L11a388

L11a387

L11a389

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a388 at Knotilus!

Link Presentations

[edit Notes on L11a388's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, 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 w} , ...)	${\frac {3uvw^{2}-4uvw+2uv+2uw^{3}-6uw^{2}+6uw-2u+2vw^{3}-6vw^{2}+6vw-2v-2w^{3}+4w^{2}-3w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-10q^{-5}+16q^{-6}-15q^{-7}+16q^{-8}-13q^{-9}+10q^{-10}-6q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT 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^{14} z^{-2} +3 a^{12} z^{-2} +4 a^{12}-6 a^{10} z^2-2 a^{10} z^{-2} -7 a^{10}+3 a^8 z^4+3 a^8 z^2-a^8 z^{-2} -a^8+3 a^6 z^4+5 a^6 z^2+a^6 z^{-2} +4 a^6+a^4 z^4+a^4 z^2} (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+9a^{15}z^{3}-7a^{15}z+2a^{15}z^{-1}+2a^{14}z^{8}-7a^{14}z^{6}+7a^{14}z^{4}-2a^{14}z^{2}-a^{14}z^{-2}+a^{14}+2a^{13}z^{9}-a^{13}z^{7}-18a^{13}z^{5}+36a^{13}z^{3}-27a^{13}z+8a^{13}z^{-1}+a^{12}z^{10}+5a^{12}z^{8}-22a^{12}z^{6}+19a^{12}z^{4}-7a^{12}z^{2}-3a^{12}z^{-2}+5a^{12}+6a^{11}z^{9}-6a^{11}z^{7}-27a^{11}z^{5}+49a^{11}z^{3}-34a^{11}z+10a^{11}z^{-1}+a^{10}z^{10}+10a^{10}z^{8}-28a^{10}z^{6}+17a^{10}z^{4}-5a^{10}z^{2}-2a^{10}z^{-2}+4a^{10}+4a^{9}z^{9}+3a^{9}z^{7}-23a^{9}z^{5}+23a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}+7a^{8}z^{8}-7a^{8}z^{6}-4a^{8}z^{4}+9a^{8}z^{2}+a^{8}z^{-2}-3a^{8}+7a^{7}z^{7}-6a^{7}z^{5}-a^{7}z^{3}+4a^{7}z-2a^{7}z^{-1}+6a^{6}z^{6}-8a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                       1 1 -5                     3 1 -2 -7                   4     4 -9                 6 3     -3 -11               10 4       6 -13             8 9         1 -15           8 7           1 -17         5 8             3 -19       5 8               -3 -21     1 5                 4 -23   1 5                   -4 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ 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}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-9$ 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}^{5}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ 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 r=-7} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ 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 r=-6} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ 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}^{8}} 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=-5} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{10}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ 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}^{3}\oplus{\mathbb Z}_2^{4}} ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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