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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X20,17,21,18 X16,5,17,6 X18,11,19,12 X12,19,13,20 X22,16,7,15 X4,21,5,22

Gauss Code:

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

Jones Polynomial:

q-23/2 - 4q-21/2 + 8q-19/2 - 13q-17/2 + 17q-15/2 - 19q-13/2 + 18q-11/2 - 16q-9/2 + 11q-7/2 - 7q-5/2 + 3q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + 2q-32 - q-30 + 2q-26 - 3q-24 + 4q-22 - q-20 + 2q-18 + 3q-16 - 2q-14 + 4q-12 - 2q-10 + q-8 + q-6 - q-4 + q-2

HOMFLY-PT Polynomial:

- 2a3z - 3a3z3 - a3z5 - a5z-1 - 2a5z + a5z3 + 3a5z5 + a5z7 + a7z-1 + a7z + 2a7z3 + 3a7z5 + a7z7 - 2a9z3 - a9z5

Kauffman Polynomial:

2a3z - 5a3z3 + 4a3z5 - a3z7 + 4a4z2 - 12a4z4 + 11a4z6 - 3a4z8 + a5z-1 - 3a5z + 3a5z3 - 9a5z5 + 12a5z7 - 4a5z9 - a6 + 10a6z2 - 28a6z4 + 25a6z6 - 3a6z8 - 2a6z10 + a7z-1 - 3a7z + 7a7z3 - 22a7z5 + 30a7z7 - 11a7z9 + 7a8z2 - 31a8z4 + 39a8z6 - 11a8z8 - 2a8z10 + 2a9z - 8a9z3 + 9a9z5 + 6a9z7 - 7a9z9 - 7a10z4 + 17a10z6 - 11a10z8 - 5a11z3 + 14a11z5 - 11a11z7 - a12z2 + 7a12z4 - 8a12z6 + 2a13z3 - 4a13z5 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                     2   j = -4                   5 1   j = -6                 7 3     j = -8               9 4       j = -10             9 7         j = -12           10 9           j = -14         7 9             j = -16       6 10               j = -18     3 8                 j = -20   1 5                   j = -22   3                     j = -24 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 212]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 212]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 8, 15, 7], > X[6, 13, 1, 14], X[20, 17, 21, 18], X[16, 5, 17, 6], X[18, 11, 19, 12], > X[12, 19, 13, 20], X[22, 16, 7, 15], X[4, 21, 5, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 7, -5}, > {4, -1, 2, -3, 8, -9, 5, -4, 10, -7, 6, -8, 9, -6, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 4 8 13 17 19 18 16 11 7 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 3 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 2 3 4 -20 2 3 2 4 2 -q + --- - q + --- - --- + --- - q + --- + --- - --- + --- - --- + 32 26 24 22 18 16 14 12 10 q q q q q q q q q -8 -6 -4 -2 > q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 212]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 5 3 7 3 9 3 -(--) + -- - 2 a z - 2 a z + a z - 3 a z + a z + 2 a z - 2 a z - z z 3 5 5 5 7 5 9 5 5 7 7 7 > a z + 3 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 212]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 4 2 6 2 -a + -- + -- + 2 a z - 3 a z - 3 a z + 2 a z + 4 a z + 10 a z + z z 8 2 12 2 3 3 5 3 7 3 9 3 11 3 > 7 a z - a z - 5 a z + 3 a z + 7 a z - 8 a z - 5 a z + 13 3 4 4 6 4 8 4 10 4 12 4 14 4 > 2 a z - 12 a z - 28 a z - 31 a z - 7 a z + 7 a z - a z + 3 5 5 5 7 5 9 5 11 5 13 5 4 6 > 4 a z - 9 a z - 22 a z + 9 a z + 14 a z - 4 a z + 11 a z + 6 6 8 6 10 6 12 6 3 7 5 7 7 7 > 25 a z + 39 a z + 17 a z - 8 a z - a z + 12 a z + 30 a z + 9 7 11 7 4 8 6 8 8 8 10 8 5 9 > 6 a z - 11 a z - 3 a z - 3 a z - 11 a z - 11 a z - 4 a z - 7 9 9 9 6 10 8 10 > 11 a z - 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 3 1 5 3 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 10 7 9 10 9 9 7 9 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 4 7 t 2 t 2 > ---- + ---- + -- + --- + t 8 6 4 2 q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a212

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