L11a36

L11a35

L11a37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a36 at Knotilus!

Link Presentations

[edit Notes on L11a36's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3uv^{4}-6uv^{3}+7uv^{2}-5uv+2u+2v^{5}-5v^{4}+7v^{3}-6v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {7}{q^{23/2}}}-{\frac {10}{q^{21/2}}}+{\frac {13}{q^{19/2}}}-{\frac {15}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {8}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+3z^{3}a^{11}+5za^{11}+a^{11}z^{-1}-2z^{5}a^{9}-4z^{3}a^{9}+2a^{9}z^{-1}-2z^{5}a^{7}-5z^{3}a^{7}-4za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-2za^{5}$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{16} z^6-3 a^{16} z^4+3 a^{16} z^2-a^{16}+3 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3-a^{15} z+4 a^{14} z^8-8 a^{14} z^6+a^{14} z^2+3 a^{13} z^9-2 a^{13} z^7-9 a^{13} z^5+6 a^{13} z^3-a^{13} z^{-1} +a^{12} z^{10}+6 a^{12} z^8-21 a^{12} z^6+22 a^{12} z^4-12 a^{12} z^2+3 a^{12}+6 a^{11} z^9-13 a^{11} z^7+10 a^{11} z^5-a^{11} z^3+a^{11} z-a^{11} z^{-1} +a^{10} z^{10}+5 a^{10} z^8-17 a^{10} z^6+21 a^{10} z^4-5 a^{10} z^2+3 a^9 z^9-5 a^9 z^7+4 a^9 z^5+7 a^9 z^3-8 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-3 a^8 z^6-2 a^8 z^4+6 a^8 z^2-3 a^8+3 a^7 z^7-6 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+a^6 z^2+a^5 z^5-3 a^5 z^3+2 a^5 z} (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     2 1 -1 -8                   3     3 -10                 5 2     -3 -12               8 3       5 -14             7 6         -1 -16           8 7           1 -18         5 7             2 -20       5 8               -3 -22     2 5                 3 -24   1 5                   -4 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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