L11a41

L11a40

L11a42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a41 at Knotilus!

Link Presentations

[edit Notes on L11a41's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X22,15,5,16 X16,7,17,8 X20,18,21,17 X14,10,15,9 X12,19,13,20 X18,13,19,14 X8,21,9,22 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 6, -2, 11, -7, 8, -6, 3, -4, 5, -8, 7, -5, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {15}{q^{9/2}}}-q^{7/2}+{\frac {21}{q^{7/2}}}+5q^{5/2}-{\frac {26}{q^{5/2}}}-12q^{3/2}+{\frac {26}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+18{\sqrt {q}}-{\frac {24}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-3a^{3}z^{5}-7a^{3}z^{3}-7a^{3}z-3a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+4az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+6a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-8a^{6}z^{6}+2a^{6}z^{4}+3a^{6}z^{2}-2a^{6}+5a^{5}z^{9}-a^{5}z^{7}-15a^{5}z^{5}+21a^{5}z^{3}-12a^{5}z+3a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-40a^{4}z^{6}+32a^{4}z^{4}-8a^{4}z^{2}+14a^{3}z^{9}-12a^{3}z^{7}-23a^{3}z^{5}+z^{5}a^{-3}+33a^{3}z^{3}-16a^{3}z+3a^{3}z^{-1}+2a^{2}z^{10}+25a^{2}z^{8}-58a^{2}z^{6}+5z^{6}a^{-2}+39a^{2}z^{4}-3z^{4}a^{-2}-11a^{2}z^{2}+2a^{2}+9az^{9}+4az^{7}+12z^{7}a^{-1}-32az^{5}-16z^{5}a^{-1}+24az^{3}+6z^{3}a^{-1}-9az-2za^{-1}+2az^{-1}+a^{-1}z^{-1}+15z^{8}-22z^{6}+9z^{4}-3z^{2}$ (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                     4   -4 4                   8 1   7 2                 10 4     -6 0               14 8       6 -2             14 12         -2 -4           12 12           0 -6         9 14             5 -8       6 12               -6 -10     2 9                 7 -12   1 6                   -5 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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