L11a198

L11a197

L11a199

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a198 at Knotilus!

Link Presentations

[edit Notes on L11a198's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-4t(2)^{3}+8t(1)^{2}t(2)^{2}-15t(1)t(2)^{2}+8t(2)^{2}-4t(1)^{2}t(2)+8t(1)t(2)-5t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-15q^{5/2}+19q^{3/2}-23{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {19}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-4a^{3}z+8az-6za^{-1}+2za^{-3}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-3a^{5}z^{5}+4z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}-a^{5}z+4a^{4}z^{8}-14a^{4}z^{6}+9z^{6}a^{-4}+16a^{4}z^{4}-7z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+5a^{3}z^{9}-12a^{3}z^{7}+14z^{7}a^{-3}+2a^{3}z^{5}-20z^{5}a^{-3}+9a^{3}z^{3}+13z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{3}z^{-1}+2a^{2}z^{10}+10a^{2}z^{8}+14z^{8}a^{-2}-46a^{2}z^{6}-20z^{6}a^{-2}+45a^{2}z^{4}+6z^{4}a^{-2}-9a^{2}z^{2}-3a^{2}+13az^{9}+8z^{9}a^{-1}-24az^{7}+3z^{7}a^{-1}-10az^{5}-39z^{5}a^{-1}+27az^{3}+35z^{3}a^{-1}-11az-12za^{-1}+3az^{-1}+2a^{-1}z^{-1}+2z^{10}+20z^{8}-61z^{6}+43z^{4}-6z^{2}-3$ (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 12                       1 -1 10                     3   3 8                   6 1   -5 6                 9 3     6 4               11 7       -4 2             12 8         4 0           10 12           2 -2         9 11             -2 -4       5 10               5 -6     3 9                 -6 -8   1 5                   4 -10   3                     -3 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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