{-2}+3z^{4}+6a^{5}z^{3}-5a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}+4a^{6}z^{2}+a^{4}z^{2}-2a^{2}z^{2}-2z^{2}a^{-2}-z^{2}-3a^{5}z+8az+4za^{-1}-za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 2   -2 4               5 1   4 2             5 2     -3 0           7 5       2 -2         7 7         0 -4       5 5           0 -6     3 7             4 -8   2 5               -3 -10   3                 3 -12 2                   -2

Integral Khovanov Homology (db, data source)

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

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). Back to the top.

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