L11a438

L11a437

L11a439

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a438 at Knotilus!

Link Presentations

[edit Notes on L11a438's Link Presentations]

Planar diagram presentation	X6172 X2,16,3,15 X10,4,11,3 X14,6,15,5 X22,12,13,11 X12,14,5,13 X4,21,1,22 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9
Gauss code	{1, -2, 3, -7}, {4, -1, 9, -10, 11, -3, 5, -6}, {6, -4, 2, -9, 8, -11, 10, -8, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)^{2}(t(3)-1)^{2}(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+14q^{5}-17q^{4}+21q^{3}-20q^{2}+18q-12+8q^{-1}-4q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-9z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-6z^{2}a^{-2}+6z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}-a^{-4}+1+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+7z^{7}a^{-7}-10z^{5}a^{-7}+2z^{3}a^{-7}+8z^{8}a^{-6}-11z^{6}a^{-6}+3z^{4}a^{-6}+z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+6z^{9}a^{-5}-5z^{7}a^{-5}-z^{5}a^{-5}-z^{3}a^{-5}+4za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+11z^{8}a^{-4}-35z^{6}a^{-4}+37z^{4}a^{-4}-11z^{2}a^{-4}+2a^{-4}z^{-2}-4a^{-4}+12z^{9}a^{-3}-28z^{7}a^{-3}+24z^{5}a^{-3}-7z^{3}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+10z^{8}a^{-2}+a^{2}z^{6}-41z^{6}a^{-2}-2a^{2}z^{4}+47z^{4}a^{-2}-16z^{2}a^{-2}+a^{-2}z^{-2}-a^{-2}+6z^{9}a^{-1}+4az^{7}-12z^{7}a^{-1}-10az^{5}+4z^{5}a^{-1}+4az^{3}+z^{3}a^{-1}+7z^{8}-20z^{6}+17z^{4}-6z^{2}+1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   5 1   -4 11                 9 3     6 9               10 7       -3 7             11 7         4 5           9 10           1 3         9 11             -2 1       5 11               6 -1     3 7                 -4 -3   1 5                   4 -5   3                     -3 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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