L11a334

L11a333

L11a335

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a334 at Knotilus!

Link Presentations

[edit Notes on L11a334's Link Presentations]

Planar diagram presentation	X10,1,11,2 X20,6,21,5 X14,3,15,4 X4,15,5,16 X22,20,9,19 X16,7,17,8 X18,12,19,11 X12,18,13,17 X2,9,3,10 X8,13,1,14 X6,22,7,21
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 $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-4t(2)^{3}t(1)^{2}+11t(2)^{2}t(1)^{2}-11t(2)t(1)^{2}+3t(1)^{2}+3t(2)^{3}t(1)-11t(2)^{2}t(1)+11t(2)t(1)-4t(1)+2t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {8}{q^{9/2}}}-4q^{7/2}+{\frac {14}{q^{7/2}}}+9q^{5/2}-{\frac {20}{q^{5/2}}}-15q^{3/2}+{\frac {22}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+20{\sqrt {q}}-{\frac {23}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+4az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-5a^{3}z+2az-2za^{-1}+za^{-3}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-6a^{3}z^{9}-12az^{9}-6z^{9}a^{-1}-8a^{4}z^{8}-14a^{2}z^{8}-7z^{8}a^{-2}-13z^{8}-6a^{5}z^{7}+15az^{7}+5z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+12a^{4}z^{6}+35a^{2}z^{6}+15z^{6}a^{-2}-z^{6}a^{-4}+36z^{6}-a^{7}z^{5}+9a^{5}z^{5}+16a^{3}z^{5}+7az^{5}+10z^{5}a^{-1}+9z^{5}a^{-3}+4a^{6}z^{4}-10a^{4}z^{4}-27a^{2}z^{4}-9z^{4}a^{-2}+2z^{4}a^{-4}-24z^{4}+2a^{7}z^{3}-8a^{5}z^{3}-21a^{3}z^{3}-13az^{3}-8z^{3}a^{-1}-6z^{3}a^{-3}-a^{6}z^{2}+2a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+6z^{2}-a^{7}z+5a^{5}z+9a^{3}z+4az+2za^{-1}+za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 9 3     6 2               11 6       -5 0             12 9         3 -2           11 12           1 -4         9 11             -2 -6       5 11               6 -8     3 9                 -6 -10   1 6                   5 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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