L11a309

L11a308

L11a310

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a309 at Knotilus!

Link Presentations

[edit Notes on L11a309's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}-t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-t(2)^{2}+t(1)^{2}t(2)-3t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {24}{q^{3/2}}}-{\frac {25}{q^{5/2}}}+{\frac {21}{q^{7/2}}}-{\frac {16}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+3a^{3}z^{5}+4a^{3}z^{3}+4a^{3}z+3a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+3az^{3}-2z^{3}a^{-1}-az-za^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-8a^{7}z^{5}+4a^{7}z^{3}+8a^{6}z^{8}-18a^{6}z^{6}+13a^{6}z^{4}-6a^{6}z^{2}+a^{6}+8a^{5}z^{9}-14a^{5}z^{7}+7a^{5}z^{5}-7a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}+3a^{4}z^{10}+11a^{4}z^{8}-37a^{4}z^{6}+31a^{4}z^{4}-13a^{4}z^{2}+3a^{4}+16a^{3}z^{9}-31a^{3}z^{7}+20a^{3}z^{5}+z^{5}a^{-3}-10a^{3}z^{3}-z^{3}a^{-3}+7a^{3}z-3a^{3}z^{-1}+3a^{2}z^{10}+13a^{2}z^{8}-36a^{2}z^{6}+4z^{6}a^{-2}+28a^{2}z^{4}-5z^{4}a^{-2}-8a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+8az^{9}-5az^{7}+8z^{7}a^{-1}-8az^{5}-12z^{5}a^{-1}+9az^{3}+7z^{3}a^{-1}+az-2za^{-1}-2az^{-1}+10z^{8}-14z^{6}+7z^{4}-z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               13 6       7 -2             13 11         -2 -4           12 11           1 -6         9 13             4 -8       7 12               -5 -10     3 9                 6 -12   1 7                   -6 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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