L11a152

L11a151

L11a153

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a152 at Knotilus!

Link Presentations

[edit Notes on L11a152's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3u^{2}v^{3}-5u^{2}v^{2}+4u^{2}v-u^{2}+3uv^{4}-8uv^{3}+11uv^{2}-8uv+3u-v^{4}+4v^{3}-5v^{2}+3v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {7}{q^{23/2}}}-{\frac {12}{q^{21/2}}}+{\frac {16}{q^{19/2}}}-{\frac {19}{q^{17/2}}}+{\frac {19}{q^{15/2}}}-{\frac {17}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+3a^{11}z^{3}+5a^{11}z+2a^{11}z^{-1}-2a^{9}z^{5}-3a^{9}z^{3}-3a^{7}z^{5}-8a^{7}z^{3}-6a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$a^{16}z^{6}-3a^{16}z^{4}+2a^{16}z^{2}+3a^{15}z^{7}-8a^{15}z^{5}+6a^{15}z^{3}-2a^{15}z+5a^{14}z^{8}-13a^{14}z^{6}+12a^{14}z^{4}-7a^{14}z^{2}+2a^{14}+4a^{13}z^{9}-4a^{13}z^{7}-6a^{13}z^{5}+7a^{13}z^{3}-a^{13}z-a^{13}z^{-1}+a^{12}z^{10}+12a^{12}z^{8}-38a^{12}z^{6}+44a^{12}z^{4}-25a^{12}z^{2}+5a^{12}+8a^{11}z^{9}-10a^{11}z^{7}-a^{11}z^{3}+6a^{11}z-2a^{11}z^{-1}+a^{10}z^{10}+13a^{10}z^{8}-32a^{10}z^{6}+30a^{10}z^{4}-13a^{10}z^{2}+3a^{10}+4a^{9}z^{9}+3a^{9}z^{7}-14a^{9}z^{5}+11a^{9}z^{3}-3a^{9}z+6a^{8}z^{8}-5a^{8}z^{6}-3a^{8}z^{4}+4a^{8}z^{2}-a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 7 3     -4 -12               10 5       5 -14             9 7         -2 -16           10 10           0 -18         7 10             3 -20       5 9               -4 -22     2 7                 5 -24   1 5                   -4 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a151 L11a153