L11a30

L11a29

L11a31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a30 at Knotilus!

Link Presentations

[edit Notes on L11a30's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-9t(1)t(2)^{3}+13t(2)^{3}+13t(1)t(2)^{2}-9t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+3q^{5/2}-{\frac {20}{q^{5/2}}}-7q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+12{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}-7a^{3}z-4a^{3}z^{-1}-za^{-3}-az^{5}+2az^{3}+2z^{3}a^{-1}+4az+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-8a^{3}z^{9}-4az^{9}-6a^{6}z^{8}-15a^{4}z^{8}-15a^{2}z^{8}-6z^{8}-4a^{7}z^{7}-4a^{5}z^{7}+a^{3}z^{7}-4az^{7}-5z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+42a^{4}z^{6}+40a^{2}z^{6}-3z^{6}a^{-2}+8z^{6}+10a^{7}z^{5}+33a^{5}z^{5}+37a^{3}z^{5}+22az^{5}+7z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-4a^{6}z^{4}-35a^{4}z^{4}-41a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}-8a^{7}z^{3}-36a^{5}z^{3}-52a^{3}z^{3}-30az^{3}-4z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}+12a^{4}z^{2}+17a^{2}z^{2}-2z^{2}a^{-2}+5z^{2}+3a^{7}z+16a^{5}z+25a^{3}z+16az+3za^{-1}-za^{-3}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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 3 4 χ 8                       1 1 6                     2   -2 4                   5 1   4 2                 7 2     -5 0               10 5       5 -2             11 8         -3 -4           9 9           0 -6         8 11             3 -8       6 9               -3 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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