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L11n351 L11n353

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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X11,20,12,21 X7,18,8,19 X9,13,10,22 X21,17,22,16 X17,8,18,9 X15,11,16,10 X19,12,20,5 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

q-8 - q-7 + q-6 + 2q-5 - 3q-4 + 6q-3 - 5q-2 + 6q-1 - 5 + 3q - q2

A2 (sl(3)) Invariant:

q-26 + 2q-24 + 2q-22 + 3q-20 + 2q-18 + 5q-16 + 3q-14 + 3q-12 + 4q-10 + 3q-6 - q-4 + q-2 - q2 + q4 - q6

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-1z - 2a-1z3 + a-1z5 - 1 + 3z2 - 7z4 + 3z6 + az - 3az3 - 5az5 + 3az7 + 4a2z2 - 9a2z4 + 2a2z6 + a2z8 - 3a3z + 7a3z3 - 7a3z5 + 3a3z7 - a4z-2 + 9a4 - 23a4z2 + 28a4z4 - 12a4z6 + 2a4z8 + 2a5z-1 - 8a5z + 5a5z3 + 11a5z5 - 7a5z7 + a5z9 - 2a6z-2 + 13a6 - 38a6z2 + 45a6z4 - 18a6z6 + 2a6z8 + 2a7z-1 - 5a7z - 3a7z3 + 12a7z5 - 7a7z7 + a7z9 - a8z-2 + 6a8 - 14a8z2 + 15a8z4 - 7a8z6 + 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             1 4 4       j = -5             3 2         j = -7         1 2 4           j = -9       1 3 3             j = -11       1 4               j = -13   1 1                   j = -15                         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, 352]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 352]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[11, 20, 12, 21], X[7, 18, 8, 19], > X[9, 13, 10, 22], X[21, 17, 22, 16], X[17, 8, 18, 9], X[15, 11, 16, 10], > X[19, 12, 20, 5], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 7, -5, 8, -3, 9}, > {11, -2, -8, 6, -7, 4, -9, 3, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-8 -7 -6 2 3 6 5 6 2 -5 + q - q + q + -- - -- + -- - -- + - + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 2 3 2 5 3 3 4 3 -4 -2 2 q + --- + --- + --- + --- + --- + --- + --- + --- + -- - q + q - q + 24 22 20 18 16 14 12 10 6 q q q q q q q q q 4 6 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 352]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 2 4 2 -1 + 2 a + 3 a - 5 a + a + -- - ---- + -- - 2 z + 5 a z + 2 a z - 2 2 2 z z z 6 2 4 2 4 2 6 > 2 a z - z + 4 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 352]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a z 3 -1 + 9 a + 13 a + 6 a - -- - ---- - -- + ---- + ---- + - + a z - 3 a z - 2 2 2 z z a z z z 3 5 7 2 2 2 4 2 6 2 8 2 2 z > 8 a z - 5 a z + 3 z + 4 a z - 23 a z - 38 a z - 14 a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 3 a z + 7 a z + 5 a z - 3 a z - 7 z - 9 a z + 28 a z + 5 6 4 8 4 z 5 3 5 5 5 7 5 6 > 45 a z + 15 a z + -- - 5 a z - 7 a z + 11 a z + 12 a z + 3 z + a 2 6 4 6 6 6 8 6 7 3 7 5 7 > 2 a z - 12 a z - 18 a z - 7 a z + 3 a z + 3 a z - 7 a z - 7 7 2 8 4 8 6 8 8 8 5 9 7 9 > 7 a z + a z + 2 a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 3 1 1 1 1 1 4 3 1 -- + - + ------ + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 3 q 17 8 13 7 13 6 11 5 9 5 11 4 9 4 7 4 q q t q t q t q t q t q t q t q t 3 2 4 3 1 2 4 2 t 2 > ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q t + q t + 9 3 7 3 7 2 5 2 3 2 5 3 q q t q t q t q t q t q t q t 3 2 5 3 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n352

L11n351 L11n353