L11a245

L11a244

L11a246

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Link Presentations

[edit Notes on L11a245's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X2,15,3,16 X18,5,19,6 X12,3,13,4 X22,11,7,12 X4,21,5,22 X14,20,15,19 X20,14,21,13 X6,9,1,10 X10,17,11,18
Gauss code	{1, -3, 5, -7, 4, -10}, {2, -1, 10, -11, 6, -5, 9, -8, 3, -2, 11, -4, 8, -9, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 t(1) t(2)^4-2 t(2)^4+3 t(1)^2 t(2)^3-11 t(1) t(2)^3+7 t(2)^3-9 t(1)^2 t(2)^2+17 t(1) t(2)^2-9 t(2)^2+7 t(1)^2 t(2)-11 t(1) t(2)+3 t(2)-2 t(1)^2+2 t(1)}{t(1) t(2)^2}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{24}{q^{7/2}}+\frac{27}{q^{9/2}}-\frac{28}{q^{11/2}}+\frac{24}{q^{13/2}}-\frac{18}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^3\right)+a^9 z+a^7 z^5-2 a^7 z^3-2 a^7 z+a^7 z^{-1} +3 a^5 z^5+4 a^5 z^3+a^5 z-a^5 z^{-1} +a^3 z^5-2 a^3 z^3-3 a^3 z-a z^3$ (db)
Kauffman polynomial	$a^{12} z^6-a^{12} z^4+5 a^{11} z^7-9 a^{11} z^5+3 a^{11} z^3+a^{11} z+10 a^{10} z^8-21 a^{10} z^6+11 a^{10} z^4-a^{10} z^2+10 a^9 z^9-16 a^9 z^7+3 a^9 z^5-a^9 z^3+4 a^8 z^{10}+13 a^8 z^8-45 a^8 z^6+33 a^8 z^4-8 a^8 z^2+21 a^7 z^9-43 a^7 z^7+27 a^7 z^5-8 a^7 z^3+3 a^7 z-a^7 z^{-1} +4 a^6 z^{10}+16 a^6 z^8-50 a^6 z^6+44 a^6 z^4-13 a^6 z^2+a^6+11 a^5 z^9-13 a^5 z^7+a^5 z^5+5 a^5 z^3+a^5 z-a^5 z^{-1} +13 a^4 z^8-23 a^4 z^6+19 a^4 z^4-6 a^4 z^2+9 a^3 z^7-13 a^3 z^5+8 a^3 z^3-3 a^3 z+4 a^2 z^6-4 a^2 z^4+a z^5-a z^3$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   7 1   6 -4                 11 4     -7 -6               13 6       7 -8             14 11         -3 -10           14 13           1 -12         10 14             4 -14       8 14               -6 -16     4 11                 7 -18   1 7                   -6 -20   4                     4 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-5$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-4$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=-3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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