L11a352

L11a351

L11a353

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a352 at Knotilus!

Link Presentations

[edit Notes on L11a352's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X14,5,15,6 X20,7,21,8 X16,13,17,14 X18,10,19,9 X10,11,1,12 X6,15,7,16 X4,21,5,22 X22,17,11,18 X2,20,3,19
Gauss code	{1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{3}v^{3}-4u^{3}v^{2}+3u^{3}v-u^{3}-5u^{2}v^{3}+13u^{2}v^{2}-11u^{2}v+4u^{2}+4uv^{3}-11uv^{2}+13uv-5u-v^{3}+3v^{2}-4v+2}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {28}{q^{9/2}}}-{\frac {28}{q^{7/2}}}+{\frac {23}{q^{5/2}}}+q^{3/2}-{\frac {17}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {19}{q^{13/2}}}-{\frac {25}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+z^{3}a^{7}-z^{7}a^{5}-2z^{5}a^{5}+2za^{5}+a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-5z^{3}a^{3}-5za^{3}-a^{3}z^{-1}+z^{5}a+2z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}+5a^{10}z^{6}-3a^{10}z^{4}+12a^{9}z^{7}-15a^{9}z^{5}+5a^{9}z^{3}-a^{9}z+16a^{8}z^{8}-24a^{8}z^{6}+11a^{8}z^{4}-3a^{8}z^{2}+11a^{7}z^{9}-3a^{7}z^{7}-20a^{7}z^{5}+13a^{7}z^{3}-2a^{7}z+3a^{6}z^{10}+24a^{6}z^{8}-58a^{6}z^{6}+36a^{6}z^{4}-6a^{6}z^{2}+18a^{5}z^{9}-24a^{5}z^{7}-8a^{5}z^{5}+19a^{5}z^{3}-7a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+15a^{4}z^{8}-43a^{4}z^{6}+31a^{4}z^{4}-5a^{4}z^{2}-a^{4}+7a^{3}z^{9}-5a^{3}z^{7}-13a^{3}z^{5}+18a^{3}z^{3}-8a^{3}z+a^{3}z^{-1}+7a^{2}z^{8}-13a^{2}z^{6}+7a^{2}z^{4}-a^{2}z^{2}+4az^{7}-9az^{5}+7az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   6 1   -5 -2                 11 3     8 -4               13 7       -6 -6             15 10         5 -8           13 13           0 -10         12 15             -3 -12       8 14               6 -14     4 11                 -7 -16   1 8                   7 -18   4                     -4 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-4$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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