L11a197

L11a196

L11a198

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a197 at Knotilus!

Link Presentations

[edit Notes on L11a197's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}+4t(1)^{2}t(2)^{3}-10t(1)t(2)^{3}+5t(2)^{3}-8t(1)^{2}t(2)^{2}+17t(1)t(2)^{2}-8t(2)^{2}+5t(1)^{2}t(2)-10t(1)t(2)+4t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{21/2}-4q^{19/2}+9q^{17/2}-15q^{15/2}+21q^{13/2}-24q^{11/2}+24q^{9/2}-22q^{7/2}+15q^{5/2}-10q^{3/2}+4{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-3z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}+2z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+z^{3}a^{-9}+5za^{-3}-6za^{-5}+2za^{-7}+2a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-9z^{5}a^{-11}+6z^{3}a^{-11}-za^{-11}+7z^{8}a^{-10}-15z^{6}a^{-10}+10z^{4}a^{-10}-3z^{2}a^{-10}+6z^{9}a^{-9}-4z^{7}a^{-9}-12z^{5}a^{-9}+11z^{3}a^{-9}-3za^{-9}+2z^{10}a^{-8}+15z^{8}a^{-8}-41z^{6}a^{-8}+29z^{4}a^{-8}-6z^{2}a^{-8}-a^{-8}+13z^{9}a^{-7}-13z^{7}a^{-7}-18z^{5}a^{-7}+25z^{3}a^{-7}-8za^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}+19z^{8}a^{-6}-46z^{6}a^{-6}+31z^{4}a^{-6}-2z^{2}a^{-6}-3a^{-6}+7z^{9}a^{-5}+4z^{7}a^{-5}-31z^{5}a^{-5}+33z^{3}a^{-5}-13za^{-5}+3a^{-5}z^{-1}+11z^{8}a^{-4}-17z^{6}a^{-4}+10z^{4}a^{-4}-3a^{-4}+9z^{7}a^{-3}-15z^{5}a^{-3}+12z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}+4z^{6}a^{-2}-4z^{4}a^{-2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   6 1   -5 16                 9 3     6 14               12 6       -6 12             12 9         3 10           12 12           0 8         10 12             -2 6       6 13               7 4     4 9                 -5 2   1 7                   6 0   3                     -3 -2 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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