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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X16,6,17,5 X20,13,21,14 X18,15,19,16 X14,19,15,20 X22,10,7,9 X4,18,5,17 X10,22,11,21 X2738 X6,11,1,12

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 10q-9/2 + 13q-7/2 - 16q-5/2 + 15q-3/2 - 14q-1/2 + 10q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - 2q-18 + q-16 + 2q-14 - 3q-12 + 3q-10 + q-8 + 2q-6 + 3q-4 - q-2 + 4 - 2q2 - q4 + 2q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z + a-1z + 2a-1z3 - az-1 - az - az5 + a3z-1 - a3z5 + a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 3a-2z2 + 6a-2z4 - 3a-2z6 + 5a-1z5 - 4a-1z7 - z2 + 2z4 + 3z6 - 4z8 + az-1 - az + az5 + az7 - 3az9 - a2 + 5a2z2 - 15a2z4 + 15a2z6 - 6a2z8 - a2z10 + a3z-1 - 3a3z + 3a3z3 - 8a3z5 + 11a3z7 - 6a3z9 + 7a4z2 - 20a4z4 + 20a4z6 - 6a4z8 - a4z10 + a5z - 6a5z3 + 6a5z5 + 3a5z7 - 3a5z9 + 2a6z2 - 6a6z4 + 10a6z6 - 4a6z8 + 2a7z - 7a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     2   j = 4                   4 1   j = 2                 6 2     j = 0               8 4       j = -2             8 7         j = -4           8 7           j = -6         6 9             j = -8       4 7               j = -10     2 6                 j = -12   1 4                   j = -14   2                     j = -16 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, 221]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 221]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[16, 6, 17, 5], X[20, 13, 21, 14], > X[18, 15, 19, 16], X[14, 19, 15, 20], X[22, 10, 7, 9], X[4, 18, 5, 17], > X[10, 22, 11, 21], X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -11}, > {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 6, -4, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 10 13 16 15 14 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 10 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 -16 2 3 3 -8 2 3 -2 2 4 - q + --- - --- + q + --- - --- + --- + q + -- + -- - q - 2 q - 20 18 14 12 10 6 4 q q q q q q q 4 6 8 12 > q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 221]][a, z]
Out[8]=	3 3 a a z z 5 7 2 z 5 3 5 3 5 -(-) + -- - -- + - - a z + a z - a z + ---- + 2 a z - a z - a z z z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 221]][a, z]
Out[9]=	3 2 2 a a z 3 5 7 2 3 z 2 2 -a + - + -- - -- - a z - 3 a z + a z + 2 a z - z - ---- + 5 a z + z z 3 2 a a 3 4 2 6 2 8 2 2 z 3 3 5 3 7 3 4 > 7 a z + 2 a z - 2 a z + ---- + 3 a z - 6 a z - 7 a z + 2 z + 3 a 4 5 5 6 z 2 4 4 4 6 4 8 4 z 5 z 5 > ---- - 15 a z - 20 a z - 6 a z + 3 a z - -- + ---- + a z - 2 3 a a a 6 3 5 5 5 7 5 6 3 z 2 6 4 6 > 8 a z + 6 a z + 9 a z + 3 z - ---- + 15 a z + 20 a z + 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 8 > 10 a z - a z - ---- + a z + 11 a z + 3 a z - 3 a z - 4 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 4 10 > 6 a z - 6 a z - 4 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 2 1 4 2 6 4 7 8 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 6 9 8 7 8 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 4 t + 6 q t + 2 q t + 4 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a221

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