L11n41

L11n40

L11n42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n41 at Knotilus!

Link Presentations

[edit Notes on L11n41's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X8493 X5,14,6,15 X15,20,16,21 X17,22,18,5 X21,16,22,17 X19,10,20,11 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, 11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+6za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{12}+a^{12}-2z^{3}a^{11}+za^{11}-z^{6}a^{10}+z^{4}a^{10}-2z^{2}a^{10}-3z^{7}a^{9}+10z^{5}a^{9}-12z^{3}a^{9}+4za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+9z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-3a^{8}-z^{9}a^{7}-3z^{7}a^{7}+23z^{5}a^{7}-27z^{3}a^{7}+8za^{7}+a^{7}z^{-1}-5z^{8}a^{6}+18z^{6}a^{6}-14z^{4}a^{6}+2z^{2}a^{6}-z^{9}a^{5}-z^{7}a^{5}+18z^{5}a^{5}-26z^{3}a^{5}+12za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}-2z^{2}a^{4}+3a^{4}-z^{7}a^{3}+5z^{5}a^{3}-9z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               4 1   3 -6             3 3     0 -8           4 2       2 -10       1 3 3         1 -12       4 4           0 -14     2 3             1 -16   1 4               -3 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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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