L11a161

L11a160

L11a162

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a161 at Knotilus!

Link Presentations

[edit Notes on L11a161's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+9uv^{3}-13uv^{2}+9uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-8q^{9/2}+{\frac {1}{q^{9/2}}}+14q^{7/2}-{\frac {4}{q^{7/2}}}-19q^{5/2}+{\frac {8}{q^{5/2}}}+21q^{3/2}-{\frac {14}{q^{3/2}}}-q^{13/2}+4q^{11/2}-22{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}+2z^{5}a^{-3}-a^{3}z^{3}+5z^{3}a^{-3}-a^{3}z+5za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-9z^{3}a^{-1}+5az-9za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-4az^{9}-8z^{9}a^{-1}-4z^{9}a^{-3}-6a^{2}z^{8}-16z^{8}a^{-2}-7z^{8}a^{-4}-15z^{8}-4a^{3}z^{7}-4az^{7}-4z^{7}a^{-1}-11z^{7}a^{-3}-7z^{7}a^{-5}-a^{4}z^{6}+12a^{2}z^{6}+28z^{6}a^{-2}+3z^{6}a^{-4}-4z^{6}a^{-6}+34z^{6}+10a^{3}z^{5}+28az^{5}+40z^{5}a^{-1}+33z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-5a^{2}z^{4}-7z^{4}a^{-2}+9z^{4}a^{-4}+6z^{4}a^{-6}-17z^{4}-8a^{3}z^{3}-28az^{3}-40z^{3}a^{-1}-27z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}-a^{4}z^{2}-a^{2}z^{2}-7z^{2}a^{-2}-8z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}+2a^{3}z+11az+16za^{-1}+9za^{-3}+2za^{-5}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 9 3     -6 6               10 5       5 4             11 9         -2 2           11 10           1 0         8 12             4 -2       6 10               -4 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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