L11a401

L11a400

L11a402

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a401 at Knotilus!

Link Presentations

[edit Notes on L11a401's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}\left(w^{2}-w+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-8}+5q^{-7}-12q^{-6}+20q^{-5}-27q^{-4}+q^{3}+32q^{-3}-5q^{2}-30q^{-2}+12q+28q^{-1}-19$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-a^{6}z^{2}+2a^{4}z^{6}+5a^{4}z^{4}+3a^{4}z^{2}+a^{4}z^{-2}-a^{2}z^{8}-4a^{2}z^{6}-6a^{2}z^{4}-3a^{2}z^{2}-2a^{2}z^{-2}-a^{2}+z^{6}+2z^{4}+z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$4a^{4}z^{10}+4a^{2}z^{10}+13a^{5}z^{9}+24a^{3}z^{9}+11az^{9}+17a^{6}z^{8}+28a^{4}z^{8}+22a^{2}z^{8}+11z^{8}+12a^{7}z^{7}-6a^{5}z^{7}-37a^{3}z^{7}-14az^{7}+5z^{7}a^{-1}+5a^{8}z^{6}-25a^{6}z^{6}-73a^{4}z^{6}-67a^{2}z^{6}+z^{6}a^{-2}-23z^{6}+a^{9}z^{5}-14a^{7}z^{5}-17a^{5}z^{5}+2a^{3}z^{5}-4az^{5}-8z^{5}a^{-1}-3a^{8}z^{4}+14a^{6}z^{4}+51a^{4}z^{4}+51a^{2}z^{4}-z^{4}a^{-2}+16z^{4}+5a^{7}z^{3}+10a^{5}z^{3}+8a^{3}z^{3}+6az^{3}+3z^{3}a^{-1}-4a^{6}z^{2}-12a^{4}z^{2}-12a^{2}z^{2}-4z^{2}-a^{3}z-az+a^{4}+a^{2}+1+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     4   -4 3                   8 1   7 1                 11 4     -7 -1               17 8       9 -3             15 13         -2 -5           17 15           2 -7         12 17             5 -9       8 15               -7 -11     4 12                 8 -13   1 8                   -7 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-2$ ${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{17}$ $r=-1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{17}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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