L11a141

L11a140

L11a142

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a141 at Knotilus!

Link Presentations

[edit Notes on L11a141's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code	{1, -4, 3, -10, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -11, 9, -7, 4, -2, 10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-3t(1)^{2}t(2)^{5}+4t(1)t(2)^{5}-t(2)^{5}+4t(1)^{2}t(2)^{4}-8t(1)t(2)^{4}+3t(2)^{4}-4t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+3t(1)^{2}t(2)^{2}-8t(1)t(2)^{2}+4t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+14q^{7/2}-19q^{5/2}+21q^{3/2}-22{\sqrt {q}}+{\frac {18}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+6z^{7}a^{-1}-z^{7}a^{-3}-4az^{5}+12z^{5}a^{-1}-4z^{5}a^{-3}-4az^{3}+7z^{3}a^{-1}-4z^{3}a^{-3}+az-3za^{-1}+za^{-3}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-7az^{9}-14z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-5z^{8}a^{-2}-8z^{8}a^{-4}-4z^{8}-4a^{3}z^{7}+16az^{7}+36z^{7}a^{-1}+9z^{7}a^{-3}-7z^{7}a^{-5}-a^{4}z^{6}+18a^{2}z^{6}+16z^{6}a^{-2}+9z^{6}a^{-4}-4z^{6}a^{-6}+22z^{6}+10a^{3}z^{5}-11az^{5}-38z^{5}a^{-1}-6z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-12a^{2}z^{4}-10z^{4}a^{-2}+2z^{4}a^{-4}+6z^{4}a^{-6}-20z^{4}-5a^{3}z^{3}+2az^{3}+14z^{3}a^{-1}+3z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+2a^{2}z^{2}-3z^{2}a^{-2}-5z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}+3az+4za^{-1}+za^{-3}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 9 3     -6 6               10 5       5 4             11 9         -2 2           11 10           1 0         8 12             4 -2       6 10               -4 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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