L11a160

L11a159

L11a161

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a160 at Knotilus!

Link Presentations

[edit Notes on L11a160's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-8q^{3/2}+13{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+3a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-9za^{3}-2a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+4za-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+4a^{6}z^{8}-6a^{6}z^{6}-3a^{6}z^{4}+7a^{6}z^{2}-3a^{6}+3a^{5}z^{9}+3a^{5}z^{7}-21a^{5}z^{5}+23a^{5}z^{3}-12a^{5}z+3a^{5}z^{-1}+a^{4}z^{10}+10a^{4}z^{8}-22a^{4}z^{6}+9a^{4}z^{4}+4a^{4}z^{2}-3a^{4}+7a^{3}z^{9}-27a^{3}z^{5}+z^{5}a^{-3}+33a^{3}z^{3}-z^{3}a^{-3}-15a^{3}z+2a^{3}z^{-1}+a^{2}z^{10}+13a^{2}z^{8}-25a^{2}z^{6}+4z^{6}a^{-2}+11a^{2}z^{4}-6z^{4}a^{-2}+2a^{2}z^{2}+2z^{2}a^{-2}+4az^{9}+7az^{7}+7z^{7}a^{-1}-26az^{5}-11z^{5}a^{-1}+24az^{3}+6z^{3}a^{-1}-8az-2za^{-1}+7z^{8}-6z^{6}-4z^{4}+4z^{2}$ (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                   5 1   4 2                 8 3     -5 0               10 5       5 -2             10 9         -1 -4           10 9           1 -6         7 11             4 -8       5 9               -4 -10     2 7                 5 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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