L11a328

L11a327

L11a329

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a328 at Knotilus!

Link Presentations

[edit Notes on L11a328's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,5,21,6 X18,9,19,10 X22,19,9,20 X16,12,17,11 X6,21,7,22 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18
Gauss code	{1, -11, 2, -9, 3, -7, 8, -10}, {4, -1, 6, -2, 9, -8, 10, -6, 11, -4, 5, -3, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-2t(2)^{2}t(1)+4t(2)t(1)-2t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones 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 -3 q^{9/2}+\frac{4}{q^{9/2}}+6 q^{7/2}-\frac{8}{q^{7/2}}-11 q^{5/2}+\frac{12}{q^{5/2}}+15 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-18 \sqrt{q}+\frac{17}{\sqrt{q}}} (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+z^{5}a^{-3}+2a^{3}z^{3}+3z^{3}a^{-3}+2za^{-3}-az^{7}-z^{7}a^{-1}-3az^{5}-4z^{5}a^{-1}-az^{3}-6z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (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^4 a^{-6} -z^2 a^{-6} +a^5 z^7-3 a^5 z^5+3 z^5 a^{-5} +3 a^5 z^3-3 z^3 a^{-5} -a^5 z+z a^{-5} +4 a^4 z^8-14 a^4 z^6+5 z^6 a^{-4} +14 a^4 z^4-4 z^4 a^{-4} -4 a^4 z^2+z^2 a^{-4} +5 a^3 z^9-14 a^3 z^7+7 z^7 a^{-3} +5 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+4 z^3 a^{-3} -3 a^3 z-z a^{-3} +2 a^2 z^{10}+6 a^2 z^8+8 z^8 a^{-2} -37 a^2 z^6-13 z^6 a^{-2} +38 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-z^2 a^{-2} +11 a z^9+6 z^9 a^{-1} -28 a z^7-6 z^7 a^{-1} +9 a z^5-10 z^5 a^{-1} +11 a z^3+15 z^3 a^{-1} -6 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}+10 z^8-41 z^6+37 z^4-9 z^2-1} (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   4 1   -3 6                 7 2     5 4               8 4       -4 2             10 7         3 0           9 10           1 -2         7 8             -1 -4       5 9               4 -6     3 7                 -4 -8   1 5                   4 -10   3                     -3 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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