L11a35

L11a34

L11a36

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a35 at Knotilus!

Link Presentations

[edit Notes on L11a35's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-7t(1)t(2)^{3}+9t(2)^{3}+9t(1)t(2)^{2}-7t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {3}{q^{9/2}}}+6q^{7/2}-{\frac {7}{q^{7/2}}}-10q^{5/2}+{\frac {10}{q^{5/2}}}+14q^{3/2}-{\frac {14}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-16{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+za^{-5}-3a^{3}z^{3}-2z^{3}a^{-3}-5a^{3}z-2a^{3}z^{-1}-za^{-3}+a^{-3}z^{-1}+2az^{5}+z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+6az+3az^{-1}-4za^{-1}-3a^{-1}z^{-1}$ (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 z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-4 a^5 z^5+3 z^5 a^{-5} +6 a^5 z^3-3 z^3 a^{-5} -4 a^5 z+z a^{-5} +a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-5 z^4 a^{-4} -4 a^4 z^2+3 z^2 a^{-4} - a^{-4} +3 a^3 z^9-5 a^3 z^7+6 z^7 a^{-3} -11 a^3 z^5-5 z^5 a^{-3} +22 a^3 z^3+z^3 a^{-3} -11 a^3 z-z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -36 a^2 z^6-5 z^6 a^{-2} +38 a^2 z^4-5 z^4 a^{-2} -14 a^2 z^2+8 z^2 a^{-2} +2 a^2-2 a^{-2} +7 a z^9+4 z^9 a^{-1} -11 a z^7+z^7 a^{-1} -18 a z^5-19 z^5 a^{-1} +35 a z^3+23 z^3 a^{-1} -18 a z-13 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +z^{10}+11 z^8-35 z^6+27 z^4-6 z^2} (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   4 1   -3 6                 6 2     4 4               8 4       -4 2             8 6         2 0           8 9           1 -2         6 7             -1 -4       4 8               4 -6     3 6                 -3 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a34 L11a36