L11a14

L11a13

L11a15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a14 at Knotilus!

Link Presentations

[edit Notes on L11a14's Link Presentations]

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

A Braid Representative
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	$22q^{9/2}-22q^{7/2}+17q^{5/2}-13q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+5q^{17/2}-10q^{15/2}+16q^{13/2}-20q^{11/2}+6{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-z^{3}a^{-7}+2za^{-7}+2a^{-7}z^{-1}+z^{7}a^{-5}+2z^{5}a^{-5}-3z^{3}a^{-5}-9za^{-5}-5a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+9za^{-3}+3a^{-3}z^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-2za^{-1}$ (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} +5 z^6 a^{-10} -5 z^4 a^{-10} + a^{-10} +10 z^7 a^{-9} -14 z^5 a^{-9} +3 z^3 a^{-9} +11 z^8 a^{-8} -14 z^6 a^{-8} +z^4 a^{-8} +7 z^9 a^{-7} -z^7 a^{-7} -12 z^5 a^{-7} +6 z^3 a^{-7} -3 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +13 z^8 a^{-6} -28 z^6 a^{-6} +12 z^4 a^{-6} +6 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -17 z^7 a^{-5} +2 z^5 a^{-5} +14 z^3 a^{-5} -13 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +6 z^8 a^{-4} -17 z^6 a^{-4} +9 z^4 a^{-4} +7 z^2 a^{-4} -5 a^{-4} +4 z^9 a^{-3} -3 z^7 a^{-3} -10 z^5 a^{-3} +20 z^3 a^{-3} -14 z a^{-3} +3 a^{-3} z^{-1} +4 z^8 a^{-2} -7 z^6 a^{-2} +3 z^2 a^{-2} +3 z^7 a^{-1} -9 z^5 a^{-1} +9 z^3 a^{-1} -4 z a^{-1} +z^6-3 z^4+2 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                     4   -4 16                   6 1   5 14                 10 4     -6 12               10 6       4 10             12 10         -2 8           10 10           0 6         7 12             5 4       6 10               -4 2     2 9                 7 0   1 4                   -3 -2   2                     2 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ 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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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