L11a325

L11a324

L11a326

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a325 at Knotilus!

Link Presentations

[edit Notes on L11a325's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,14,19,13 X14,8,15,7 X20,5,21,6 X4,19,5,20 X6,21,7,22 X16,9,17,10 X22,15,9,16 X8,18,1,17
Gauss code	{1, -2, 3, -7, 6, -8, 5, -11}, {9, -1, 2, -3, 4, -5, 10, -9, 11, -4, 7, -6, 8, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-2t(1)^{3}t(2)^{4}+4t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+2t(1)^{3}t(2)^{3}-4t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+4t(1)^{2}t(2)^{2}-4t(1)t(2)^{2}+2t(2)^{2}-2t(1)^{2}t(2)+4t(1)t(2)-2t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {11}{q^{13/2}}}+{\frac {9}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+5a^{7}z^{5}+8a^{7}z^{3}+5a^{7}z-a^{5}z^{9}-7a^{5}z^{7}-18a^{5}z^{5}-21a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+a^{3}z^{7}+5a^{3}z^{5}+7a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}-3z^{4}a^{12}-6z^{5}a^{11}+4z^{3}a^{11}-za^{11}-9z^{6}a^{10}+13z^{4}a^{10}-5z^{2}a^{10}-10z^{7}a^{9}+19z^{5}a^{9}-7z^{3}a^{9}+za^{9}-9z^{8}a^{8}+21z^{6}a^{8}-10z^{4}a^{8}+4z^{2}a^{8}-6z^{9}a^{7}+14z^{7}a^{7}-2z^{5}a^{7}-z^{3}a^{7}-2za^{7}-2z^{10}a^{6}-3z^{8}a^{6}+33z^{6}a^{6}-43z^{4}a^{6}+16z^{2}a^{6}-9z^{9}a^{5}+40z^{7}a^{5}-55z^{5}a^{5}+29z^{3}a^{5}-7za^{5}-a^{5}z^{-1}-2z^{10}a^{4}+5z^{8}a^{4}+8z^{6}a^{4}-25z^{4}a^{4}+11z^{2}a^{4}+a^{4}-3z^{9}a^{3}+16z^{7}a^{3}-28z^{5}a^{3}+18z^{3}a^{3}-3za^{3}-a^{3}z^{-1}-z^{8}a^{2}+5z^{6}a^{2}-8z^{4}a^{2}+4z^{2}a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     2   2 -2                   3 1   -2 -4                 6 2     4 -6               5 4       -1 -8             8 5         3 -10           5 5           0 -12         6 8             -2 -14       4 6               2 -16     2 5                 -3 -18   1 4                   3 -20   2                     -2 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a324 L11a326