L11a45

L11a44

L11a46

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Link Presentations

[edit Notes on L11a45's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X18,9,19,10 X20,15,21,16 X16,19,17,20 X8,17,9,18 X4,21,1,22 X12,6,13,5 X10,4,11,3 X22,12,5,11 X2,14,3,13
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -6, 3, -9, 10, -8, 11, -2, 4, -5, 6, -3, 5, -4, 7, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)^2}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+3 a z^5-z^5 a^{-1} -2 a^5 z^3+3 a^3 z^3+3 a z^3-2 z^3 a^{-1} -a^5 z+2 a^3 z-z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1}$ (db)
Kauffman polynomial	$-2 a^4 z^{10}-2 a^2 z^{10}-6 a^5 z^9-12 a^3 z^9-6 a z^9-7 a^6 z^8-12 a^4 z^8-14 a^2 z^8-9 z^8-4 a^7 z^7+7 a^5 z^7+15 a^3 z^7-4 a z^7-8 z^7 a^{-1} -a^8 z^6+16 a^6 z^6+31 a^4 z^6+27 a^2 z^6-4 z^6 a^{-2} +9 z^6+9 a^7 z^5+2 a^5 z^5+20 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-11 a^6 z^4-19 a^4 z^4-10 a^2 z^4+5 z^4 a^{-2} +z^4-5 a^7 z^3+a^3 z^3-13 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+2 a^6 z^2+5 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2-2 a^5 z-3 a^3 z+a z+2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               12 6       6 -2             12 11         -1 -4           12 10           2 -6         8 12             4 -8       6 12               -6 -10     3 8                 5 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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