L11a170

L11a169

L11a171

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a170 at Knotilus!

Link Presentations

[edit Notes on L11a170's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, 4, -7, 10, -8, 11, -9, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-6u^{2}v+2u^{2}-3uv^{4}+11uv^{3}-15uv^{2}+11uv-3u+2v^{4}-6v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {10}{q^{9/2}}}-5q^{7/2}+{\frac {18}{q^{7/2}}}+11q^{5/2}-{\frac {24}{q^{5/2}}}-19q^{3/2}+{\frac {28}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+24{\sqrt {q}}-{\frac {29}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+2az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+az^{3}-2z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-a^{3}z-az+2za^{-1}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+9a^{5}z^{7}-12a^{5}z^{5}+7a^{5}z^{3}-a^{5}z+13a^{4}z^{8}-21a^{4}z^{6}+z^{6}a^{-4}+17a^{4}z^{4}-z^{4}a^{-4}-6a^{4}z^{2}+11a^{3}z^{9}-11a^{3}z^{7}+5z^{7}a^{-3}-9z^{5}a^{-3}+2a^{3}z^{3}+4z^{3}a^{-3}+a^{3}z+za^{-3}-a^{-3}z^{-1}+4a^{2}z^{10}+17a^{2}z^{8}+10z^{8}a^{-2}-48a^{2}z^{6}-20z^{6}a^{-2}+38a^{2}z^{4}+10z^{4}a^{-2}-12a^{2}z^{2}-z^{2}a^{-2}+a^{-2}+21az^{9}+10z^{9}a^{-1}-39az^{7}-14z^{7}a^{-1}+21az^{5}-z^{5}a^{-1}-10az^{3}+5az+4za^{-1}-a^{-1}z^{-1}+4z^{10}+14z^{8}-44z^{6}+28z^{4}-6z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   7 1   -6 4                 12 4     8 2               13 8       -5 0             16 11         5 -2           13 14           1 -4         11 15             -4 -6       7 13               6 -8     3 11                 -8 -10   1 7                   6 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{16}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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