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L11n337 L11n339

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X22,20,13,19 X20,16,21,15 X16,22,17,21 X17,12,18,5 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, 7, -8, -9, 3, 6, -7, 8, -6}}

Jones Polynomial:

q-8 + q-6 + 3q-5 - 4q-4 + 6q-3 - 6q-2 + 6q-1 - 5 + 3q - q2

A2 (sl(3)) Invariant:

q-26 + 3q-24 + 5q-22 + 7q-20 + 6q-18 + 7q-16 + 2q-14 - 3q-8 + q-6 - 2q-4 + q-2 - q2 + q4 - q6

HOMFLY-PT Polynomial:

- 1 - 2z2 - z4 - a2z-2 + a2 + 5a2z2 + 4a2z4 + a2z6 + 4a4z-2 + 6a4 + 3a4z2 - 5a6z-2 - 7a6 - 2a6z2 + 2a8z-2 + a8

Kauffman Polynomial:

a-1z - 2a-1z3 + a-1z5 - 1 + 3z2 - 7z4 + 3z6 + az-1 - 2az3 - 5az5 + 3az7 - a2z-2 - a2 + 8a2z2 - 12a2z4 + 3a2z6 + a2z8 + 5a3z-1 - 15a3z + 17a3z3 - 11a3z5 + 4a3z7 - 4a4z-2 + 12a4 - 12a4z2 + 9a4z4 - 3a4z6 + a4z8 + 9a5z-1 - 35a5z + 42a5z3 - 15a5z5 + 2a5z7 - 5a6z-2 + 23a6 - 41a6z2 + 35a6z4 - 11a6z6 + a6z8 + 5a7z-1 - 21a7z + 25a7z3 - 10a7z5 + a7z7 - 2a8z-2 + 12a8 - 24a8z2 + 21a8z4 - 8a8z6 + a8z8

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     2   j = 1                   3 1   j = -1                 3 2     j = -3               4 4       j = -5           1 3 2         j = -7           2 4           j = -9       1 3 3             j = -11         4               j = -13     1                   j = -15 1                       j = -17 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 338]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 338]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 18, 12, 19], X[7, 14, 8, 15], > X[13, 8, 14, 9], X[22, 20, 13, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[17, 12, 18, 5], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, > {-5, 4, 7, -8, -9, 3, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -6 3 4 6 6 6 2 -5 + q + q + -- - -- + -- - -- + - + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 3 5 7 6 7 2 3 -6 2 -2 2 4 6 q + --- + --- + --- + --- + --- + --- - -- + q - -- + q - q + q - q 24 22 20 18 16 14 8 4 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 338]][a, z]
Out[8]=	2 4 6 8 2 4 6 8 a 4 a 5 a 2 a 2 2 2 -1 + a + 6 a - 7 a + a - -- + ---- - ---- + ---- - 2 z + 5 a z + 2 2 2 2 z z z z 4 2 6 2 4 2 4 2 6 > 3 a z - 2 a z - z + 4 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 338]][a, z]
Out[9]=	2 4 6 8 3 5 2 4 6 8 a 4 a 5 a 2 a a 5 a 9 a -1 - a + 12 a + 23 a + 12 a - -- - ---- - ---- - ---- + - + ---- + ---- + 2 2 2 2 z z z z z z z 7 5 a z 3 5 7 2 2 2 4 2 > ---- + - - 15 a z - 35 a z - 21 a z + 3 z + 8 a z - 12 a z - z a 3 6 2 8 2 2 z 3 3 3 5 3 7 3 > 41 a z - 24 a z - ---- - 2 a z + 17 a z + 42 a z + 25 a z - a 5 4 2 4 4 4 6 4 8 4 z 5 3 5 > 7 z - 12 a z + 9 a z + 35 a z + 21 a z + -- - 5 a z - 11 a z - a 5 5 7 5 6 2 6 4 6 6 6 8 6 > 15 a z - 10 a z + 3 z + 3 a z - 3 a z - 11 a z - 8 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 8 8 > 3 a z + 4 a z + 2 a z + a z + a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 1 1 4 3 3 2 -- + - + ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + 3 q 17 8 15 8 13 6 9 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t q t 1 4 3 2 4 2 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 5 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n338

L11n337 L11n339