L11a234

L11a233

L11a235

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a234 at Knotilus!

Link Presentations

[edit Notes on L11a234's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X18,6,19,5 X22,12,7,11 X20,17,21,18 X16,21,17,22 X14,10,15,9 X10,16,11,15 X4,20,5,19 X2738 X6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 7, -8, 4, -2, 11, -7, 8, -6, 5, -3, 9, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-3t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-4t(2)^{3}+4t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+4t(2)^{2}-4t(1)^{2}t(2)+8t(1)t(2)-3t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+16q^{9/2}-19q^{7/2}+18q^{5/2}-17q^{3/2}+12{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-7za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+8za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+az-3za^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-2z^{3}a^{-9}+6z^{6}a^{-8}-6z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+9z^{7}a^{-7}-14z^{5}a^{-7}+13z^{3}a^{-7}-5za^{-7}+a^{-7}z^{-1}+9z^{8}a^{-6}-12z^{6}a^{-6}+z^{4}a^{-6}+7z^{2}a^{-6}-3a^{-6}+6z^{9}a^{-5}-z^{7}a^{-5}-23z^{5}a^{-5}+27z^{3}a^{-5}-13za^{-5}+3a^{-5}z^{-1}+2z^{10}a^{-4}+11z^{8}a^{-4}-38z^{6}a^{-4}+26z^{4}a^{-4}-2z^{2}a^{-4}-3a^{-4}+11z^{9}a^{-3}-25z^{7}a^{-3}+3z^{5}a^{-3}+15z^{3}a^{-3}-10za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+6z^{8}a^{-2}-34z^{6}a^{-2}+32z^{4}a^{-2}-7z^{2}a^{-2}+5z^{9}a^{-1}+az^{7}-14z^{7}a^{-1}-3az^{5}+6z^{5}a^{-1}+3az^{3}+6z^{3}a^{-1}-az-3za^{-1}+4z^{8}-14z^{6}+14z^{4}-3z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               9 5       -4 8             10 7         3 6           9 10           1 4         8 9             -1 2       5 10               5 0     3 7                 -4 -2   1 5                   4 -4   3                     -3 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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