L11a113

L11a112

L11a114

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a113 at Knotilus!

Link Presentations

[edit Notes on L11a113's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+6t(2)^{4}+8t(1)t(2)^{3}-10t(2)^{3}-10t(1)t(2)^{2}+8t(2)^{2}+6t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {13}{q^{9/2}}}-q^{7/2}+{\frac {16}{q^{7/2}}}+4q^{5/2}-{\frac {20}{q^{5/2}}}-8q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+13{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-10za^{3}-4a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+4za+az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-7a^{3}z^{9}-4az^{9}-4a^{6}z^{8}-10a^{4}z^{8}-13a^{2}z^{8}-7z^{8}-3a^{7}z^{7}-4a^{5}z^{7}-a^{3}z^{7}-7az^{7}-7z^{7}a^{-1}-a^{8}z^{6}+5a^{6}z^{6}+18a^{4}z^{6}+22a^{2}z^{6}-4z^{6}a^{-2}+6z^{6}+8a^{7}z^{5}+23a^{5}z^{5}+26a^{3}z^{5}+23az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+6a^{6}z^{4}+3a^{4}z^{4}-3a^{2}z^{4}+6z^{4}a^{-2}+3z^{4}-8a^{7}z^{3}-25a^{5}z^{3}-27a^{3}z^{3}-17az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-11a^{6}z^{2}-16a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}-4z^{2}+4a^{7}z+15a^{5}z+14a^{3}z+4az+za^{-1}+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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 3 4 χ 8                       1 1 6                     3   -3 4                   5 1   4 2                 8 3     -5 0               10 5       5 -2             11 9         -2 -4           9 9           0 -6         7 11             4 -8       6 9               -3 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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