L11a6

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a6 at Knotilus!

Link Presentations

[edit Notes on L11a6's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X20,12,21,11 X22,18,5,17 X18,22,19,21 X10,20,11,19 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -10, 7, -4, 11, -2, 3, -6, 8, -9, 10, -7, 9, -8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(2v^{4}-4v^{3}+5v^{2}-4v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$22q^{9/2}-22q^{7/2}+18q^{5/2}-15q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-8q^{15/2}+14q^{13/2}-19q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-2z^{3}a^{-7}+a^{-7}z^{-1}+z^{7}a^{-5}+3z^{5}a^{-5}+2z^{3}a^{-5}-2za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+3z^{3}a^{-3}+2za^{-3}-z^{5}a^{-1}-2z^{3}a^{-1}+a^{-1}z^{-1}$ (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^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -6 z^4 a^{-10} +2 z^2 a^{-10} +7 z^7 a^{-9} -10 z^5 a^{-9} +4 z^3 a^{-9} +8 z^8 a^{-8} -9 z^6 a^{-8} +5 z^2 a^{-8} -2 a^{-8} +6 z^9 a^{-7} -2 z^7 a^{-7} -6 z^5 a^{-7} +4 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +11 z^8 a^{-6} -22 z^6 a^{-6} +6 z^4 a^{-6} +9 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -15 z^7 a^{-5} +z^5 a^{-5} +3 z^3 a^{-5} -3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +9 z^8 a^{-4} -21 z^6 a^{-4} +4 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -2 z^7 a^{-3} -14 z^5 a^{-3} +12 z^3 a^{-3} -4 z a^{-3} +6 z^8 a^{-2} -11 z^6 a^{-2} +2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -10 z^5 a^{-1} +8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4+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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   5 1   4 14                 9 3     -6 12               10 5       5 10             12 9         -3 8           10 10           0 6         8 12             4 4       7 10               -3 2     3 10                 7 0   1 5                   -4 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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